76 o N O PTIMA Mathematical Programming Society Newsletter March 2008

Structure Prediction and A new column Global Optimization for Optima by Alberto Caprara Marco Locatelli Andrea Lodi Dip. Informatica, Univ. di Torino (Italy) Katya Scheinberg Fabio Schoen Dip. Sistemi e Informatica, Univ. di Firenze (Italy) This is the first issue of 2008 and it is a very dense one with a Scientific February 26, 2008 contribution on computational biology, some announcements, reports of “Every attempt to employ mathematical methods in the study of chemical conferences and events, the MPS Chairs questions must be considered profoundly irrational and contrary to the spirit in chemistry. If mathematical analysis should ever hold a prominent place Column. Thus, the extra space is limited in chemistry — an aberration which is happily almost impossible — it but we like to use few additional lines would occasion a rapid and widespread degeneration of that science.” for introducing a new feature we are — Auguste Comte particularly happy with. Starting with Cours de philosophie positive issue 76 there will be a Discussion Column whose content is tightly related “It is not yet clear whether optimization is indeed useful for biology — surely biology has been very useful for optimization” to the Scientific contribution, with ­ — Alberto Caprara a purpose of making every issue of private communication Optima recognizable through a special topic. The Discussion Column will take the form of a comment on the Scientific 1 Introduction contribution from some experts in Many new problem domains arose from the study of biological and the field other than the authors or chemical systems and several mathematical programming models an interview/discussion of a couple as well as algorithms have been developed. We are slightly more of experts in the area or some other optimistic than Comte and Caprara and still believe that the use short contribution which may reflect of mathematical programming tools can be valuable in biology, chemical-physics as well as in the study of innovative materials. Of alternative points of view related to the course we perfectly agree on the fact that a lot of research stimuli special topic. for the optimization community originated from those fields. In this paper we concentrate our attention on the problem of structure We hope our readers will enjoy the new prediction: given some information on the composition of a complex column and we strongly encourage molecule we would like to predict the structure that the molecule feedbacks especially in terms of will most likely assume. Such a problem is a very relevant one as the properties of, e.g., biomolecules are intimately related to their three- suggestions for topics to be covered in future issues.

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76Structure Prediction and Global Optimization 2 A Celebration of 50 Years of Integer Programming 9 op t i m a 7 6 March 2008 page  Structure Prediction and Global Optimization

dimensional conformation; it is quite opposite is true for pairs of atoms with In the following we will present in some well accepted nowadays that this most charges of the same sign. In most energy detail the most important models for atomic stable conformation corresponds to a models, covalent bonds are considered to and molecular clusters and will give a global minimum of a suitable function be too strong to be broken or modified short introduction to the more challenging which represents the free energy of the at physiological temperatures, so the only problems of protein docking and protein molecule. Given our only limited capability degrees of freedom of complex molecules folding. Then, in Section 3 we will of capturing the essential phenomena can be considered as those associated with introduce some basic ideas underlying many in a manageable energetic model, and non bonded pairs. This is the reason why in of the global optimization approaches used given the fact that various factors (e.g., many applications terms (1)-(2) are the only to solve these problems. thermodynamic and kinetic factors) concur ones taken into account in optimization, as to determine the actual structure, it is all the other terms are assumed to contribute 2 Structure prediction problems widely believed that being able to detect a constant term to the total energy. the global minimum as well as other low- Globally minimizing E turns out to be In this section we review some well lying local minima is an important issue. a very challenging task. Several discretized known structure prediction problems. This observation immediately leads to the and simplified versions of the problem application of global optimization: in order have been proven to be NP-hard (see, 2.1 Cluster optimization to predict the structure, first a mathematical e.g., [8]). The main source of difficulty is In cluster optimization we are given N model of the total potential energy, the not necessarily the dimensionality of the particles (atoms or molecules) and an energy energy function E, is defined, and then problem (in some cases the number of function E, which depends on the relative the function E is (globally) minimized. control variables is very small) but the huge positions of the particles in the 3D-space; we Defining a reasonable model E for the number of local (and not global) minima of aim at detecting the global minimum of the energy is in general an extremely complex the energy function E, which rules out any energy function. Different energy functions task; many classical models include sums of trivial Multistart approach. Moreover, in have been proposed in the literature. Within terms which account for various interactions some cases a single function and/or gradient the field of atomic clusters usually only non- inside the molecule. In general, some terms evaluation of E may be extremely expensive. bonded interactions are accounted for and in are related to bonded interactions (forcing A very deep survey on the models which the simplest models, particles are considered pairs of bonded atoms to stabilize around a are currently used to describe the energy of to be charge-free. Such potentials only fixed distance, the bond length, or favoring complex molecules can be found in [49]; depend on the distance Rij between pairs i, j triplets of bonded atoms to form specific an in–depth analysis on the characteristics of atoms, thus, for a cluster of N atoms, the angles or quadruplets to find an equilibrium of nanoclusters can be also found in [2]. function to be minimized is the following: around some known dihedral angles). We cannot close this introduction without Other terms account for weaker interactions citing some recent developments in the between non-bonded pairs; usually these field of energy modeling, which we cannot interactions have the following form: survey here but which are of great interest for mathematical programming. We refer to approaches which, differently from classical Different models only differ from each ones which start “ab initio” and try to form other for the definition of the pair potential models according to first principles, are function Epair. In the field of atomic based on the desire of finding a model for clusters the most popular energy potential where term (1) represents the van der Waals which the structures which are observed is the Lennard-Jones (LJ) one. The LJ interaction and depends on the distance Rij in nature are indeed global minima of the pair potential can be defined as follows: between any pair of non-bonded atoms; as model, while structures obtained through Aij > 0 and Bij > 0, this term is composed perturbation of the observed ones are of a repulsive term and an attractive one. not. Models can be built through linear Term (2) represents the Coulomb, or combination of suitable base functions; electrostatic, interaction and, again, depends given the enormous amount of knowledge This potential produces very accurate on the pair distance Rij and on the electric already available in protein databases, representations of real clusters like, e.g., charges qi and qj of the two atoms: as it is the parameters of these models can be some noble gases or some metals like well known, atoms whose electric charges obtained through the solution of huge linear gold and nickel. But the interest of the are opposite in sign contribute a negative programs. We do not comment any more on LJ potential also lies in the fact that (attractive) term to the energy, while the this subject, but refer the interested reader to [47]. op t i m a 7 6 March 2008 page 

it has been widely employed as a test system to develop and gain insight into new algorithmic techniques to be later extended to other molecular conformation problems. Another popular potential for atomic clusters is the Morse one. The Morse pair potential is defined as follows:

; Epair(Rij) = VM(Rij ρ) = 2 (exp{ρ(1 - Rij)} - 1) - 1.

