70 OP11 Abstracts IP1 general. Moreover, box constraints on the state function Sparse Optimization are also admitted. Later, the model is improved by in- cluding Maxwell equations for induction heating and state Many computational problems of recent interest can be constraints on the generated temperature. Also the com- formulated as optimization problems that contain an un- pletion of the model by Navier-Stokes equations in a melt is derlying objective together with regularization terms that briefly mentioned. For these problems, the well-posedness promote a particular type of structure in the solution. of the underlying systems of PDEs and the principal form Since a commonly desired property is that the vector of of optimality conditions are sketched. Numerical examples unknowns should have relatively few nonzeros (a “sparse illustrate the theory, justify certain simplifications and and vector’), the term “sparse optimization’ is used as a broad motivate the necessity of more complex models. Finally, label for the area. These problems have arisen in machine ongoing research on applications to industrial processes of learning, image processing, compressed sensing, and sev- crystal growth is briefly addressed. eral other fields. This talk surveys several applications that can be formulated and solved as sparse optimization Fredi Tr¨oltzsch problems, highlighting the novel ways in which algorithms Technische Universit¨at, Berlin have been assembled from a wide variety of optimization [email protected] techniques, old and new. Stephen Wright IP4 University of Wisconsin Matrix Optimization: Searching Between the First Dept. of Computer Sciences andSecondOrderMethods [email protected] During the last two decades, matrix optimization prob- lems (MOPs) have undergone rapid developments due to IP2 their wide applications as well as their mathematical ele- Optimizing Radiation Therapy - Past Accomplish- gance. In the early days, second order methods, in par- ments and Future Challenges ticular the interior point methods (IPMs), seemed to be a natural choice for solving MOPs. This choice has been Optimization has found a successful and broad application perfectly justified over the years by the success of using in intensity-modulated radiation therapy (IMRT). We will IPMs to solve small and medium sized semi-definite pro- briefly review the standard model of IMRT optimization. gramming (SDP) problems. On the other hand, since at There is still a lot of space for improvement in IMRT and each iteration the memory requirement and the computa- elsewhere in radiation therapy: 1. Optimization of the tional cost for second order methods grow too fast with irradiation geometry including multiple fixed beams and the number of constraints and the matrix dimensions in dynamic arcs. 2. The adaptation (re-optimization) of the MOPs, second order methods have recently become less treatment during the course of treatment. 3. The han- popular. The current trend is to apply first order methods dling of motion and uncertainties, particularly in proton to MOPs. This comes no surprise as first order methods therapy, and 4. The direct optimization of treatment out- normally need less computational cost at each iteration. come rather than physical (dosimetric) surrogates of it. A To say the least while first order methods can often run for general challenge is that physicians often cannot provide some iterations even for large scale problems, the second clear mathematical objectives and strict constraints. In- order methods may fail even at the first iteration. However, stead, they pursue an I know it when I see it approach. We for many MOPs the overall cost of first order methods for try to address this issue through interactive multi-criteria obtaining a reasonably good solution is often prohibitive, optimization. if possible at all. To break this deadlock, we need to con- struct algorithms that possess two desirable properties: 1) Thomas Bortfeld at each iteration the computational cost is affordable; and Massachusetts General Hospital and Harvard Medical 2) the convergence speed is fast but may not be as fast School as second order methods (no free lunch). In this talk, by Department of Radiation Oncology using least squares correlation matrix and SDP problems [email protected] as examples we demonstrate how these two properties can be potentially achieved simultaneously. Variational anal- ysis, in particular semi-smooth analysis, will be empha- IP3 sized and inexact proximal point algorithms (PPAs) will be On Some Optimal Control Problems of Electro- recommended for solving large scale symmetric and non- magnetic Fields symmetric MOPs. In this survey, a sequence of optimal control problems for Defeng Sun systems of partial differential