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convex semi-infinite programming: Implicit Optimality Criterion Based on the Concept of Immobile Indices

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2010年第2期145卷 325-342页

作者： Kostyukova, O. I. Tchemisova, T. V. Yermalinskaya, S. A. Natl Acad Sci Belarus Inst Math Minsk 220072 BELARUS Univ Aveiro Dept Math P-3810193 Aveiro Portugal Belorussian State Univ Informat & Radioelect Dept Informat Minsk 220013 BELARUS

We state a new implicit optimality criterion for convex semi-infinite programming (SIP) problems. This criterion does not require any constraint qualification and is based on concepts of immobile index and immobility order. Given a convex SIP problem with a continuum of constraints, we use an information about its immobile indices to construct a nonlinear programming (NLP) problem of a special form. We prove that a feasible point of the original infinite SIP problem is optimal if and only if it is optimal in the corresponding finite NLP problem. This fact allows us to obtain new efficient optimality conditions for convex SIP problems using known results of the optimality theory of NLP. To construct the NLP problem, we use the DIO algorithm. A comparison of the optimality conditions obtained in the paper with known results is provided.

关键词： convex semi-infinite programming Nonlinear programming Optimality criteria Constraint qualifications

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Locally Farkas-Minkowski systems in convex semi-infinite programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1999年第2期103卷 313-335页

作者： Fajardo, MD López, MA Univ Alicante Fac Sci Dept Stat & Operat Res E-03080 Alicante Spain

A pair of constraint qualifications in convex semi-infinite programming, namely the locally Farkas-Minkowski constraint qualification and generalized Slater constraint qualification, are studied in the paper. We analyze the relationship between them, as well as the behavior of the so-called active and sup-active mappings, accounting for the tightness of the constraint system at each point of the variables space. The generalized Slater constraint qualification guarantees a regular behavior of the supremum function (defined as supremum of the infinitely many functions involved in the constraint system), giving rise to the well-known Valadier formula.

关键词： convex semi-infinite programming constraint qualifications subdifferential mappings Valadier formula monotone operators locally Farkas-Minkowski systems

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Comparative study of RPSALG algorithm for convex semi-infinite programming

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2015年第1期60卷 59-87页

作者： Auslender, A. Ferrer, A. Goberna, M. A. Lopez, M. A. Univ Lyon CNRS Inst Camille Jordan Lyon France Ecole Polytech Dept Econ Palaiseau France Univ Politecn Cataluna Dept Matemat Aplicada 1 Catalonia Spain Univ Alicante Dept Stat & Operat Res E-03080 Alicante Spain Federat Univ Sch Sci Informat Technol & Engn Ballarat Vic Australia

The Remez penalty and smoothing algorithm (RPSALG) is a unified framework for penalty and smoothing methods for solving min-max convex semi-infinite programing problems, whose convergence was analyzed in a previous paper of three of the authors. In this paper we consider a partial implementation of RPSALG for solving ordinary convex semi-infinite programming problems. Each iteration of RPSALG involves two types of auxiliary optimization problems: the first one consists of obtaining an approximate solution of some discretized convex problem, while the second one requires to solve a non-convex optimization problem involving the parametric constraints as objective function with the parameter as variable. In this paper we tackle the latter problem with a variant of the cutting angle method called ECAM, a global optimization procedure for solving Lipschitz programming problems. We implement different variants of RPSALG which are compared with the unique publicly available SIP solver, NSIPS, on a battery of test problems.