The shape of this potential is quite similar to the LJ one, but it allows for a greater ρ flexibility: a small value models those Figure 1: Illustration of Lennard-Jones and Morse pair potentials situations where the repulsive force as the distance between two atoms is driven the fundamental interactions among water to 0 is a mild one, while large ρ values Jones, other potential energy model have molecules are based on the assumption models situations where such a force is very been proposed in the literature like, e.g., that a single molecule has a prescribed strong. Also, large ρ values correspond to the Gupta model analyzed in [43, 44]. shape which cannot be altered; the energy short range forces, which quickly vanish All the above mentioned potentials contribution is thus dependent on the outside a restricted neighborhood of the are extremely challenging for global relative positions of H and O atoms of minimum. In Figure 1 we plot both the optimization methods. The number of different molecules. Among the best known LJ and a few Morse pair potentials. local minima is conjectured to grow at models for water clusters we cite the TIP4P Besides producing an accurate least exponentially with the number N of and TIP5P potentials. These molecular representation for some real clusters such atoms and, for Morse potential, the problem clusters are much more difficult to optimize as those of C molecules (ρ = 13.6) and of becomes more and more difficult as the 60 ρ with respect to atomic clusters because alkali metals (ρ = 3.1), the Morse potential parameter increases because the energy of the additional orientational degrees of also offers a more varied test system with landscape becomes more and more rugged. freedom; indeed, the energy contribution of respect to the LJ one. While the above In the binary LJ clusters things are also a pair of water molecules does not depend models always assume that all atoms in the complicated by the combinatorial aspects only on the distance between the geometric cluster are equal, other models have also introduced by distinct atom types. In the centers of the molecules, but also on the been proposed for clusters where atoms of Cambridge Cluster Database (CCDB) ([7]), relative rotation of one with respect to the different types are present, in particular the main database in the field of cluster other. In other words, while the potential for binary clusters, i.e., clusters with two optimization, putative global minima are is still given as a sum of contributions (van distinct atom types. As an example, we reported for up to N ≤ 150 and for 300 ≤ N der Waals and electrostatic) due to the mention binary Lennard-Jones clusters ([18, ≤ 1000 atoms for LJ clusters; for the more relative distances between pairs of atoms, 19, 37]), i.e., clusters formed by a mixture challenging Morse clusters only results up as each molecule is considered to be rigid, of two different atom types, which can be to N = 80 atoms are reported; finally, for the “natural” degrees of freedom of each modeled through binary LJ clusters results for N up to 100 and different σij values are reported. molecule are the position of its center and Many other cluster optimization problems the rotation with respect to a prefixed together with lists of their putative global orientation. Thus, all pairwise distances minima are reported in the CCDB. Among can be seen as functions of these degrees of them we recall Dzugutov clusters (see [17, freedom. The CCDB reports putative global where εij and σij are suitable constants 21]) and C60 (fullerene) clusters: the latter minima for water clusters with no more which depend only on the types of are indeed molecular clusters, but C60 than N = 21 molecules and, according to atom i and atom j. Note that from the molecules are extremely close to spheres and the current literature on the subject, there optimization point of view, binary clusters thus many optimization methods consider are still some doubts that the published turn out to be particularly interesting them just as single particles in space. structure at N = 21 (represented in Figure 2) because they mix continuous aspects In the field of molecular clusters we is indeed the global minimum. (atom positions) and combinatorial ones mention water clusters, in which some water (atom types). Besides binary Lennard- molecules interact; models that capture op t i m a 7 6 March 2008 page 

complexity of rigid protein-protein docking is formidable, both because of the very rugged energy landscape, characterized by an enormous number of local optima, and or, alternatively, the cumulative as a consequence of the large number of absolute error: interactions which have to be computed for each energy evaluation: proteins are usually composed of thousands of atoms and thus, given the relative position of two proteins, millions of pairwise interactions (van der Waals and Coulomb) have to be computed. As it can be easily understood, when relaxing the assumption of rigidity, the Figure 2: Putative optimal custer for TIP5P21 Here X1, . . . ,XN are the positions of the atoms, ℓij and uij are respectively the known number of degrees of freedom enormously lower and upper bound for the distance increases and the (flexible) docking problem 2.2 Molecular distance geometry between atoms i and j, and D is the subset of becomes a large scale global optimization The three-dimensional structure of real-life distances for which lower and upper bounds one. In [5] an approach is presented in proteins is usually determined by means are available. which rigid docking is used as a tool for of Nuclear Magnetic Resonance (NMR) generating good starting conformations for and X-ray diffraction experiments. From 2.3 Protein-protein docking flexible docking. A survey of some these experiments we obtain a subset and protein folding global optimization approaches for of all possible pairwise distances, from docking can be found in [12]. Protein-protein docking is the process by which we aim at finding a conformation Protein folding is concerned with the which a large and complex biomolecule of all the atoms in such a way that determination of the three-dimensional interacts with another one by forming a the constraints imposed by the subset structure of a protein given the so-called single complex; such complexes form the of known distances is satisfied. More primary structure, i.e., the aminoacid bases of most activities of all living bodies. complex versions of this problem also sequence. A protein is, in fact, composed Being able to predict the correct docking include information about angles. of a linear set of aminoacids; it is widely of two specific proteins is considered to be If complete information were available accepted and confirmed by several one of the most important challenges in (all distances were known), the problem experiments and observations that the 3D 3 computational biology for the next years. In could be solved in O(N ) time (where N structure of a protein is largely determined rigid docking, the two interacting molecules is the number of atoms) by eigenvalue by the sequence of its aminoacids. In other are considered as rigid bodies; of course, decomposition of the distance matrix; words, proteins which are composed by the this cannot be true in practice, but often however, when incomplete information on same sequence of aminoacids are considered the relative position of two proteins docked distances is available the problem becomes equal and all assume roughly the same in this way can be used as a starting point strongly NP-hard (see [45]). three-dimensional shape. Being able to for a flexible docking phase, in which both To further complicate matters, distances predict the conformation of a protein built proteins are allowed to change their shape. are not usually known exactly but only from a specific sequence of aminoacids is In rigid docking we can formulate the lower and upper bounds can be retrieved considered as a fundamental problem in problem as one of minimizing the energetic from the experiments. computational biology; if we were able to contribution of pairs composed of atoms Since the introduction of the EMBED solve this problem, we could simulate the belonging to the two different proteins algorithm in [9], many different techniques folding of many synthetic proteins until – in fact internal contributions account for have been proposed to solve this problem, we find one which folds in a prescribed a constant term in the energy if the shape including graph reduction [30], geometric and stable shape. The literature on protein of each molecule is kept fixed. It can be build-up [13], Semidefinite Programming folding methods is enormous; here we easily understood that rigid protein docking [4]. Global optimization techniques just refer the reader to [11, 20, 23, 46, 48] is a low-dimensional global optimization have also been proposed ( see, e.g., [36, for some general references and to [22] problem: we can formulate the problem 51]). Indeed, the problem can be easily for an interesting example of how global assuming that one of the two molecules reformulated as that of globally minimizing optimization and integer programming is kept fixed, and thus there are only six the cumulative relative error can be effectively used to design a novel degrees of freedom: three translation and biomolecule which might lead to the design three rotation parameters which enable of a more effective treatment for specific to identify the relative position of the two diseases. molecules. Despite the low dimension, the op t i m a 7 6 March 2008 page 

