The Remez penalty and smoothing algorithm (RPSALG) is a unified framework for penalty and smoothing methods for solving min-max convexsemi-infinite programing problems, whose convergence was analyzed in a previous pa...
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The Remez penalty and smoothing algorithm (RPSALG) is a unified framework for penalty and smoothing methods for solving min-max convexsemi-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.
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 th...
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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.
We give new characterizations for the subdifferential of the supremum of an arbitrary family of convex functions, dropping out the standard assumptions of compactness of the index set and upper semi-continuity of the ...
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We give new characterizations for the subdifferential of the supremum of an arbitrary family of convex functions, dropping out the standard assumptions of compactness of the index set and upper semi-continuity of the functions with respect to the index (J. convex Anal. 26, 299-324, 2019). We develop an approach based on the compactification of the index set, giving rise to an appropriate enlargement of the original family. Moreover, in contrast to the previous results in the literature, our characterizations are formulated exclusively in terms of exact subdifferentials at the nominal point. Fritz-John and KKT conditions are derived for convex semi-infinite programming.
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 ...
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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.
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 constrain...
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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.
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 met...
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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.
We extend some recent developments on metric regularity in convexsemi-infinite optimization from the case of a compact metric index set (for the constraint system) to the case of a compact Hausdorff one. The latter i...
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We extend some recent developments on metric regularity in convexsemi-infinite optimization from the case of a compact metric index set (for the constraint system) to the case of a compact Hausdorff one. The latter is the continuous setting where different contributions on stability in semi-infiniteprogramming were developed since the 1980s (see for instance 1, 7 and 8). In contrast, 3 and 4 (see also 2 for the linear case) deal with a compact metric index set when analysing the stable behaviour of KKT conditions. The fact of removing the 'metric' assumption yields some technical difficulties which are tackled in the present article via basic topological tools.
We are concerned with the Lipschitz modulus of the optimal set mapping associated with canonically perturbed convexsemi-infinite optimization problems. Specifically, the paper provides a lower and an upper bound for ...
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We are concerned with the Lipschitz modulus of the optimal set mapping associated with canonically perturbed convexsemi-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.
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 trans...
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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 convexsemi-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.
We consider convex semi-infinite programming (SIP) problems with a continuum of constraints. For these problems we introduce new concepts of immobility orders and immobile indices. These concepts are objective and imp...
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We consider convex semi-infinite programming (SIP) problems with a continuum of constraints. For these problems we introduce new concepts of immobility orders and immobile indices. These concepts are objective and important characteristics of the feasible sets of the convex SIP problems since they make it possible to formulate optimality conditions for these problems in terms of optimality conditions for some NLP problems (with a finite number of constraints). In the paper we describe a finite algorithm (DIO algorithm) of determination of immobile indices together with their immobility orders, study some important properties of this algorithm, and formulate the Implicit Optimality Criterion for convex SIP without any constraint qualification conditions (CQC). An example illustrating the application of the DIO algorithm is provided.
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