关键词： convex semi-infinite programming Remez-type methods Penalty methods Smoothing methods Cutting angle method

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A NEW EXCHANGE METHOD FOR convex semi-infinite programming

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SIAM JOURNAL ON OPTIMIZATION 2010年第6期20卷 2959-2977页

作者： Zhang, Liping Wu, Soon-Yi Lopez, Marco A. Natl Cheng Kung Univ Dept Math Tainan 70101 Taiwan Tsinghua Univ Dept Math Sci Beijing 100084 Peoples R China Univ Alicante Dept Stat & Operat Res E-03080 Alicante Spain

In this paper we propose a new exchange method for solving convex semi-infinite programming (CSIP) problems. We introduce a new dropping-rule in the proposed exchange algorithm, which only keeps those active constraints with positive Lagrange multipliers. Moreover, we exploit the idea of looking for eta-infeasible indices of the lower level problem as the adding-rule in our algorithm. Hence the algorithm does not require to solve a maximization problem over the index set at each iteration;it only needs to find some points such that a certain computationally-easy criterion is satisfied. Under some reasonable conditions, the new adding-dropping rule guarantees that our algorithm provides an approximate optimal solution for the CSIP problem in a finite number of iterations. In the numerical experiments, we apply the proposed algorithm to solve some test problems from the literature, including some medium-sized problems from complex approximation theory and FIR filter design. We compare our algorithm with an existing central cutting plane algorithm and with the semi-infinite solver fseminf in MATLAB toolbox, and we find that our algorithm solves the CSIP problem much faster. For the FIR filter design problem, we show that our algorithm solves the problem better than some algorithms that were technically established for the problem.

关键词： convex semi-infinite programming exchange methods adding-dropping rules complex approximation theory filter design

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Penalty and Smoothing Methods for convex semi-infinite programming

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MATHEMATICS OF OPERATIONS RESEARCH 2009年第2期34卷 303-319页

作者： Auslender, Alfred Goberna, Miguel A. Lopez, Marco A. Univ Lyon Inst Camille Jordan CNRS UMR 5208 F-69622 Villeurbanne France Ecole Polytech Dept Econ F-91128 Palaiseau France Univ Alicante Dept Stat & Operat Res E-03080 Alicante Spain

In this paper we consider min-max convex semi-infinite programming. To solve these problems we introduce a unified framework concerning Remez-type algorithms and integral methods coupled with penalty and smoothing methods. This framework subsumes well-known classical algorithms, but also provides some new methods with interesting properties. Convergence of the primal and dual sequences are proved under minimal assumptions.

关键词： convex semi-infinite programming asymptotic functions penalty methods smoothing methods duality

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Proximal interior point method for convex semi-infinite programming

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OPTIMIZATION METHODS & SOFTWARE 2001年第2期15卷 87-119页

作者： Kaplan, A Tichatschke, R Univ Trier FB Math 4 D-54286 Trier Germany

A regularized logarithmic barrier method for solving (ill-posed) convex semi-infinite programming problems is considered. In this method a multi-step proximal regularization is coupled with an adaptive discretization strategy in the framework of an interior point approach. Termination of the proximal iterations at each discretization level is controlled by means of estimates, characterizing the efficiency of these iterations. A special deleting rule permits to use only a part of the constraints of the discretized problems. Convergence of the method and its stability with respect to data perturbations in the cone of convex C-1-functions are studied as well as some numerical experiments are presented.

关键词： interior point method convex semi-infinite programming proximal point methods ill-posed problems

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Lipschitz behavior of convex semi-infinite optimization problems:: a variational approach

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JOURNAL OF GLOBAL OPTIMIZATION 2008年第1期41卷 1-13页

作者： Canovas, Maria J. Hantoute, Abderrahim Lopez, Marco A. Parra, Juan Miguel Hernandez Univ Elche Ctr Operat Res Alicante 03202 Spain Univ Alicante Dept Stat & Operat Res Alicante 03071 Spain

In this paper we make use of subdifferential calculus and other variational techniques, traced out from [Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55, 3(333), 103-162;Engligh translation Math. Surveys 55, 501-558 (2000);Ioffe, A.D.: On rubustness of the regularity property of maps. Control cybernet 32, 543-554 (2003)], to derive different expressions for the Lipschitz modulus of the optimal set mapping of canonically perturbed convex semi-infinite optimization problems. In order to apply this background for obtaining the modulus of metric regularity of the associated inverse multifunction, we have to analyze the stable behavior of this inverse mapping. In our semi-infinite framework this analysis entails some specific technical difficulties. We also provide a new expression of a global variational nature for the referred regularity modulus.