3 Computational approaches Note that the global minimum of an 3.2 Geometric properties and energy function is always the funnel landscape deformation In this section we report some observations bottom of one of its funnels. Therefore, the Local and global minima of the energy and related techniques which allowed to problem of detecting the global minimum functions have particular geometrical greatly increase the efficiency in solving is equivalent to the problem of detecting the structures in the 3D-space. Some approaches some of the problems presented above. lowest funnel bottom. The key observation are based on conjectures about the 3.1 Funnel structure is that the number of funnels is usually geometrical structure of global minima. In spite of their huge number, the local very small compared to the number of local Some care is needed when making explicit minima of the energy functions are not minima. The simplest instances are those use of such conjectures within an algorithm. randomly displaced. Indeed, we can group with even a single funnel (a typical example One of the first methods to solve LJ

them into a few (not necessarily disjoint) is LJ55, i.e., the LJ instance with N = 55 instances ([38], later refined by [53]) was large sets called funnels. If we denote a atoms). Hard instances are those with many based on the conjecture that global minima funnel by S, then every local minimum funnels (this is typical for Morse instances for instances with a relatively small number X0 ∈ S is the starting point of (at least) with large ρ values) but also those with few of atoms have an icosahedral structure. Based one sequence of local minima within S funnels when the funnel whose bottom on this conjecture, the search for global is the global minimum is very narrow minima was carried on an icosahedral lattice. … X0 → X1 → → Xk = Xfinal Xi ∈ S ∀ i (typical examples are LJ38 and LJ75). Simple Though successful on many instances, the algorithms, such as Basin Hopping and its limit of this approach, as well as of any such that ∀ i it holds that: variants (see [32, 50]), are able to reach quite other approach making a priori assumptions • E(Xi ) > E(Xi+1), i.e., the sequence efficiently a funnel bottom starting from about the structure of global minima, is its is monotonically decreasing; a given local minimum. In the simplest biasedness: it only explores a portion of the • Xi+1 is “reachable” from Xi, which instances with a single funnel (whose funnel search space, the one containing minima basically means that a small bottom must be the global minimum) every with icosahedral structure, but is completely neighborhood of Xi has a nonempty run of these algorithms quickly leads to the unable to detect global minima with a intersection with the region of global minimum, no matter which is the different structure. This was confirmed by attraction of the local minimum Xi+1 starting point, in spite of the huge number the later discovery of new putative global of local minima. In the hardest cases it may minima with non-icosahedral structure: All the sequences with the above properties be necessary to run the algorithms many LJ38 (FCC structure, see [14, 24, 40]), within a given funnel have a common times in a Multistart fashion from different LJ75−77,102−104 (decahedral structure, see [14, end point, denoted by Xfinal in (4), (usually randomly sampled) starting points 15]), LJ98 (a new and quite unexpected called the funnel bottom. In Figure 3 we before reaching the global minimum. structure, the Leary tetrahedron, see [31]). report an example of a one dimensional Another possible way to exploit function with a funnel structure. geometrical properties comes from observing that global minima of LJ and Morse instances (no matter if they have icosahedral, decahedral, close-packed or any other structure) are compact figures with particular shapes related to the three eigenvalues of their moment-of-inertia ten-sor. These shapes are usually spherical (all eigenvalues are equal), prolate (one eigenvalue is larger with respect to the other two), or oblate (one eigenvalue is smaller with respect to the other two). Figure 4 reports an example of each of these three shapes.

Figure 3: Illustration of a funnel op t i m a 7 6 March 2008 page 

is introduced by making a priori assumption starting from the local optimizer obtained on the structure of global minima with in the previous iteration. In the last iteration the risk of being unable to detect some the parameter is fixed to 0 and the function global minima whose structure is not one corresponds to the original one. of the a priori assumed, here we do not restrict to particular shapes, but, by means 3.3 Population-based approaches

of properly chosen bias parameters w1 and and dissimilarity measures w2, it is possible to drive the search towards Population-based approaches, where a all possible shapes. In practice it has been population of clusters (basically, local Figure 4: Cluster shapes: (a) spherical for observed that two-phase local searches minima of the energy function) is grown, (with few choices of the parameters w1 have been widely employed in cluster LJ98, (b) prolate for M30, (c) oblate for M61. and w2) employed within the monotonic optimization (see e.g. [3, 10, 28, 33, 41] Based on this observation, the following variant of Basin Hopping (see [32]) reduces for LJ clusters, [26, 42] for Morse clusters, modification of the energy function was by orders of magnitudes the effort for [29] for water clusters, [25, 39, 43, 44] for introduced (see [16, 33]): detecting the hardest (non-icosahedral) LJ binary clusters). Besides the usual operations F = E + h (5) instances (see [35]) and the very difficult (mutation, crossover), a key element in these i.e., F is the sum of the original Morse instances with ρ = 14 (see [16]). approaches is the dissimilarity measure, energy function E and of a geometric Here we have only discussed the geometry which measures the dissimilarity between penalization term h. For LJ and Morse of LJ and Morse instances. While the above two given clusters. One of the limits of all clusters h is defined as follows discussion can be extended to other clusters, those approaches, like Basin Hopping and some cases need a special attention. For its variants, where at each iteration a single instance, another potential, the Dzugutov cluster is kept in memory, is the fact that in one (see [21]), has global minima which some cases different runs of these approaches do not have compact shapes but have often converge to the same funnel bottom, polytetrahedral structures. Of course, in this although this might not correspond to case the geometric penalization (6), which the global minimum, but only to one where (xi, yi, zi) are the three coordinates of atom i, D is a parameter underestimating favors compact shapes, is not suitable, but it which is more easily reached. Population- the diameter of the cluster (largest distance is possible to think about other definitions based approaches are able to avoid this between atoms within the cluster), and for h which are suitable for this potential. phenomenon by keeping the population Water clusters represent a particularly diversified through a dissimilarity measure. w1,w2 are parameters belonging to the interval [0, 1]. This penalization term has interesting case from the geometrical point A basic requirement for such a measure the effect of compressing the cluster favoring of view because of the strong competition is to be able to recognize the equivalence its compactness, but the compression is between different three-dimensional (prism, (dissimilarity measure equal to 0) between a different along the three axes. The modified cage) and also bi-dimensional (book, 6- cluster and every possible result of rotations function F was employed to define two- ring) structures. Also in this case a suitable and/or translations of the cluster itself. In phase local searches: during the first definition for the geometric penalization population-based approaches each newly phase a local minimum of the modified term h is a possible subject for future generated cluster is compared only to similar function F is detected; then, in the second research. clusters within the current population phase the local minimum of F detected Finally, we remark that the introduction and replaces one of them if it has a lower in the first phase becomes the starting of the geometric penalization term h induces energy. The choice of the dissimilarity point of a local search with the original a deformation of the energy landscape, measure is essential for the efficiency of the which somehow “simplifies” it (in particular, approach. In [33] a measure for LJ instances energy function E. Parameters w1,w2 are the key ones. These are strictly related to it reduces the number of local minimizers is proposed. In [26] different measures are the eigenvalues of the moment-of-inertia and, even more important, of funnels). tested and compared over some hard LJ tensor (see [16]) and can be used to favor This is not the only possible way to induce and Morse instances and the results make different shapes. For instance, if we want deformations. Among others, we recall clear the high impact of the choice of the here the smoothing technique employed in measure on the efficiency of the approach. to favor spherical shapes we set w1 = w2 = 1 so that the first phase of two-phase [36] to solve molecular distance geometry We also refer to [27] for an experimental local searches will favor local minima with problems. This is basically a deformation analysis giving some insight into the results a spherical shape with respect to local of function (3) depending on a parameter. of the population based approach presented minima with other shapes. Therefore, by For some initial value of the parameter in [26]. Some measures have also been the function is deformed into a convex introduced to discriminate between different appropriately selecting parameters w1 and one, while in the following iterations the geometrical structures. For instance, the g w2 we can bias the search towards special geometrical shapes. However, while in the parameter is progressively reduced and value in [28], based on a projection of the previously mentioned approaches biasedness the resulting function is locally optimized cluster over a plane, is small for close-packed op t i m a 7 6 March 2008 page 