equations of increasing com- Dept of Mathematics plexity is discussed. The audience is guided step by step National University of Singapore from an academic heating problem to fairly complex con- [email protected] trol problems of induction heating. While the first prob- lem is modeled by the Poission equation, the last one is related to industrial applications. It includes equations for IP5 heat conduction, heat radiation, and Maxwell equations Replacing Spectral Techniques for Expander Ratio, modeling induction heating. To introduce into main prin- Normalized Cut and Conductance by Combinato- ciples of PDE control, the talk begins with a simplified rial Flow Algorithms optimal control problem for a linear elliptic equation with box constraints on the control function. It can be inter- We address challenging problems in clustering, partition- preted as a heating problem. Next, nonlinear local and ing and imaging including the normalized cut problem, nonlocal radiation boundary conditions are considered and graph expander, Cheeger constant problem and conduc- the geometry of the computational domain will be more tance problem. These have traditionally been solved using OP11 Abstracts 71 the “spectral technique”. These problems are formulated IP8 here as a quadratic ratio (Raleigh) with discrete constraints Title Not Available at Time of Publication and a single sum constraint. The spectral method solves a relaxation that omits the discreteness constraints. A new Abstract not available at time of publication. relaxation, that omits the sum constraint, is shown to be solvable in strongly polynomial time. It is shown, via an Thomas F. Coleman experimental study, that the results of the combinatorial The Ophelia Lazaridis Research Chair algorithm often improve dramatically on the quality of the University of Waterloo results of the spectral method in image segmentation and [email protected] image denoising instances. Dorit Hochbaum CP1 Industrial Engineering and Operations Research Chance-Constrained Second-Order Cone Program- University of California, Berkeley ming: A Definition [email protected] Second-order cone programs (SOCPs) are class of con- vex optimization problems. Stochastic SOCPs have been IP6 defined recently to handle uncertainty in data defining Nonsmooth Optimization: Thinking Outside of the SOCPs. A prominent alternative for stochastic program- Black Box ming for handling uncertain data is chance constrained programming (CCP). In this presentation, we introduce In many optimization problems, nonsmoothness appears an extension of CCPs termed chance-constrained second- in a structured manner, because the objective function has order cone programs (CCSOCPs) for handling uncertainty some special form. For example, in compressed sensing, in data defining SOCPs. Some applications of this new regularized least square problems have composite objective paradigm of CCP will be described. functions. Likewise, separable functions arise in large-scale stochastic or mixed-integer programming problems solved Baha M. Alzalg,AriAriyawansa by some decomposition technique. The last decade has Washington State University seen the advent of a new generation of bundle methods, [email protected], [email protected] capable of fully exploiting structured objective functions. Such information, transmitted via an oracle or black box, can be handled in a highly versatile manner, depending on CP1 how much data is given by the black box. If certain first- On Semidefinite Programming Relaxations of Max- order information is missing, it is possible to deal with imum K-Section inexactness very efficiently. But if some second-order in- formation is available, it is possible to mimic a Newton We derive a semidefinite programming bound for max k- algorithm and converge rapidly. We outline basic ideas equipartition problem, which is, for k =2,atleastas strong as the well-known bound by Frieze and Jerrum. For and computational questions, highlighting the main fea- ≥ tures and challenges in the area on application examples. k 3 this new bound dominates a bound of Karish and Rendl [S.E. Karish, F. Rendl. Semidefinite Programming and Graph Equipartition. In: Topics in Semidefinite and Claudia Sagastiz´abal Interior-Point Methods, P.M. Pardalos and H. Wolkowicz, CEPEL Eds., 1998.] The new bound coincides with a recent bound [email protected] by De Klerk and Sotirov for the more general quadratic assignment problem. IP7 Cristian Dobre Nonlinear Integer Optimization: Incomputability, TiUniversity of Groningen Computability and Computation Groningen, The Netherlands [email protected] Within the realm of optimization, the Nonlinear Integer Optimization problem is, in some sense, the mother of all Etienne De Klerk deterministic optimization problems.