Continuous Optimization
Minimax programming under (p,r)-invexity

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Abstract

Minimax programming problems involving (p,r)-invex functions with respect to η are considered. Parametric and non-parametric necessary and sufficient optimality conditions and duality results for a class of minimax programming problems are obtained under (p,r)-invexity assumption on objective and constraint functions. Some different types of dual problems are introduced and duality theorems is established for each dual problem.

Introduction

We consider the following programming problem:f(x):=minx∈X0max1⩽i⩽kfi(x)subjecttogj(x)⩽0,j=1,…,m,(P)where fi:X0R, i=1,…,k, gj:X0R, j=1,…,m, are differentiable functions, and X0 is a non-empty open subset of Rn. Let D:={x∈X0:gi(x)⩽0,i=1,…,m} be the set of all feasible solutions of (P).

Problems of this type are known in the area of the mathematical programming as general minimax programming problems, and have been the subject of immense interest in the past few years. The importance of minimax models and methods is well-known in a great variety of optimal decision making situations. The minimax problems was earlier considered by Danskin [8] for which he obtained necessary conditions as a Lagrange multipler rule. Later this problem was studied by Chew [5] with pseudolinear functions. Tanimoto [14] derived duality theorems for the some minimax type problems involving convex functions. Mond and Weir [12], and also Bector and Bhatia [4] proved sufficient optimality conditions and duality results for (P) with the assumption of pseudo-convexity on the functions involved. Crouzeix et al. in [7] have shown that the minimax fractional programming problem can be solved by solving a minimax non-linear parametric program.

Hanson [9] extended the concept of convex functions and applied them to prove first-order sufficient optimality conditions and duality theorems in a more general setting. Craven [6] later on named these functions, introduced by Hanson [9], as invex. In [2], it was introduced to the optimization theory a new class of functions, named (p,r)-invex, which is an extension of invex functions.

The purpose of the paper is to discuss the application of (p,r)-invexity for a class of minimax programming problems. We shall establish necessary and sufficient optimality conditions and construct several parametric and parametric-free duality models for the considered general minimax programming problem in which occurring functions are (p,r)-invex.

Section snippets

Definition and classes of (p,r)-invex vector functions

In this section we provide some definitions and results that we shall use in the sequel. The following definition is generalizing of the definition of a class of (p,r)-invex functions [2] to the case of a class of (p,r)-invex vector functions.

Definition 1

Let a function f:X0Rk be a differentiable function on a non-empty set X0Rn. If there exists a function η:X0×X0Rn such that for each xX0 and for i=1,…,k, one of the relations1rerfi(x)1rerfi(u)1+rpfi(u)(epη(x,u)1)(>)(x≠u)forp≠0,r≠0,1rerfi(x)1rerfi(u)

Connection with non-linear programming

It is well-known (see, for example, Tanimoto [14], Schmitendorf [13], and others) that the minimax problem is often associated with necessary conditions of Kuhn–Tucker type [10]. It will be shown that this holds under fairly general conditions, and that such necessary conditions become also sufficient for a minimax, under a suitable (p,r)-invexity assumption.

Definition 3

A programming problem (P) will be called a (p,r)-invex minimax programming problem (with respect to η) if each of the function f1,…,fk and

Duality results

In this section, with the help of problem (EP) we shall introduce parametric dual problems for (P) and prove appropriate duality theorems. Moreover, we shall demonstrate non-parametric duals type for (P) in direct way, that is, without help of problem (EP).

Making use of the first-order necessary conditions of Theorem 6, Theorem 12, Remark 11, in this section we shall present a number of dual problems and establish appropriate duality relations under (p,r)-invexity assumptions. These duality

Applications to OR problems

In this section we discuss the potential applications of the defined (p,r)-invexity notion to certain OR problems. Minimax problems arise in various OR applications. The considered classes of (p,r)-invex functions are useful to prove optimality and duality results in different OR minimax problems, above all with logarithmic and exponential functions (of course, not only). It is well-known that logarithmic functions and exponents are used in many different ways, for example, in economics, game

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