structures, larger for decahedral structures, types are exchanged), and the change one shapes; second, the model is a constrained and even larger for icosahedral structures. (the type of a single atom i is changed). one and thus local optimization, sampling We remark that these moves are extremely and perturbation have to take into account 3.4 Direct mutation important when dealing with binary (or, the constraints. Finally, there is no “energy” more generally, multi-atomic) clusters, to be minimized – we might think that Direct mutation is a special mutation because they allow to take into account the non-overlapping circles contribute 0 to operator employed in population-based combinatorial nature of these problems. the energy and overlapping ones have +∞ approaches (see, e.g., [28, 41]). We discuss penalty, but, in any case, the energetic it separately because of its relevance model is radically different from those seen especially when the number N of atoms 4 Conclusions and further remarks in this paper. Despite these differences, the increases. In spite of its importance in the In the previous section on computational authors successfully applied some of the field of cluster optimization, it has been approaches we did not mention methods techniques described in this paper within observed that the performance of the Basin for protein docking and protein folding the “Circle packing contest” (see [1, 54]); Hopping algorithm degrades as N increases. problems. There are several reasons for this in that contest participants were required The main reason for this behavior is the omission. First, the literature on methods to guess the positions of N non overlapping mechanism to generate a new candidate for protein conformation is so large that circles of prescribed different radii included local minimum in the neighborhood of the we cannot include even a short survey in in a smallest circular container. A quite current local minimum. In Basin Hopping this paper. Second, when dealing with effective algorithm was obtained by properly this is obtained by randomly perturbing proteins not only methods are different mixing the idea of funnel exploration, the all the coordinates of the current local but, perhaps more important, models are use of populations to avoid too greedy minimum. This way Basin Hopping often widely different - it is not clear which model searches and combinatorial moves (direct quickly reaches a local minimum which is good enough for protein prediction and mutations). only slightly differs from the global one, docking; quite often structures generated We are quite confident that many other but then the final improvement towards the by the minimization of an energy model structure determination problems might be global minimum is a very difficult, time- have to be “manually” refined by expert tackled with success through clever use of consuming step, because of the perturbation biologists who have sufficient experience global optimization techniques. at each iteration of all the atoms in the to visually analyze the shape of complex current solution, which disrupts the whole proteins. The difficulties associated with structure of the solution. Therefore, the global optimization of different models, key to improve the performance is to find each one depending on suitably calibrated other more structured and less random parameter sets, makes the comparison References (or even deterministic) moves, which are between algorithms quite a difficult task. defined as direct mutations. Often a direct This is the reason why in this survey we [1] B. Addis, M. Locatelli, and F. Schoen, mutation simply removes a single atom chose to give substantial space only to Efficiently packing unequal disks in a from a position (typically a position where the treatment of cluster optimization, circle: a computational approach which exploits the continuous and combinatorial the atom does not give a “good enough” where models are well defined and widely contribution to the total energy) and tries structure of the problem, Operations accepted. We may also add that many of the Research Letters, 36 (1), 37-42 (2008) to place it in a new and better position. ideas we find in the optimization of clusters [2] Baletto F., Ferrando R., Structural Direct mutation can thus be regarded as and, in particular, the notion of “funnel properties of nanoclusters: Energetic, a “fault correction” mechanism. In [28] landscape” and the methods to explore thermodynamic, and kinetic effects, Hartke observes that if direct mutation is funnel bottoms, are commonly found in the Reviews of Modern Physics, 77, 371- 423 employed “the resulting overall speedup literature on protein conformation. Thus, (2005) can be so large that it makes all the it seems that a good method for cluster [3] Barron C., Gomez S., Romero D., The difference between an efficient solution and optimization coupled with a good model optimal geometry of Lennard-Jones impractically long computation times”. for the evaluation of the free energy of a clusters:148-309, Computer Physics Dynamic Lattice Searching (DLS) protein will yield a promising approach Communications, 123, 87-96 (1999) [4] Biswas P., Toh K.-C., Ye Y., A distributed [6], where only atoms with high energy to solve protein conformation problems. contribution are moved over a (dynamic) SDP approach for large-scale noisy anchor- Although this paper was focused on free graph realization with applications to lattice made up by their own positions plus molecular conformation problems, we are molecular conformation, available at www. all possible vacant sites, can be viewed as a confident that some of the ideas presented stanford.edu/~yyye/ SISC-molecule-revised- direct mutation operator. here might find an application in other 2.pdf (2007) Finally, we include in the field of direct fields. As an example, let us consider [5] Camacho C.J., Vajda S., Protein Docking mutation operators also the combinatorial the classical disk packing problem: Along Smooth Association Path ways, moves employed with binary clusters (see, although vaguely resembling a molecular PNAS, 98, 10636-10641 (2001) e.g., [19]). Such moves are the swap one (if conformation problem, this one is indeed [6] Cheng L., Cai W., Shao X., A atoms i and j are of different types, their quite different. First, it is based on 2D conformational analysis method for op t i m a 7 6 March 2008 page 

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(2002) folding problem: global optimization of force fields, op t i m a 7 6 March 2008 page  WOSP2007 By Tom Luo, University of Minnesota Shuzhong Zhang, The Chinese University of Hong Kong

A first of its kind workshop on Optimization and Signal Processing on-going work has substantially enlarged the set of signal processing (WOSP2007) was recently held on the campus of the Chinese problems that can be reliably solved in an efficient manner. For University of Hong Kong (CUHK) during the period December 19 the optimization community, signal processing provides a rich - 21, 2007. With financial support from the Department of Systems source of application problems to which the advanced optimization Engineering and Engineering Management, The Shun Hing knowledge and algorithms can bring a strong and immediate Institute of Advanced Engineering at CUHK, and the Huawei impact. Some of the signals processing problems have led to Technologies Ltd., the workshop has brought together some of significant theoretical advances in optimization. Through close the world’s leading experts from both signal processing and the collaboration with researchers from signal processing, optimizers optimization communities, as well as technical representatives from can help recognizing and solving convex problem formulations; leading information technology industry. It provided a valuable utilizing the theory of convex optimization to characterize and gain forum for the algorithm developers and engineering practitioners to insight into the optimal solution structure and to derive bounds share research ideas and identify important topics of future research. on performance; deriving convex relaxations of hard, non-convex In the past thirty years, the work-horse algorithms in the field problems; and developing powerful general purpose or application- of digital signal processing and communication have been the driven specific algorithms, including those that enable large scale gradient descent and the least squares algorithms. While these optimization by exploiting the problem structure. algorithms have served their purpose well, they suffer from slow The goal of WOSP2007 was to promote this burgeoning field of convergence and sensitivity to the algorithm initialization and interdisciplinary research. This small workshop has attracted 187 stepsize selection, especially when applied to ill-conditioned or registered participants; most of them are graduate students from nonconvex problem formulations. This is unfortunate since many various universities of Hong Kong. This plus a delegation of 20 design and implementation problems in signal processing and technical representatives from Huawei Technologies Ltd located in digital communication naturally lead to nonconvex optimization Shenzhen, as well as many non-registered participants, created a formulations, the solution of which by the gradient descent large audience that easily exceeded the maximum seating capacity algorithm usually works poorly. Moreover, some applications (245) of the lecture theatre. Some participants had to stand or sat require real-time implementation in DSP chips or large scale on the floor to listen to the talks. The technical program consisted deployment across a distributed network. Simply put, the need of tutorial lectures as well as in-depth technical presentations for efficient and robust optimization algorithms is greater than showcasing the success of applications of optimization in signal ever in the field of signal processing and communications. processing. The lecture materials, including the slides of the In recent years, the field of optimization has witnessed a presentations, can be found in the website of the workshop: significant surge in the research of interior point methods and www.se.cuhk.edu.hk/~zhang/WOSP2007/program.html convex conic optimization. A set of extremely powerful algorithms and highly reliable software packages have been developed. This

