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A new approach to multiobjective programming with a modified objective function

JOURNAL OF GLOBAL OPTIMIZATION 2003年第4期27卷 485-495页

作者： Antczak, T Univ Lodz Fac Math PL-90238 Lodz Poland

In this paper, optimality for multiobjective programming problems having invex objective and constraint functions (with respect to the same function eta) is considered. An equivalent vector programming problem is constructed by a modification of the objective function. Furthermore, an eta-Lagrange function is introduced for a constructed multiobjective problem and modified saddle point results are presented.

关键词： efficient point invex function with respect to eta modified vector valued saddle point multiobjective programming vector valued eta-Lagrange function

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Exact penalty functions method for mathematical programming problems involving invex functions

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2009年第1期198卷 29-36页

作者： Antczak, Tadeusz Univ Lodz Fac Math & Comp Sci PL-90238 Lodz Poland

In this paper, some new results on the exact penalty function method are presented. Simple optimality characterizations are given for the differentiable nonconvex optimization problems with both inequality and equality constraints via exact penalty function method. The equivalence between sets of optimal solutions in the original mathematical programming problem and its associated exact penalized optimization problem is established under suitable invexity assumption. Furthermore, the equivalence between a saddle point in the invex mathematical programming problem and an optimal point in its exact penalized optimization problem is also proved. (C) 2008 Elsevier B.V. All rights reserved.

关键词： Exact penalty function method Absolute value penalty function invex function with respect to eta Karush-Kuhn-Tucker optimality conditions Penalized optimization problem

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An η-approximation method in nonlinear vector optimization

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2005年第2期63卷 225-236页

作者： Antczak, T Univ Lodz Fac Math PL-90238 Lodz Poland

In this paper, a new approach for a solution of a nonlinear multiobjective programming problem is introduced. An equivalent eta-approximated vector optimization problem is constructed by a modification of the objective and the constraint functions in the original multiobjective programming problem. The connection between (weak) efficient points in the original multiobjective programming problem and its equivalent eta-approximated vector optimization problem is proved. In this way, optimality conditions for nonlinear constrained multiobjective programming problems having invex and/or generalized invex objective and constraint functions (with respect to the same functions eta) are obtained. (c) 2005 Elsevier Ltd. All rights reserved.

关键词： multiobjective programming invex function with respect to eta eta-approximated vector optimization problem (weak) Pareto optimal point

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An η-approximation approach for nonlinear mathematical programming problems involving invex functions

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NUMERICAL functionAL ANALYSIS AND OPTIMIZATION 2004年第5-6期25卷 423-438页

作者： Antczak, T Univ Lodz Fac Math PL-90238 Lodz Poland

A new approach to a solution of a nonlinear constrained mathematical programming problem and its Mond-Weir duals is introduced. An eta-approximated problem associated with a primal nonlinear programming problem is presented that involves eta-approximated functions constituting the primal problem. The equivalence between the original mathematical programming problem and its associated eta-approximated optimization problem is established under invexity assumption. Furthermore, eta-approximated dual problems in the sense of Mond-Weir are introduced for the obtained eta-approximated optimization problem in this method. By the help of eta-approximated dual problems some duality results are established for the original mathematical programming problem and its original duals.

关键词： eta-approximated optimization problem invex function with respect to eta optimality duality

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Penalty function methods and a duality gap for invex optimization problems

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2009年第7-8期71卷 3322-3332页

作者： Antczak, Tadeusz Univ Lodz Fac Math PL-90238 Lodz Poland

In this paper, the penalty function method is used to study duality in nonconvex mathematical programming problems. in particular, we prove the zero duality gap between optimization problems involving invex functions with respect to the same function eta and their Lagrangian dual problems. The results proved in the paper are illustrated by suitable examples of nonconvex optimization problems. (C) 2009 Elsevier Ltd. All rights reserved.

关键词： Penalty function method Quadratic penalty function invex function with respect to eta Duality gap Penalized optimization problem

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Saddle-point criteria in an η-approximation method for nonlinear mathematical programming problems involving invex functions

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2007年第1期132卷 71-87页

作者： Antczak, T. Univ Lodz Fac Math PL-90131 Lodz Poland

In this paper, the eta-approximation method introduced by Antczak (Ref. 1) for solving a nonlinear constrained mathematical programming problem involving invex functions with respect to the same function eta is extended. In this method, a so-called eta-approximated optimization problem associated with the original mathematical programming problems is constructed;moreover, an eta-saddle point and an eta-Lagrange function are defined. By the help of the constructed eta-approximated optimization problem, saddle-point criteria are obtained for the original mathematical programming problem. The equivalence between an eta-saddle point of the eta-Lagrangian of the associated eta-approximated optimization problem and an optimal solution in the original mathematical programming problem is established.

关键词： eta-approximated optimization problem eta-saddle point eta-Lagrange function invex function with respect to eta optimality

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Mean value in invexity analysis

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2005年第8期60卷 1473-1484页

作者： Antczak, T Univ Lodz Fac Math PL-90238 Lodz Poland

In this paper, a generalization of the mean value theorem is considered in the case of functions defined on an invex set with respect to n (which is not necessarily connected). (C) 2004 Elsevier Ltd. All rights reserved.

关键词： invex set with respect to eta invex function with respect to eta eta-path mean-value inequality mean-value theorem

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