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Optimality conditions for DC programming problems with reverse convex constraints

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2013年第1期80卷 18-27页

作者： Saeki, Yusuke Kuroiwa, Daishi Shimane Univ Interdisciplinary Grad Sch Sci & Engn Matsue Shimane Japan Shimane Univ Interdisciplinary Fac Sci & Engn Matsue Shimane Japan

In this paper, we give necessary and sufficient optimality conditions for DC programming problems with reverse convex constraints defined by a finite family of pseudoconvex functions. As application, we obtain necessary and sufficient optimality conditions for DC programming problems and fractional programming problems. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Optimality conditions DC programming reverse convex constraints convex programming Fractional programming

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Conditions for boundedness in concave programming under reverse convex and convex constraints

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MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2007年第2期65卷 261-279页

作者： Obuchowska, Wieslawa T. E Carolina Univ Dept Math Greenville NC 27858 USA

In this paper, we are concerned with the problem of boundedness in the constrained global maximization of a convex function. In particular, we present necessary and sufficient conditions for boundedness of a feasible region defined by reverse convex constraints and we establish sufficient and necessary conditions for existence of an upper bound for a convex objective function defined over the system of concave inequality constraints. We also address the problem of boundedness in the global maximization problem when a feasible region is convex and unbounded.

关键词： concave minimization reverse convex constraints unboundedness

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convex MINIMIZATION UNDER LIPSCHITZ constraints

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1990年第3期64卷 595-614页

作者： THACH, PT Institute of Mathematics Bo Ho Hanoi Vietnam

We consider the problem of minimizing a convex functionf(x) under Lipschitz constraintsf i (x)≤0,i=1,...,m. By transforming a system of Lipschitz constraintsf i ... 详细信息

We consider the problem of minimizing a convex functionf(x) under Lipschitz constraintsf i (x)≤0,i=1,...,m. By transforming a system of Lipschitz constraintsf i (x)≤0,i=l,...,m, into a single constraints of the formh(x)-∥x∥2≤0, withh(·) being a closed convex function, we convert the problem into a convex program with an additional reverse convex constraint. Under a regularity assumption, we apply Tuy's method for convex programs with an additional reverse convex constraint to solve the converted problem. By this way, we construct an algorithm which reduces the problem to a sequence of subproblems of minimizing a concave, quadratic, separable function over a polytope. Finally, we show how the algorithm can be used for the decomposition of Lipschitz optimization problems involving relatively few nonconvex variables.

关键词： convex minimization Global optimization Lipschitz constraints reverse convex constraints

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Unboundedness in reverse convex and concave integer programming

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MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2010年第2期72卷 187-204页

作者： Obuchowska, Wieslawa T. E Carolina Univ Dept Math Greenville NC 27858 USA

In this paper we are concerned with the problem of unboundedness and existence of an optimal solution in reverse convex and concave integer optimization problems. In particular, we present necessary and sufficient conditions for existence of an upper bound for a convex objective function defined over the feasible region contained in Z(n). The conditions for boundedness are provided in a form of an implementable algorithm, showing that for the considered class of functions, the integer programming problem is unbounded if and only if the associated continuous problem is unbounded. We also address the problem of boundedness in the global optimization problem of maximizing a convex function over a set of integers contained in a convex and unbounded region. It is shown in the paper that in both types of integer programming problems, the objective function is either unbounded from above, or it attains its maximum at a feasible integer point.

关键词： Concave integer minimization reverse convex constraints Unboundedness

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A geometric approach to global optimization

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JOURNAL OF GLOBAL OPTIMIZATION 1999年第2期15卷 109-126页

作者： Nickel, S Schöbel, A Univ Kaiserslautern Fachbereich Math D-67653 Kaiserslautern Germany

In this paper we consider the problem of optimizing a piecewise-linear objective function over a non-convex domain. In particular we do not allow the solution to lie in the interior of a prespecified region R. We discuss the geometrical properties of this problems and present algorithms based on combinatorial arguments. In addition we show how we can construct quite complicated shaped sets R while maintaining the combinatorial properties.

关键词： reverse convex constraints geometric approach discretization piecewise linear programs

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THE DESIGN CENTERING PROBLEM AS A DC PROGRAMMING PROBLEM

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MATHEMATICAL PROGRAMMING 1988年第2期41卷 229-248页

作者： THACH, PT Institute of Mathematics Bo Ho Vietnam

The following problem is studied: Given a compact setS inR n and a Minkowski functionalp(x), find the largest positive numberr for which there existsx ∈ S such that the set of ally ∈ R n satisfyingp(y−x) ≤ r is contained inS. It is shown that whenS is the intersection of a closed convex set and several complementary convex sets (sets whose complements are open convex) this “design centering problem” can be reformulated as the minimization of some d.c. function (difference of two convex functions) overR n . In the case where, moreover,p(x) = (x T Ax)1/2, withA being a symmetric positive definite matrix, a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set.

关键词： complementary convex sets d.c. programming Design centering problem difference of two convex functions global minimization of concave functions outer approximation algorithm reverse convex constraints

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Lagrange duality and partitioning techniques in nonconvex global optimization

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1997年第2期95卷 347-369页

作者： Dur, M Horst, R Department of Mathematics University of Trier Trier Germany

It is shown that, for very general classes of nonconvex global optimization problems, the duality gap obtained by solving a corresponding Lagrangian dual in reduced to zero in the limit when combined with suitably refined partitioning of the feasible set. A similar result holds for partly convex problems where exhaustive partitioning is applied only in the space of nonconvex variables. Applications include branch-and-bound approaches for linearly constrained problems where convex envelopes can be computed, certain generalized bilinear problems, linearly constrained optimization of the sum of ratios of affine functions, and concave minimization under reverse convex constraints.

关键词： global optimization duality gap branch-and-bound techniques partly convex programs bilinear constraints sum of ratios reverse convex constraints

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convex-PROGRAMS WITH AN ADDITIONAL reverse convex CONSTRAINT

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1987年第3期52卷 463-486页

作者： TUY, H Institute of Mathematics Hanoi Vietnam

A method is presented for solving a class of global optimization problems of the form (P): minimizef(x), subject tox∈D,g(x)≥0, whereD is a closed convex subset ofR n andf,g are convex finite functionsR n . Under suitable stability hypotheses, it is shown that a feasible point $$\bar x$$ is optimal if and only if 0=max{g(x):x∈D,f(x)≤f( $$\bar x$$ )}. On the basis of this optimality criterion, the problem is reduced to a sequence of subproblemsQ k ,k=1, 2, ..., each of which consists in maximizing the convex functiong(x) over some polyhedronS k . The method is similar to the outer approximation method for maximizing a convex function over a compact convex set.

关键词： concave minimization convex maximization outer approximation methods reverse convex constraints

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Feasibility in reverse convex mixed-integer programming

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2012年第1期218卷 58-67页

作者： Obuchowska, Wieslawa T. E Carolina Univ Dept Math Greenville NC 27858 USA

In this paper we address the problem of the infeasibility of systems defined by reverse convex inequality constraints, where some or all of the variables are integer. In particular, we provide a polynomial algorithm that identifies a set of all constraints critical to feasibility (CF), that is constraints that may affect a feasibility status of the system after some perturbation of the right-hand sides. Furthermore, we will investigate properties of the irreducible infeasible sets and infeasibility sets, showing in particular that every irreducible infeasible set as well as infeasibility sets in the considered system, are subsets of the set CF of constraints critical to feasibility. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Integer programming Feasibility Concave integer minimization reverse convex constraints Sensitivity analysis Irreducible infeasible sets

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