Frontiers in Bioscience, 9, Science, 285, 1368-1372, A, 101, 5111–5116, (1997). 108, 9516-9520 (2004) 3296–323, (2004). (1999). [49] Williams G.A., Dugan J.M., [51] G. L. Xue, Improvements [45] M. Wagner, J. Meller, R. [47] D. J.Wales, Energy Landscapes Altman R.B., Constrained on the Northby Algorithm Elber, Large-scale linear with Applications to Clusters, global optimization for for molecular conformation: programming techniques for Biomolecules and Glasses, estimating molecular Better solutions, Journal of the design of protein folding Cambridge University Press, structure from atomic Global Optimization, 4, 425– potentials, Mathematical Cambridge, (2003). distances, J.Comp.Biol., 8, 440 (1994). Programming B, 101, 301–318, [48] D. J. Wales and J. P. K. Doye, 523-547 (2001) [52] Al Zimmermann’s Circle (2004). Global optimization by [50] Xiang Y., Cheng L., Cai Packing Contest, www. [46] David J. Wales and Harals basin-hopping and the lowest W., Shao X., Structural recmath.org/contest/ A. Scheraga, Global energy structures of Lennard- distribution of Lennard-Jones CirclePacking, 2006. Optimization of Clusters, Jones clusters containing up clusters containing 562 to Crystals, and Biomolecules, to 110 atoms, J. Phys. Chem. 1000 atoms, J. Phys. Chem. A, op t i m a 7 6 March 2008 page 10

A Celebration of

Before getting into the specifics of the 50 Years of Integer workshop, some brief words about the venue are in order. Centre Paul Langevin is located at Aussois, a small Savoyardvillage type ski resort in the French Alps close Programming to the Italian border. It is situated in the Jon Lee Maurienne valley, at the gateway of the IBM T.J. Watson Research Center Vanoise National Park, starting from Yorktown Heights, New York, USA The year 2008 marks the fiftieth 1500m of altitude (to 2750m high). Besides anniversary of the birth of integer providing a measure of isolation, which e-mail: [email protected] programming. Naturally, you are now helps make for a good workshop, on the wondering what seminal event occurred occasion that participants need a short in 1958 that we now refer to as the birth diversion from the mathematics, there are of the subject. During that year, Dr. opportunities for skiing and snowboarding Ralph E. Gomory devised and published (a passion of the present author!). a short paper [26] that really set the field of integer programming in motion. In case The center has a conference room with 196 you follow mathematical programming seats, enough lodging for most participants, only very casually, in that paper Gomory and dining facilities. The center is named described his cutting-plane algorithm for for Paul Langevin (born 23 January 1872), pure integer programs, and he announced who was a prominent French physicist. that the method could be refined to give a Besides his scientific activities, Langevin finite algorithm for integer programming. was a founder of the Comité de vigilance A published proof of his finiteness result is des intellectuels antifascistes, and he was contained in [32]. Gomory gave a cutting- also president of the Ligue des droits de plane algorithm for the mixed integer l’homme (Human Rights League) from problem [29], and this approach was only 1944 to 1946. He died in Paris on 19 shown to be quite effective many years later December 1946, two years after living [2]. It is interesting to note that now, half to see the Liberation of Paris. Langevin a century after they were first introduced, was buried at the Panthéon (in Paris). even Gomory’s finite cutting-plane method for the pure integer case is being re- The organizers of the workshop were: examined and is showing new promise [5]. Michael Jünger (Universität zu Köln), To commemorate this occasion, on 7 Thomas Liebling (École Polytechnique January 2008 a special workshop was held Fédérale de Lausanne), Denis Naddef at Centre Paul Langevin, Aussois, France. (Ecole Nationale Supriéure d’Informatique This special workshop was part of the 12th et de Mathématiques Appliquées de Combinatorial Optimization Workshop, Grenoble), William Pulleyblank (IBM 7–11 January 2008, which is held each Corporation), Gerhard Reinelt (Universit¨at year at Aussois. It is fair to say that the Heidelberg), Giovanni Rinaldi (Istituto Aussois workshop is the yearly event for di Analisi dei Sistemi ed Informatica, presenting and keeping up with the latest Roma), and Laurence Wolsey (Université developments in integer programming and catholique de Louvain). Already one can combinatorial optimization coming from see that the organizers themselves are the operations research community. In light leaders in the subject, providing some of this, it was natural for the celebration of of the leading work over the course of a fifty years of integer programming to be few decades, spanning such key areas as staged as part of an Aussois workshop. polyhedral combinatorics, branch and op t i m a 7 6 March 2008 page 11

Left - Right: Balinski, Gomory, Karp

cut, matching, the TSP, lot sizing, and of G.B. Dantzig, D.R. Fulkerson and S. Alan J. Hoffman, Harold Kuhn and Ailsa computational integer programming. Johnson [23] on cutting planes for the TSP H. Land were also invited to be panelists, and the fact that counting is hard (even but unfortunately they were not able to The workshop began after lunch, with for integer programmers). It is noteworthy attend. Susan Powell was kind enough to a session, chaired by Tom Liebling, of that Gomory acknowledged in [26] the deliver some remarks on behalf of Land. invited survey talks on some themes in influence of [23] and [17] on his work: integer programming that have withstood Egon Balas is University Professor of the test of time. In this session, there were “The algorithm closely resembles the Industrial Administration and Applied talks by Gérard Cornuéjols (“Polyhedral procedures already used by Dantzig, Mathematics and the Thomas Lord Approaches to Mixed Integer Linear Fulkerson and Johnson, and Markowitz Professor of Operations Research at Programming”), Bill Cook (“50+ Years of and Manne to obtain solutions to Carnegie Mellon University. He was Combinatorial Integer Programming”), discrete variable programming problems. awarded the John von Neumann and Laurence Wolsey (“Decomposition and Their procedure is essentially this. Given Theory Prize (INFORMS) in 1995. Reformulation in Integer Programming”). the linear program, first maximize the Some of the fundamental work of Balas objective function using the simplex includes implicit enumeration [35] and In the early evening, George Nemhauser method, then examine the solution. If disjunctive programming [1]. Much led us through the history of the first the solution is not in integers, ingenuity more about him can be found in [20]. twenty years of integer programming, is used to formulate a new constraint walking us through a list of milestone that can be shown to be satisfied by Michel Balinski is a Directeur de Recherche papers in that time period. Nemhauser the still unknown integer solution but (émérite), CNRS, École Polytechnique, kindly agreed to allow us to reprint that not by the noninteger solution already Paris. He was founding Editor-in-Chief of list here (see references [23]–[61]). Of attained.... What has been needed the journal Mathematical Programming and course such a list is subjective and will to transform this procedure into an participated in founding the Mathematical inevitably suffer from some omissions, algorithm is a systematic method for Programming Society. Balinski was awarded but all of us interested in the field would generating the new constraints.” a Lester R. Ford Award (Mathematical be well served by studying these papers. Association of America) in 1976 for his Further information regarding the early Nemhauser’s review nicely set up a panel paper [3]. He is well known for his work days of integer programming and associated session, led by Bill Pulleyblank, with six of on routing, apportionment and voting, topics can be found in [16] and [22]. the pioneers who have been so influential, and set partitioning approaches. especially during that early period. The Nemhauser took 1954 as his starting panelists were: Egon Balas, Michel Balinski, Jack Edmonds was a professor in the point, highlighting both the seminal paper Jack Edmonds, Arthur M. Geoffrion, Department of Combinatorics and Ralph E. Gomory and Richard M. Karp. Optimization at the University of op t i m a 7 6 March 2008 page 12

