Lagrangian Approach to Quasiconvex Programing

For a mathematical programming problem, we consider a Lagrangian approach inspired by quasiconvex duality, but as close as possible to the usual convex Lagrangian. We focus our attention on the set of multipliers and we look for their interpretation as generalized derivatives of the performance function associated with a simple perturbation of the given problem. We do not use quasiconvex dualities, but simple direct arguments.     

Sandwich theorem for quasiconvex functions and its applications

In convex programming, sandwich theorem is very important because it is equivalent to Fenchel duality theorem. In this paper, we investigate a sandwich theorem for quasiconvex functions. Also, we consider some applications for quasiconvex programming.     

Duality Theorems for Separable Convex Programming Without Qualifications

In the research of mathematical programming, duality theorems are essential and important elements. Recently, Lagrange duality theorems for separable convex programming have been studied. Tseng proves that there is no duality gap in Lagrange duality for separable convex programming without any qualifications. In other words, although the infimum value of the primal problem equals to the supremum value of the Lagrange dual problem, Lagrange multiplier does not always exist. Jeyakumar and Li prove that Lagrange multiplier always exists without any qualifications for separable sublinear programming. Furthermore, Jeyakumar and Li introduce a necessary and sufficient constraint qualification for Lagrange duality theorem for separable convex programming. However, separable convex constraints do not always satisfy the constraint qualification, that is, Lagrange duality does not always hold for separable convex programming. In this paper, we study duality theorems for separable convex programming without any qualifications. We show that a separable convex inequality system always satisfies the closed cone constraint qualification for quasiconvex programming and investigate a Lagrange-type duality theorem for separable convex programming. In addition, we introduce a duality theorem and a necessary and sufficient optimality condition for a separable convex programming problem, whose constraints do not satisfy the Slater condition.     

非光滑多目标广义本性凸规划的最优性条件与对偶

In this paper, we introduce generalized essentially pseudoconvex function and generalized essentially quasiconvex function, and give sufficient optimality conditions of the nonsmooth generalized convex multi-objective programming and its saddle point theorem about cone efficient solution. We set up Mond-Weir type duality and Craven type duality for nonsmooth multiobjective programming with generalized essentially convex functions, and prove them.     

Duality theorems for a class of nonconvex programming problems

The object of this paper is to prove duality theorems for quasiconvex programming problems. The principal tool used is the transformation introduced by Manas for reducing a nonconvex programming problem to a convex programming problem. Duality in the case of linear, quadratic, and linear-fractional programming is a particular case of this general case.The authors are grateful to the referees for their kind suggestions.     

On sufficient optimality conditions for a quasiconvex programming problem

Some remarks are made on a paper by Bector, Chandra, and Bector (see Ref. 1) concerning the Fritz John and Kuhn-Tucker sufficient optmality conditions as well as duality theorems for a nonlinear programming problem with a quasiconvex objective function.This research was supported by the Italian Ministry of University Scientific and Technological Research.     

Some constraint qualifications for quasiconvex vector-valued systems

In this paper, we consider minimization problems with a quasiconvex vector-valued inequality constraint. We propose two constraint qualifications, the closed cone constraint qualification for vector-valued quasiconvex programming (the VQ-CCCQ) and the basic constraint qualification for vector-valued quasiconvex programming (the VQ-BCQ). Based on previous results by Benoist et al. (Proc Am Math Soc 13:1109–1113, 2002), and Suzuki and Kuroiwa (J Optim Theory Appl 149:554–563, 2011), and (Nonlinear Anal 74:1279–1285, 2011), we show that the VQ-CCCQ (resp. the VQ-BCQ) is the weakest constraint qualification for Lagrangian-type strong (resp. min–max) duality. As consequences of the main results, we study semi-definite quasiconvex programming problems in our scheme, and we observe the weakest constraint qualifications for Lagrangian-type strong and min–max dualities. Finally, we summarize the characterizations of the weakest constraint qualifications for convex and quasiconvex programming.     

Surrogate duality for robust optimization

Robust optimization problems, which have uncertain data, are considered. We prove surrogate duality theorems for robust quasiconvex optimization problems and surrogate min–max duality theorems for robust convex optimization problems. We give necessary and sufficient constraint qualifications for surrogate duality and surrogate min–max duality, and show some examples at which such duality results are used effectively. Moreover, we obtain a surrogate duality theorem and a surrogate min–max duality theorem for semi-definite optimization problems in the face of data uncertainty.     

Higher order duality in vector optimization over cones

In this paper higher order cone convex, pseudo convex, strongly pseudo convex, and quasiconvex functions are introduced. Higher order sufficient optimality conditions are given for a weak minimum, minimum, strong minimum and Benson proper minimum solution of a vector optimization problem. A higher order dual is associated and weak and strong duality results are established under these new generalized convexity assumptions.     

An extension of proximal methods for quasiconvex minimization on the nonnegative orthant

In this paper we propose an extension of proximal methods to solve minimization problems with quasiconvex objective functions on the nonnegative orthant. Assuming that the function is bounded from below and lower semicontinuous and using a general proximal distance, it is proved that the iterations given by our algorithm are well defined and stay in the positive orthant. If the objective function is quasiconvex we obtain the convergence of the iterates to a certain set which contains the set of optimal solutions and convergence to a KKT point if the function is continuously differentiable and the proximal parameters are bounded. Furthermore, we introduce a sufficient condition on the proximal distance such that the sequence converges to an optimal solution of the problem.     

