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Technical Sessions TECHNICAL SESSIONS Friday, 8:30 - 9:00 Friday, 9:00 - 9:45 FA-01 FB-01 Friday, 8:30 - 9:00 Friday, 9:00 - 9:45 23.1.5 23.1.5 Opening Session Invited Talk 1 Plenary session Plenary session Chair: Marco A. López-Cerdá, Statistics and Operations Research, Alicante University, Ctra. San Vicente de Raspeig s/n, 3071, Alicante, Spain, [email protected] 1 - Theory and Applications of Degeneracy in Cone Optimization Henry Wolkowicz, Faculty of Mathematics, University of Waterloo, N2L3G1, Waterloo, Ontario, Canada, [email protected] The elegant theoretical results for strong duality and strict comple- mentarity for linear programming, LP, lie behind the success of cur- rent algorithms. In addition, preprocessing is an essential step for efficiency in both simplex type and interior-point methods. How- ever, the theory and preprocessing techniques can fail for cone pro- gramming over nonpolyhedral cones. We take a fresh look at known and new results for duality, opti- mality, constraint qualifications, CQ, and strict complementarity, for linear cone optimization problems in finite dimensions. One theme is the notion of minimal representation of the cone and the constraints. This provides a framework for preprocessing cone op- timization problems in order to avoid both the theoretical and nu- merical difficulties that arise due to the (near) loss of the strong CQ, strict feasibility. Surprisingly, many instances of semidefinite programming, SDP, problems that arise from relaxations of hard combinatorial prob- lems are degenerate (CQ fails). Rather than being a disadvantage, we show that this degeneracy can be exploited. In particular, several huge instances of SDP completion problems can be solved quickly and to extremely high accuracy. 19 FC-03 EUROPT 2010 Friday, 10:15 - 12:15 FC-04 Friday, 10:15 - 12:15 FC-03 23.3.5 Friday, 10:15 - 12:15 Generalized differentiation and 23.3.4 applications Optimal Control 1 Invited session Contributed session Chair: Vera Roshchina, CIMA, Universidade de Evora, Colégio Luís Verney, Rua Romão Ramalho, 59, 7000-671, Chair: Peter Roebeling, Department of Environment and Planning, CESAM - University of Aveiro, Campus Évora, Portugal, [email protected] Universitário de Santiago, Universidade de Aveiro, 1 - Minimizing irregular convex functions: Ulam 3810-193, Aveiro, Portugal, [email protected] stability for approximate minima 1 - Necessary optimality conditions for optimal Michel Théra, Maths-Info, XLIM, UMR-CNRS control problems with discontinuous right 6172, 123, Avenue Albert Thomas, 87060, Limoges Cedex, France, [email protected] hand side This presentation summarizes a recent joint work with Emil Ernst. Werner Schmidt, Ernst-Moritz-Arndt-University Our main objective is to characterize the subclass of those convex Greifswald, D-17489, Greifswald, Germany, lower semicontinuous proper functions bounded below for which [email protected], Olga Kostyukova, the set-valued mapping which assigns to a function in this class, the Ekaterina Kostina set of its epsilon-minima is upper semi-continuous. Despite its ab- stract appearance, this type of stability turns out to be essential in We consider an optimal control problem with discontinuous right- numerical optimization, namely in answering the natural question hand side. It is assumed that the system dynamics is switched when of defining the largest class of functionals convex lower semicontin- system crosses a given surface described by a smooth function de- uous proper and bounded below for which minimization algorithms pendent on system states. Attention is paid to the situation when exist. optimal trajectory slides on switching surface during nontrivial in- tervals. Necessary optimality conditions in a form the Maximum 2 - The Directed Subdifferential Principle are proved. The principle includes new essential condi- Elza Farkhi, School of Math. Sciences, Tel-Aviv tions. Comparison of the optimality conditions obtained with some other known results is carried out. Illustrative examples are pre- University, Haim Levanon Str., 69978, Tel Aviv, sented. [email protected], Robert Baier For differences of convex functions the directed subdifferential is 2 - Invexity in mathematical programming and introduced as the difference of two embedded convex subdifferen- control problems tials in the Banach space of directed sets. Basic axioms of subd- ifferentials and nice calculus rules are established for the directed Manuel Arana-Jiménez, Estadistica e Invesitigacion subdifferential. Its visualization, called Rubinov subdifferential, is Operativa, University of Cadiz, C/Chile, 1, 11002, a non-empty, generally non-convex set in Rn. Optimality conditions Jerez de la Frontera, Cadiz, Spain, are formulated, minimizers, maximizers