关键词： convex semi-infinite programming metric regularity optimal set Lipschitz modulus

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Solving convex programs with infinitely many linear constraints by a relaxed cutting plane method

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 1999年第3-4期38卷 23-33页

作者： Wu, SY Fang, SC Natl Cheng Kung Univ Dept Math Tainan Taiwan N Carolina State Univ Raleigh NC 27695 USA

One of the major computational bottlenecks of using the conventional cutting plane approach to solve convex programming problems with infinitely many linear constraints lies in finding a global optimizer of a nonlinear and nonconvex program. This paper presents a relaxed scheme to generate a new cut. In each iteration, the proposed scheme chooses a point at which the constraints are violated to a degree rather than at which the violation is maximized. A convergence proof is provided. The proposed scheme also exhibits the capability of generating an approximate solution to any level of accuracy in a finite number of iterations. (C) 1999 Elsevier Science Ltd. All rights reserved.

关键词： convex semi-infinite programming cutting plane method duality theory

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LIPSCHITZ MODULUS IN convex semi-infinite OPTIMIZATION VIA D. C. FUNCTIONS

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2009年第4期15卷 763-781页

作者： Canovas, Maria J. Hantoute, Abderrahim Lopez, Marco A. Parra, Juan Miguel Hernandez Univ Elche Ctr Operat Res Alicante 03202 Spain Univ Alicante Dept Stat & Operat Res Alicante 03071 Spain

We are concerned with the Lipschitz modulus of the optimal set mapping associated with canonically perturbed convex semi-infinite optimization problems. Specifically, the paper provides a lower and an upper bound for this modulus, both of them given exclusively in terms of the problem's data. Moreover, the upper bound is shown to be the exact modulus when the number of constraints is infinite. In the particular case of linear problems the upper bound (or exact modulus) adopts a notably simplified expression. Our approach is based on variational techniques applied to certain difference of convex functions related to the model. Some results of [M.J . Canovas et al., J. Optim. Theory Appl. (2008) Online First] (which go back to [M. J. Canovas, J. Global Optim. 41 (2008) 1-13] and [Ioffe, Math. Surveys 55 (2000) 501-558;Control Cybern. 32 (2003) 543-554]) constitute the starting point of the present work.

关键词： convex semi-infinite programming modulus of metric regularity d.c. functions

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Subdifferential of the supremum function: moving back and forth between continuous and non-continuous settings

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MATHEMATICAL programming 2021年第1-2期189卷 217-247页

作者： Correa, R. Hantoute, A. Lopez, M. A. Univ OHiggins Rancagua Chile Univ Chile DIM CMM Santiago Chile Univ Chile Ctr Math Modeling CMM Santiago Chile Univ Alicante Alicante Spain Federat Univ CIAO Ballarat Vic Australia

In this paper we establish general formulas for the subdifferential of the pointwise supremum of convex functions, which cover and unify both the compact continuous and the non-compact non-continuous settings. From the non-continuous to the continuous setting, we proceed by a compactification-based approach which leads us to problems having compact index sets and upper semi-continuously indexed mappings, giving rise to new characterizations of the subdifferential of the supremum by means of upper semicontinuous regularized functions and an enlarged compact index set. In the opposite sense, we rewrite the subdifferential of these new regularized functions by using the original data, also leading us to new results on the subdifferential of the supremum. We give two applications in the last section, the first one concerning the nonconvex Fenchel duality, and the second one establishing Fritz-John and KKT conditions in convex semi-infinite programming.

关键词： Supremum of convex functions Subdifferentials Stone– Č ech compactification convex semi-infinite programming Fritz-John and KKT optimality conditions

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