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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 Stream: EUROPT 2010 Stream: EUROPT 2010 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. 1 FC-03 EUROPT 2010 Catchment agriculture leads to water pollution, downstream envi- Friday, 10:15 - 12:15 ronmental degradation and marine value depreciation. Sustainable development entails balancing of marginal costs and benefits from pollution abatement. As abatement costs differ across agricultural FC-03 industries and abatement benefits are nonlinear, we explore effi- Friday, 10:15 - 12:15 cient abatement across industries. Using an optimal control ap- proach with application to cane and cattle industries in Tropical 23.3.4 Australia, we show that efficient abatement per industry is depen- dent on abatement by all industries when marine abatement benefits Optimal Control 1 are non-linear. Stream: EUROPT 2010 Invited session FC-04 Chair: Peter Roebeling, Department of Environment and Friday, 10:15 - 12:15 Planning, CESAM - University of Aveiro, Campus 23.3.5 Universitário de Santiago, Universidade de Aveiro, 3810-193, Aveiro, Portugal, [email protected] Generalized differentiation and applications 1 - Necessary optimality conditions for optimal control problems with discontinuous right Stream: EUROPT 2010 hand side Invited session Chair: Vera Roshchina, CIMA, Universidade de Evora, Werner Schmidt, Ernst-Moritz-Arndt-University Colégio Luís Verney, Rua Romão Ramalho, 59, 7000-671, Greifswald, D-17489, Greifswald, Germany, Évora, Portugal, [email protected] [email protected], Olga Kostyukova, Ekaterina Kostina 1 - Minimizing irregular convex functions: Ulam stability for approximate minima We consider an optimal control problem with discontinuous right- Michel Théra, Maths-Info, XLIM, UMR-CNRS 6172, hand side. It is assumed that the system dynamics is switched when system crosses a given surface described by a smooth function de- 123, Avenue Albert Thomas, 87060, Limoges Cedex, pendent on system states. Attention is paid to the situation when France, [email protected] optimal trajectory slides on switching surface during nontrivial in- This presentation summarizes a recent joint work with Emil Ernst. tervals. Necessary optimality conditions in a form the Maximum Our main objective is to characterize the subclass of those convex Principle are proved. The principle includes new essential condi- lower semicontinuous proper functions bounded below for which tions. Comparison of the optimality conditions obtained with some the set-valued mapping which assigns to a function in this class, the other known results is carried out. Illustrative examples are pre- set of its epsilon-minima is upper semi-continuous. Despite its ab- sented. stract appearance, this type of stability turns out to be essential in numerical optimization, namely in answering the natural question of defining the largest class of functionals convex lower semicontin- 2 - Invexity in mathematical programming and uous proper and bounded below for which minimization algorithms exist. control problems 2 - The Directed Subdifferential Manuel Arana-Jiménez, Estadistica e Invesitigacion Elza Farkhi, School of Math. Sciences, Tel-Aviv Operativa, University of Cadiz, C/Chile, 1, 11002, University, Haim Levanon Str., 69978, Tel Aviv, Jerez de la Frontera, Cadiz, Spain, [email protected], Robert Baier [email protected], Antonio Rufián-Lizana, Gabriel For differences of convex functions the directed subdifferential is Ruiz-Garzón, Rafaela Osuna-Gómez introduced as the difference of two embedded convex subdifferen- tials in the Banach space of directed sets. Basic axioms of subd- This communication is focused on the study of optimal solutions for ifferentials and nice calculus rules are established for the directed mathematical programming problems and control problems and the subdifferential. Its visualization, called Rubinov subdifferential, is properties of the functions, as well as the relationship between these a non-empty, generally non-convex set in Rn. Optimality conditions types of optimization problems, from recent published results. KT- are formulated, minimizers, maximizers and saddle points are dis- invexity has been introduced in control problems, and it is a nec- tinguished, directions of descent and ascent are identified using the essary and suffcient condition in order for a Kuhn-Tucker critical directed and Rubinov subdifferential. point to be an optimal solution. Recenty, a weaker condition, FJ- invexity, has been proposed, which is characterized by a Fritz John 3 - Calculating Known Subdifferentials from the point being an optimal solution for the control problem. Rubinov Subdifferential Robert Baier, Department of Mathematics, University of Bayreuth, Chair of Applied Mathematics, D-95440, 3 - Efficient inter-industry water pollution abate- Bayreuth, Germany, [email protected], ment in linked terrestrial and marine ecosys- Elza Farkhi, Vera Roshchina tems The visualization of the directed subdifferential - defined for dif- ferences of convex functions - is the Rubinov subdifferential. This Peter Roebeling, Department of Environment and set is usually non-convex and splits into three parts. The relation Planning, CESAM - University of Aveiro, Campus between these parts and the Dini, Michel-Penot and Clarke subdif- Universitário de Santiago, Universidade de Aveiro, ferential are discussed. In 2D the Rubinov subdifferential is closely 3810-193, Aveiro, Portugal, [email protected], linked to the Mordukhovich one and offers the calculation of var- Eligius M.T. Hendrix, Arjan Ruijs, Martijn van ious subdifferentials based on simple differences of sets. Several visualizations in examples indicate the connections to other subdif- Grieken ferentials and the advantage of nonconvex subdifferentials. 2 EUROPT 2010 FC-06 Semi-infinite programming problems can be efficiently solved by 4 - Subgradient sampling algorithms for nons- reduction type methods. Here, we present a new global reduction mooth nonconvex functions method for Semi-infinite programming, where the multi-local opti- mization is carried out with a stretched simulated annealing algo- Adil Bagirov, School of Information Technology & rithm, the reduced problem is approximately solved by a primal- Mathematical Sciences, University of Ballarat, dual interior point method combined with a three-dimensional filter University Drive, Mount Helen, P.O. Box 663, 3353, line search strategy, and the global convergence is promoted through Ballarat, Victoria, Australia, [email protected] a two-dimensional filter line search. Numerical experiments with a set of well-known problems are shown. In this talk we will present an algorithm for computation of subgra- dients of nonsmooth nonconvex functions. Then we demonstrate how this algorithm can be applied to approximate subdifferentials 4 - Regularity modulus of intersection mappings: and quasidifferential. We also discuss an algorithm for the com- Application to linear semi-infinite systems of putation of descent directions of such functions. Examples wiil be equations and inequalities given to demonstrate the performance of the algorithms. Francisco J. Gómez-Senent, Operations Research Center, Miguel Hernández University, Orihuela, Alicante, Spain, [email protected], Maria Josefa FC-05 Cánovas, Juan Parra Friday, 10:15 - 12:15 The first part of this talk is devoted to relate the (metric) regular- 23.3.9 ity modulus of the intersection mapping associated with a given fi- nite family of set-valued mappings to the maximum of moduli of Convex Analysis and Applications 1 this family. We specifically refer to the so-called linear regularity and equirregularity properties. In the second part we determine the Stream: EUROPT 2010 Lipschitz modulus of the feasible set mapping associated with a pa- rameterized linear semi-infinite system containing a finite amount Invited session
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