Left - Right: Edmonds, Karp Geoffrion Balas

Waterloo, Ontario, from 1969. He retired Department in 1965, and IBM Director addition, he is a Research Scientist at the from that position in 1999. Edmonds of Research in 1970, a position which he International Computer Science Institute at was awarded the John von Neumann held until 1986. Gomory became IBM Vice Berkeley. Karp was awarded the Frederick Theory Prize (INFORMS) in 1985. President in 1973 and Senior Vice President W. Lanchester Prize (INFORMS) in Edmonds began his fundamental work in 1985. In 1986 he was named IBM Senior 1977, the Delbert Ray Fulkerson Prize at the National Bureau of Standards. Vice President for Science and Technology, in (AMS and MPS) in 1979, the Turing He is particularly well known for his retiring in 1989. Gomory was awarded the Award (ACM) in 1985, the John von contributions to polyhedral combinatorics Frederick W. Lanchester Prize (INFORMS) Neumann Theory Prize (INFORMS) in [39], branchings [44], matroids [51], in 1963, the John von Neumann Theory 1990, and the National Medal of Science matching [38], flows [4] and the notion Prize (INFORMS) in 1984, and the in 1996. Karp’s work that relates to integer of polynomial time [38]. For more details National Medal of Science (by the President programming includes Lagrangian duality, on Edmonds’ illustrious career, see [14]. of the United States) in 1988. In addition subgradient optimization and the TSP to his work on cutting-plane methods and [49, 53], reducibility of combinatorial Arthur M. Geoffrion is the James A. Collins also corner polyhedra [10, 11, 12], Gomory problems [56], and efficient algorithms Professor of Management Emeritus at the also made fundamental contributions for network-flow problems [4]. UCLA Anderson School of Management, to column generation [6, 7] and to the Los Angeles. In 2000 he was awarded concept of providing a structure that Harold W. Kuhn is a Professor Emeritus the George E. Kimball Medal, for service solves many closely related instances of an of Mathematics at Princeton University. to INFORMS and to the profession of optimization problem [9]. Much more on He was awarded the John von Neumann operations research. Geoffrion is well Gomory’s career can be found in [13]. Theory Prize (INFORMS) in 1980. known for his contributions to implicit He is particularly well known for his enumeration [45] and to decomposition Alan J. Hoffman is a Fellow Emeritus contributions to game theory and to schemes and their connections to of IBM Research. He has been a nonlinear programming, and for the Lagrangian duality [42, 55]. In 1978 Member of the National Academy of Hungarian method for the assignment Geoffrion co-founded INSIGHT, Inc., a Sciences since 1982 and was awarded problem [15]. More information management consulting firm specializing the John von Neumann Theory Prize about Kuhn can be found in [19]. in optimization-based applications in (INFORMS) in 1992. Hoffman has made supply-chain management and production fundamental contributions to the theory Ailsa H. Land and Alison G. Doig proposed planning. In 1982 he founded what has of total unimodularity [24], polyhedral in 1957 and published in 1960 [28] what become the INFORMS Roundtable. combinatorics and our understanding of is considered the origin of branch and greedy algorithm; [18] and [21] provide bound as a general technique. Land is Ralph E. Gomory is President Emeritus of wonderful opportunities to find out much Professor Emeritus of Operational Research the Sloan Foundation, which he led 1989- more about the man and his work. at the London School of Economics. 2008. Before that, Gomory spent nineteeen years at IBM, beginning in the summer of Richard M. Karp is the Class of 1939 Many of the panelists noted the strong 1959. He was named IBM Fellow in 1964, Chair and University Professor at the support, in the early days, of the Rand Director of the Mathematical Sciences University of California at Berkeley. In Corporation and Princeton University. op t i m a 7 6 March 2008 page 13

As a result of the Cold War, Government concerning nonlinearity and discrete funding was quite high for science and On Tuesday, the program was completely choices: Robert Weismantel (“Nonlinear mathematics, so there were ample resources filled with talks concerning cutting planes. Integer Programming”) and Michel to support optimization. Many of the Some highlights included: Ralph Gomory Balinski (“The Majority Judgement”). panelists told how the influence of George (“40 Years of Corner Polyhedra”), Matteo Dantzig, Ray Fulkerson, Alan Goldman, Fischetti (“Looking inside Gomory”) and Friday’s program included excellent Alan Hoffman and Alan Tucker should Jean-Philippe Richard (“Group Relaxations talks, rewarding those who could stay not be underestimated. For some, the for Integer Programming”). A bit tiring for for the full week. For example: Fritz passion came from the beauty of what those of us with a short attention span, but Eisenbrand (“Integer Programming and they were investigating, for others the ample testimony to the fact that the subject Number Theory”) Richard Karp (“How applications and the associated drive to remains a hot area even after fifty years. Hard are the NP-hard Problems?”), make a difference were motivating factors. Andrea Lodi (“Computation in Integer On Wednesday, topics included some Programming”) and Tom McCormick After a long day of listening, we reached very exciting and less traditional topics (“Strongly Polynomial Algorithms for our fill of food for thought, and the whose computational value is only just Bi-Submodular Minimization”). evening concluded with a banquet. now being well explored. For example: François Margot (“Symmetry in Integer The workshop was generously sponsored From Tuesday through Friday, the workshop Programming”), Franz Rendl (“Semidefinite by ADONET —Algorithmic Discrete was run on its traditional model. That is, Relaxations for Integer Programming”) Optimization Network (a Marie there were no parallel sessions, and the and Kurt Anstreicher (“A Computable Curie Research Training Network program was made during the meeting. Characterization of the Convex Hull of of the European Union), ALMA, As usual, there were two morning sessions Low-Dimensional Quadratic Forms”). ILOG, IBM and Springer Verlag. and one late afternoon session to leave time in the afternoon for “collaboration” On Thursday, some highlights included (broadly defined to include joint efforts at: two very different aspects of issues proving theorems, enjoying the beautiful landscape, and skiing and snowboarding).