Lagrangian Duality for Minimization of Nonconvex Multifunctions

Two alternative type theorems for nearly convexlike or * quasiconvex multifunctions are presented. They are used to derive Lagrangian conditions and duality results for vector optimization problems when the objectives and the constraints are nearly convexlike or * quasiconvex multifunctions.     

Minimax Fractional Programming for <Emphasis Type="Italic">n</Emphasis>-Set Functions and Mixed-Type Duality under Generalized Invexity

We establish the sufficient optimality conditions for a minimax programming problem involving p fractional n-set functions under generalized invexity. Using incomplete Lagrange duality, we formulate a mixed-type dual problem which unifies the Wolfe type dual and Mond-Weir type dual in fractional n-set functions under generalized invexity. Furthermore, we establish three duality theorems: weak, strong, and strict converse duality theorem, and prove that the optimal values of the primal problem and the mixed-type dual problem have no duality gap under extra assumptions in the framework. This research was partly supported by the National Science Council, NSC 94-2115-M-033-003, Taiwan.     

A minimization algorithm for equilibrium problems with polyhedral constraints

Our aim in this paper is to study an infeasible interior proximal method for solving equilibrium problems with polyhedral constrains in a quasiconvex setting. Under mild assumptions, we show that the sequence generated by our algorithm is well defined and it converges to a solution of the problem when regularization parameters converge to zero.     

On Set Containment Characterization and Constraint Qualification for Quasiconvex Programming

Dual characterizations of the containment of a convex set with quasiconvex inequality constraints are investigated. A new Lagrange-type duality and a new closed cone constraint qualification are described, and it is shown that this constraint qualification is the weakest constraint qualification for the duality.     

Dominators for Multiple-objective Quasiconvex Maximization Problems

In this paper we address the problem of finding a dominator for a multiple-objective maximization problem with quasiconvex functions. The one-dimensional case is discussed in some detail, showing how a Branch-and-Bound procedure leads to a dominator with certain minimality properties. Then, the well-known result stating that the set of vertices of a polytope S contains an optimal solution for single-objective quasiconvex maximization problems is extended to multiple-objective problems, showing that, under upper-semicontinuity assumptions, the set of (k 21)-dimensional faces is a dominator for k-objective problems. In particular, for biobjective quasiconvex problems on a polytope S, the edges of S constitute a dominator, from which a dominator with minimality properties can be extracted by Branch-and Bound methods.     

Duality for a class of quasiconvex programs

This paper derives a duality relation between a pair of explicitly quasiconvex and quasiconcave programs.     

Trust-region problems with linear inequality constraints: exact SDP relaxation,global optimality and robust optimization

The trust-region problem, which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trust-region problem having extra linear constraints. This paper shows that two useful and powerful features of the classical trust-region problem continue to hold for an extended trust-region problem with linear inequality constraints under a new dimension condition. First, we establish that the class of extended trust-region problems has an exact SDP-relaxation, which holds without the Slater constraint qualification. This is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range-convexity whenever the dimension condition is fulfilled. Second, we show that the dimension condition together with the Slater condition ensures that a set of combined first and second-order Lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust-region problem and consequently for strong duality. Through simple examples we also provide an insightful account of our development from SDP-relaxation to strong duality. Finally, we show that the dimension condition is easily satisfied for the extended trust-region model that arises from the reformulation of a robust least squares problem (LSP) as well as a robust second order cone programming model problem (SOCP) as an equivalent semi-definite linear programming problem. This leads us to conclude that, under mild assumptions, solving a robust LSP or SOCP under matrix-norm uncertainty or polyhedral uncertainty is equivalent to solving a semi-definite linear programming problem and so, their solutions can be validated in polynomial time.     

Ein Beitrag zur quasi-konvexen Optimierung in topologischen linearen Räumen

Zusammenfassung In dieser Arbeit wird für das Problem, ein quasikonvexes Funktional unter konkaven Ungleichungs- und affin linearen Gleichungsrestriktionen auf einer konvexen Teilmenge eines linearen Raumes zu minimieren, eine notwendige globale Optimalitätsbedingung in Form einer verallgemeinerten Multiplikatorenregel hergeleitet und einige einfache Folgerungen wie eine Dualitätsaussage und alternative Sätze in endlich dimensionalen Räumen diskutiert.

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Summary In this paper we present a global necessary optimality condition in form of a generalized multiplier rule for the problem of minimizing quasiconvex functionals subject to concave inequality — and affine linear equality — constraints over a convex subset of a linear space and discuss some consequences as a duality theorem and alternative conditions for finite dimensional problems.

     

Complete Duality for Quasiconvex and Convex Set-Valued Functions

This paper provides a unique dual representation of set-valued lower semi-continuous quasiconvex and convex functions. The results are based on a duality result for increasing set-valued functions.     

无限维空间拟凸映射多目标最优化问题解集的连通性

本文在一个无限格中引入了拟凸、强拟凸和严格拟凸映射。并在约束集为紧凸条件下，证明了相应的多目标规划问题之有效解集和弱有效解集连通性结果。