In this tutorial paper we present some recent developments in parametric semi-infinite programming. We mainly focus our attention to two problems: The determination of one-sided derivatives of the value-function with ...
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We consider generalized semi-infinite programming problems. Second order necessary and sufficient conditions for local optimality are given. The conditions are derived under assumptions such that the feasible set can ...
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This paper is devoted to the study of perturbed semi-infinite optimization problems, i.e., minimization over R(n) with an infinite number of inequality constraints. We obtain the second-order expansion of the optimal ...
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This paper is devoted to the study of perturbed semi-infinite optimization problems, i.e., minimization over R(n) with an infinite number of inequality constraints. We obtain the second-order expansion of the optimal value function and the first-order expansion of approximate optimal solutions in two cases: (i) when the number of binding constraints is finite and (ii) when the inequality constraints are parametrized by a real scalar. These results are partly obtained by specializing the sensitivity theory for perturbed optimization developed in part I (cf. [SIAM J. Control Optim., 34 (1996), pp. 1151-1171]) and deriving specific sharp lower estimates for the optimal value function which take into account the curvature of the positive cone in the space C(Omega) of continuous real-valued functions.
This paper emphasizes the great potential applicability of the so-called Haar's dual problem, in linear semi-infinite programming, and analyzes its properties in order to its reduction to an ordinary linear progra...
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This paper emphasizes the great potential applicability of the so-called Haar's dual problem, in linear semi-infinite programming, and analyzes its properties in order to its reduction to an ordinary linear program, its sequential approximation through finite subprograms, as well as to its numerical solution by feasible directions strategies.
A common strategy for achieving global convergence in the solution of semi-infinite programming (SIP) problems, and in particular continuous minimax problems, is to (approximately) solve a sequence of discretized prob...
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A common strategy for achieving global convergence in the solution of semi-infinite programming (SIP) problems, and in particular continuous minimax problems, is to (approximately) solve a sequence of discretized problems, with progressively finer discretization meshes. Finely discretized minimax and SIP problems, as well as other problems with many more objectives/constraints than variables, call for algorithms in which successive search directions are computed based on a small but significant subset of the objectives/constraints, with ensuing reduced computing cost per iteration and decreased risk of numerical difficulties. In this paper, a sequential quadratic programming (SQP)-type algorithm is proposed that incorporates this idea in the particular case of minimax problems. The general case will be considered in a separate paper. The quadratic programming subproblem that yields the search direction involves only a small subset of the objective functions. This subset is updated at each iteration in such a way that global convergence is ensured. Heuristics are suggested that take advantage of a possible close relationship between ''adjacent'' objective functions. Numerical results demonstrate the efficiency of the proposed algorithm.
In this paper we study a form of convex quadratic semi-infinite programming problems with finitely many variables and infinitely many constraints over a compact metric space. An entropic path-following algorithm is in...
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In this paper, we propose an “inexact solution” approach to dea! with linear semi-infinite programming problems with finitely many variables and infinitely many constraints over a compact metric space. A general con...
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A vector subdifferential is constructed for various classes of Lipschitz vector functions on Banach spaces The existence proved by compactness arguments, not needing explicit construction of sequences. This subdiffent...
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A vector subdifferential is constructed for various classes of Lipschitz vector functions on Banach spaces The existence proved by compactness arguments, not needing explicit construction of sequences. This subdiffential agrees with some definitions of Thibault and Xu, but appears to be more widely applicable. An application is given to semi-infinite programming.
The equivalence of multinomial maximum likelihood and the isotonic projection problem: {Mathematical expression} can be established using Fenchel's Duality Theorem and subgradient and complementary slackness relat...
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The central cutting plane algorithm for linear semi-infinite programming (SIP) is extended to nonlinear convex SIP of the form min {f(x)vertical bar x is an element of H, g(x, t) <= 0 all t is an element of S}. Und...
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The central cutting plane algorithm for linear semi-infinite programming (SIP) is extended to nonlinear convex SIP of the form min {f(x)vertical bar x is an element of H, g(x, t) <= 0 all t is an element of S}. Under differentiability assumptions that are weaker than those employed in superlinearly convergent algorithms, a linear convergence rate is established that has additional important features. These features are the ability to (i) generate a cut from any violated constraint;(ii) invoke efficient constraint-dropping rules for management of linear programming (LP) subproblem size;(iii) provide an efficient grid management scheme to generate cuts and ultimately to test feasibility to a high degree of accuracy, as well as to provide an automatic grid refinement for use in obtaining admissible starting solutions for the nonlinear system of first-order conditions;and, (iv) provide primal and dual (Lagrangian) SIP feasible solutions in a finite number of iterations.
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