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Kernighan of an algorithm for integer search algorithm for mixed Combinatorial Mathematics (1973). An effective heuristic solutions to linear programs, integer programming and its Applications, algorithm for the traveling Bull. of the American problems, The Computer R.C. Bose et al. eds. University salesman problem, Operations Mathematical Society 64, Journal 8, 250–254. of North Carolina Press, Research 21, 498–516. 275–278. [38] J. Edmonds (1965). Paths, 171–175. [61] M.W. Padberg (1973). On the [27] G.B. Dantzig (1960). On the trees, and flowers, Canad. J. [49] M. Held and R.M. Karp facial structure of set packing significance of solving linear Math. 17, 449–467. (1970). The traveling salesman polyhedra, Mathematical programs with some integer [39] J. Edmonds (1965). Maximum problem and minimum Programming 5, 199–215. variables, Econometrica 28, matching and a polyhedron spanning trees, Operations IBM T.J. Watson Research Center 30– 34. with 0,1-vertices, J. Res. Nat. Research 18, 1138–1162. Bur. Standards 69B, 125–130. e-mail address: [email protected] op t i m a 7 6 March 2008 page 15

MPS Chair’s Column Steve Wright 17 February 2008

It’s a pleasure to contribute my second inventor of the product-form inverse of the level, due in large part to new members who column to Optima and an even greater basis matrix. Gene H. Golub of Stanford joined through their participation in the pleasure to see our society’s newsletter University died on 16 November 2007 most recent ISMP and ICCOPT meetings. back on a regular production schedule. after a short illness. Gene was a giant of Part of our challenge is to serve these new We are much obliged to Andrea Lodi and numerical analysis and scientific computing, members well enough that they will renew the editorial and publishing teams for and his research laid important foundations in future years! If you have any suggestions making this happen. This issue of Optima for numerical optimization. Even more concerning services that you would like contains an article by Jon Lee on the importantly, in both his leadership capacities to see MPS provide, or new roles that we 2008 Aussois workshop, which included a and personal life, he was a great supporter of should be playing, please contact Rolf or commemoration of the 50th anniversary the optimization community and of many myself. And please, if you have not already of the publication of Ralph E. Gomory’s individual optimization researchers. done so, renew your MPS membership for 1958 paper that founded the area of integer I salute our former Chair Rolf Moehring, 2008! programming. Jon’s article makes clear that who celebrated his 60th birthday on 16 Finally, I draw your attention to integer programming remains a thriving February 2008. The occasion was marked provisions in the society’s by-laws (posted field, with many of the founders remaining by a symposium organized by his colleagues on our web site mathprog.org) about the actively engaged in research alongside in Berlin. Rolf continues to work hard establishment of regional and technical extremely talented younger generations. on behalf of MPS by heading an ad hoc sections of MPS. If you and a group of like- I note with sadness the passing of two membership committee whose charge is to minded optimization colleagues see benefits individuals who meant a great deal to find new ways for the society to serve its in organizing yourselves, possibly with a our community. Alex Orden, who was members and the optimization community. view to holding regional meetings or topical Chair of MPS in 1983-86 and a founding With the vast changes to the academic conferences, or to establishing a web-based council member of the society, passed landscape brought on by electronic community around some interest area, feel away in Chicago on 9 February 2008. publishing, and with the increasing size and free to consult with us about the possibility Alex was renowned for his work on linear diversity of our own research community, of MPS affiliation. The arrangements programming, in particular, as co-author it is time for us to step back and think hard outlined in our by-laws are quite flexible. with Dantzig and Wolfe of the paper on about how MPS should adapt and evolve. the generalized simplex method and as the Our membership stands at 1150, a record

Application for Membership Mail to: Mathematical Programming Society I wish to enroll as a member of the Society. 3600 University City Sciences Center My subscription is for my personal use and not for the benefit of any library or institution. Philadelphia, PA 19104-2688 USA c I will pay my membership dues on receipt of your invoice. c I wish to pay by credit card (Master/Euro or Visa). Cheques or money orders should be made payable to The Mathematical Programming Society, Inc. Dues for credit card no. expiration date 2008, including subscription to the journal Mathematical Programming, family name are US $85. Retired are $40. mailing address Student applications: Dues are $20. Have a faculty member verify your student status and send application with dues to above address. telephone no. telefax no.

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signature4 Institution op t i m a 7 6 March 2008 page 16 Call for nomination of the 2009 George B. Dantzig Prize in Mathematical Programming

Nominations are solicited for the George B. Dantzig Prize, administered jointly by the Mathematical Programming Society (MPS) and the Society for Industrial and Applied Mathematics (SIAM). This prize is awarded to one or more individuals for original research which by its originality, breadth and depth, is having a major impact on the field of mathematical programming. The contribution(s) for which the award is made must be publicly available and may belong to any aspect of mathematical programming in its broadest sense. Preference will be given to candidates who have not reached their 50th birthday in the year of the award.

The prize will be presented at the 2009 International Symposium on Mathematical Programming, to be held August 23-28, 2009, in Chicago, Illinois, U.S.A. Past prize recipients are listed on the MPS Web site. The members of the prize committee are Jong-Shi Pang (Chair), Yuri Nesterov, Alexander Schrijver, and Eva Tardos.

Nominations should consist of a letter describing the nominee’s qualifications for the prize, and a current curriculum vitae of the nominee including a list of publications. They should be sent to

Jong-Shi Pang Department of Industrial and Enterprise Systems Engineering University of Illinois at Urbaba-Champaign 117 Transportation Building MC-238 104 S. Mathews Ave. Urbana Illinois 61801 U.S.A.

e-mail: [email protected]

and received by 15 November 2008. Submission of nomination materials in electronic form is strongly encouraged. op t i m a 7 6 March 2008 page 17 Discussion Column Commentary on “Structure prediction and global optimization”