and saddle points are dis- [email protected], Antonio Rufián-Lizana, tinguished, directions of descent and ascent are identified using the directed and Rubinov subdifferential. Gabriel Ruiz-Garzón, Rafaela Osuna-Gómez This communication is focused on the study of optimal solutions for 3 - Calculating Known Subdifferentials from the mathematical programming problems and control problems and the Rubinov Subdifferential properties of the functions, as well as the relationship between these Robert Baier, Department of Mathematics, types of optimization problems, from recent published results. KT- University of Bayreuth, Chair of Applied invexity has been introduced in control problems, and it is a nec- essary and suffcient condition in order for a Kuhn-Tucker critical Mathematics, D-95440, Bayreuth, Germany, point to be an optimal solution. Recenty, a weaker condition, FJ- [email protected], Elza Farkhi, Vera invexity, has been proposed, which is characterized by a Fritz John Roshchina point being an optimal solution for the control problem. The visualization of the directed subdifferential - defined for dif- ferences of convex functions - is the Rubinov subdifferential. This 3 - Efficient inter-industry water pollution abate- set is usually non-convex and splits into three parts. The relation ment in linked terrestrial and marine ecosys- between these parts and the Dini, Michel-Penot and Clarke subdif- ferential are discussed. In 2D the Rubinov subdifferential is closely tems linked to the Mordukhovich one and offers the calculation of var- Peter Roebeling, Department of Environment and ious subdifferentials based on simple differences of sets. Several visualizations in examples indicate the connections to other subdif- Planning, CESAM - University of Aveiro, Campus ferentials and the advantage of nonconvex subdifferentials. Universitário de Santiago, Universidade de Aveiro, 3810-193, Aveiro, Portugal, [email protected], 4 - Subgradient sampling algorithms for nons- Eligius M.T. Hendrix, Arjan Ruijs, Martijn van mooth nonconvex functions Grieken Adil Bagirov, School of Information Technology & Catchment agriculture leads to water pollution, downstream envi- Mathematical Sciences, University of Ballarat, ronmental degradation and marine value depreciation. Sustainable University Drive, Mount Helen, P.O. Box 663, 3353, development entails balancing of marginal costs and benefits from pollution abatement. As abatement costs differ across agricultural Ballarat, Victoria, Australia, industries and abatement benefits are nonlinear, we explore effi- [email protected] cient abatement across industries. Using an optimal control ap- In this talk we will present an algorithm for computation of subgra- proach with application to cane and cattle industries in Tropical dients of nonsmooth nonconvex functions. Then we demonstrate Australia, we show that efficient abatement per industry is depen- how this algorithm can be applied to approximate subdifferentials dent on abatement by all industries when marine abatement benefits and quasidifferential. We also discuss an algorithm for the com- are non-linear. putation of descent directions of such functions. Examples wiil be given to demonstrate the performance of the algorithms. 20 EUROPT 2010 FC-06 The first part of this talk is devoted to relate the (metric) regular- ity modulus of the intersection mapping associated with a given fi- nite family of set-valued mappings to the maximum of moduli of this family. We specifically refer to the so-called linear regularity FC-05 and equirregularity properties. In the second part we determine the Friday, 10:15 - 12:15 Lipschitz modulus of the feasible set mapping associated with a pa- rameterized linear semi-infinite system containing a finite amount 23.3.9 of equations, via the strategy of splitting them and applying results from linear inequality systems. Convex Analysis and Applications 1 Contributed session FC-06 Chair: Edite M.G.P. Fernandes, Production and Systems, Friday, 10:15 - 12:15 University of Minho, School of Engineering, Campus de 23.3.10 Gualtar, 4710-057, Braga, Portugal, [email protected] Conic and Semidefinite Programming 1 - Lipschitz modulus of the feasible set map- 1 ping for linear and convex semi-infinite sys- tems under different perturbation settings Contributed session Juan Parra, Operations Research Center, Miguel Chair: Miguel Anjos, Management Sciences, University of Hernández University, Avda. del Ferrocarril s/n, Waterloo, 200 University Avenue West, N2L 3G1, (Edif. Torretamarit), 03202, Elche, Alicante, Spain, Waterloo, Ontario, Canada, [email protected] [email protected], Maria Josefa Cánovas, Francisco J. Gómez-Senent 1 - An Improved Characterisation of the Interior Our most particular
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