by *Roy L. Johnston

Global optimization is of undoubted and for certain classes of problems, no one minimization. The danger of assuming a increasing importance in most areas of approach is guaranteed to work in every growth sequence (i.e. extrapolating from science and engineering. The problem of application. known global minima to predict those of global optimization – the determination of In the article “Structure prediction larger clusters) is pointed out, but it is also the absolute maximum or minimum of a and global optimization”, Locatelli and shown that biasing can be used in a positive function (or indeed a process) depending Shoen describe the application of global sense by modifying the energy or “penalty” on a large number of variables – is very optimization to several problems, taken function (deforming the landscape) to difficult, even for discrete integer-valued from chemistry and biology, involving include a geometrical term in order to direct problems, let alone for continuous, real- the prediction of structure: finding the the optimization towards structures with valued problems. Global optimization can minimum energy configuration of a cluster a certain desired packing arrangement or be regarded as the process of searching (or nanoparticle) composed of atoms or overall shape. a multi-dimensional landscape for the molecules; determining the lowest energy The efficiency of population-based search highest “mountain peaks” or lowest “valley folded conformation of a model protein methods (such as Genetic Algorithms, and bottoms”. While this does not present molecule; and the docking of protein parallel tempering) is due to the parallel too many problems in the everyday 3- molecules. In these cases, the inter- exploration of the configuration landscape. dimensional world in which we live, in the atomic interactions are described by quite Locatelli and Shoen show that these higher dimensions (i.e. number of variables) simple potential energy functions, which, methods are most successful when coupled commonly encountered in important nevertheless, reproduce essential aspects with the use of dissimilarity measures, so as scientific and engineering problems, the of the underlying physics. The difficulty to ensure (or at least encourage) exploration situation is far harder. Thus, apart from in finding the global mimimum energy of diverse regions of the landscape. The the cases of simple convex mathematical arises due to the high dimensionality of process of “direct mutation”, which involves functions, or discrete problems small the problem (for example, the number more deterministic permutations, perhaps enough to be grid searched (or for which of variables for an N-atom cluster is 3N, utilising prior knowledge about the system, a branch-and-bound search tree-pruning corresponding to the Cartesian coordinates can also be used within a population-based algorithm can be applied), one can never of all the component atoms) and the search method to move (ideally) towards be certain that the lowest (or highest) value consequent very high number of possible the global minimum. The Dynamic Lattice that is found is really the global optimum. configurations (local minima). Searching method (wherein high energy This is to be contrasted with the certainty As well as providing an important survey atoms in the clusters are preferentially with which we can locate and identify and bibliography of applications of global moved) is one such example of this local minima and maxima, utilising (either optimization to the cluster optimization approach. analytical or numerical) gradients and problem (which they concentrate on), In conclusion, I believe that Locatelli curvatures of the function in question. Locatelli and Shoen describe a number and Schoen have presented a concise yet Over the past 30 years or so, many of approaches and algorithms which they informative introduction to the field of approaches have been developed to increase (among others) have employed to improve structure prediction as an exercise in global the likelihood of finding global optima, the success rate and efficiency of global optimization, along with recent techniques including techniques such as simulated optimization techniques. For example, for understanding and improving the annealing, Monte Carlo methods (e.g. the the recognition of funnel-like features on optimization procedure. Applications of Basin Hopping approach) and the growing energy landscapes is used to rationalise the many of these methods to other problems class of Evolutionary Algorithms (e.g. differences between certain clusters whose are already widespread and likely to grow Genetic Algorithms). Of course, the “No (putative) global minima are easy to find considerably in the future. In their final free lunch theorem” ensures that while with those which are far more difficult to section, by way of an example, the authors certain methods may be particularly good find. Consideration of the energy landscape show that their approach has been very also explains the relatively high success of successfully applied to the 2-D disk packing methods, such as Basin Hopping and hybrid problem. *School of Chemistry, University of Genetic Algorithms which incorporate local Birmingham,Edgbaston, Birmingham B15 2TT,United Kingdom op t i m a 7 6 March 2008 page 18 Ettore Majorana Centre for Scientific Culture

International School of Mathematics PURPOSE OF THE WORKSHOP “G. Stampacchia” The need of providing satisfactory answers to several questions posed by diverse advanced Erice - Sicily, Italy application fields, basically in Engineering, Mechanics and Economics, has been the 48st Workshop: Nonsmooth Analysis, strongest motivation to tackle mathematical problems where differentiability of the involved Optimization and Applications functions is no longer guaranteed. Nonsmooth Analysis has then grown considerably and May 9 - 17, 2008 it is now a well established area of modern Mathematics. The main battlefield, where the Lecture-Hall: San Rocco theoretical findings of Nonsmooth Analysis are tested, is the design of effective algorithms for solving a wide range of optimisation problems. Starting from the pioneering works in Sponsored by the: the Sixties on the minmax problems, several application fields have benefited from the development of Nonsmooth Analysis. We list here the study of large scale programming – Italian Ministry of University problems via decomposition techniques, the Lagrangian relaxation of integer extremum and Scientific Research problems, the numerical solution of variational inequalities, the extremum problems with – Sicilian Regional Government equilibrium constraints, several classification and approximation problems, structural – University of Calabria design, nonsmooth mechanics, etc. The aim of the Workshop is to bring together people – Centro “Enrico Fermi” working on both sides of Nonsmooth Analysis and Optimization and Applications to – Italian National Research Council, discuss the state-of-the-art and the possible future developments. Some tutorials will also Institute of High Performance be given to encourage young scientists to approach such an exciting research area. The – Computing and Networking, Workshop is dedicated to Vladimir F. Demyanov, on the occasion of his 70th birthday. Rende (CS) LOCATION Young people with only limited experience should enclose a scientific curriculum The workshop will be held in Erice, vitae and a letter of recommendation Sicily, Italy at the “E. Majorana” Centre from the head of their research group or for Scientific Culture. The Centre is from a senior person active in the field. located in the pre-mediaeval city of Application by e-mail is strongly Erice and the lecture halls are located encouraged. Closing date for application: in two restored monasteries and the March 15, 2008. Participants are ancient Palazzo Ventimiglia - former expected to arrive in Erice on residence of Viceroys of Sicily. May 9, no later than 5 p.m.

APPLICATIONS M. Gaudioso and D. Pallaschke Persons wishing to attend the Directors of the Workshop Workshop and possibly to contribute a lecture should contact: F. Giannessi Director of the School Professor Manlio Gaudioso D.E.I.S. - Università della Calabria A. Zichichi Via Pietro Bucci, Cubo 41C Director of the Centre 87036 Rende (CS), Italy. e-mail: [email protected] TOPICS Numerical methods for nonsmooth Specifying: optimization 1. Data and place of birth, together Nonsmooth optimization and integer with present nationality; programming 2. Affiliation; Nonsmooth dynamics 3. Address, e-mail address. Nonsmooth analysis Learning methods op t i m a 7 6 March 2008 page 19

continued Ettore Majorana Centre for Scientific Culture

Minmax problems Antonio Frangioni Juan Enrique Martinez Legaz Decomposition methods Università di Pisa, I Universitat Autònoma de Barcelona, E Variational inequalities Online and incremental methods Masao Fukushima Boris Mordukhovich Nonsmooth mechanics University of Kyoto, J Wayne State University, USA

LECTURES Alexei Gaivoronski Massimo Pappalardo Norwegian University of Science and Università di Pisa, I The workshop will consist of invited lectures Technology, Trondheim, NO and contributed lectures. Invited lecturers Panos Pardalos who have confirmed the participation are: Giorgio Giorgi University of Florida, Gainesville, USA Università di Pavia, I Adil Bagirov J.P. Penot University of Ballarat, A Angelo Guerraggio University of Pau, F Università dell’Insubria, I J.-P. Crouzeix R.T. Rockafellar Université Blaise Pascal, Clermont-Ferrand, F Alexander Ioffe University of Washington, USA Technion, Haifa, IL Asen L. Dontchev Stefan Scholtes University of Michigan, USA H.-Th. Jongen University of Cambridge, UK RWTH Aachen University, D Rosalind Elster Michel Thera Universitat Autònoma de Barcelona, E Alexander Kurzhanski Université de Limoges, F University of California, Berkeley, USA Francisco Facchinei Xiaoqi Yang Università di Roma “La Sapienza”, I Hong Kong Polytechnic University, PRC OP T I M A mathematical programming society

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Katya Scheinberg Journal contents are subject to IBM T.J. Watson Research Center change by the publisher. PO Box 218 Yorktown Heights, NY 10598, USA [email protected]