This paper is concerned with the optimality for multi-objective programming problems with nonsmooth and nonconvex (but directionally differentiable) objective and constraint functions. The main results are Kuhn-Tucker...
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This paper is concerned with the optimality for multi-objective programming problems with nonsmooth and nonconvex (but directionally differentiable) objective and constraint functions. The main results are Kuhn-Tucker type necessary conditions for properly efficient solutions and weakly efficient solutions. Our proper efficiency is a natural extension of the Kuhn-Tucker one to the nonsmooth case. Some sufficient conditions for an efficient solution to be proper are also given. As an application, we derive optimality conditions for multi-objective programming problems including extremal-value functions.
In this paper we assume that a deterministic multiobjective programming problem is approximated by surrogate problems based on estimations for the objective functions and the constraints. Making use of a large deviati...
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In this paper we assume that a deterministic multiobjective programming problem is approximated by surrogate problems based on estimations for the objective functions and the constraints. Making use of a large deviations approach, we investigate the behaviour of the constraint sets, the sets of efficient points and the solution sets if the size of the underlying sample tends to infinity. The results are illustrated by applying them to stochastic programming with chance constraints, where (i) the distribution function of the random variable is estimated by the empirical distribution function, (ii) certain parameters have to be estimated.
We establish the following theorems: (i) an existence theorem for weak type generalized saddle points;(ii) an existence theorem for strong type generalized saddle points;(iii). a generalized minimax theorem for a vect...
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We establish the following theorems: (i) an existence theorem for weak type generalized saddle points;(ii) an existence theorem for strong type generalized saddle points;(iii). a generalized minimax theorem for a vector-valued function. These theorems are generalizations and extensions of the author's recent results. For such extensions, we propose new concepts of convexity and continuity of vector-valued functions, which are weaker than ordinary ones. Some of the proofs are bawd on a few key observations and also on the Browder coincidence. theorem or the Tychonoff fixed-point theorem. Also, the minimax theorem follows from the existence theorem for weak type generalized saddle points. The main spaces with mathematical structures considered are real locally convex spaces and real ordered topological vector spaces.
The economic goals and the resulting locational objectives of a franchisor and its individual franchisees are frequently in conflict. For example, one goal of the franchisor is to maximize system-wide market coverage,...
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The economic goals and the resulting locational objectives of a franchisor and its individual franchisees are frequently in conflict. For example, one goal of the franchisor is to maximize system-wide market coverage, while the corresponding goal of the franchisee is to maximize his or her individual market share. Consequently, the optimal facility siting scheme from one perspective may be suboptimal from the other. That is, the facility siting scheme which maximizes system-wide coverage will not necessarily maximize the market shares of the individual franchises which make up the system. In this paper we introduce a multiobjective integer programming model to design franchise networks. The model selects franchise locations and identifies individual franchise market areas. Constraints in the formulation guarantee that all franchise locations are assigned at least a minimal threshold market area with sufficient demand to ensure economic survival. An underlying assumption of the model is that a rationing mechanism exists to assign demand to various franchise locations. Consequently, the model is most appropriate for service delivery franchises in which the franchisor can define and enforce exclusive franchise territories for the various franchise outlets. The purpose of this model is to generate alternative siting configurations which demonstrate the inherent trade-offs between the objectives of the franchisor and the individual franchisees. Given these various location alternatives, it is expected that the franchisor will then evaluate them in terms of other criteria such as the likelihood of the individual franchisee's success, pricing strategies for the various sites, total costs, total profit, and the effects of the response of competitors. Consequently, the proposed model should be viewed as an aid for one aspect of the decision process, i.e. the generation of alternative courses of action.
In the design of a new urban retail network, the size of each office can be determined-once the number and the location of outlets have been fixed-by means of a location-allocation model. In order to carry out this ta...
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In the design of a new urban retail network, the size of each office can be determined-once the number and the location of outlets have been fixed-by means of a location-allocation model. In order to carry out this task, two different solutions have been considered: the best solution in the opinion of the firm's managers and the solution obtained by maximizing the outlets' accessibility, based on a spatial interaction model. Our biobjective program bridges the gap between both solutions by enabling the generation of a finite set of non-inferior points, and constitutes, therefore, a valuable decision-support tool. The paper closes with a case study in the banking sector.
Necessary Kuhn-Tucker conditions up to precision epsilon without constraint qualification for epsilon-Pareto optimality of multiobjective programming are derived. This article suggests the establishment of a Wolfe-typ...
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Necessary Kuhn-Tucker conditions up to precision epsilon without constraint qualification for epsilon-Pareto optimality of multiobjective programming are derived. This article suggests the establishment of a Wolfe-type epsilon-duality theorem for nondifferentiable, nonconvex, multi-objective minimization problems. The epsilon-vector Lagrangian and the generalized epsilon-saddle point for Pareto optimality are studied.
In this paper, Lagrange multiplier theorems are developed for the cases of single-objective and multiobjective programming problems with set functions. Properly efficient solutions are also characterized by subdiffere...
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In this paper, Lagrange multiplier theorems are developed for the cases of single-objective and multiobjective programming problems with set functions. Properly efficient solutions are also characterized by subdifferentials and zero-like functions.
In this paper we study second-order optimality conditions for the multi-objective programming problems with both inequality constraints and equality constraints. Two weak second-order constraint qualifications are int...
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In this paper we study second-order optimality conditions for the multi-objective programming problems with both inequality constraints and equality constraints. Two weak second-order constraint qualifications are introduced, and based on them we derive several second-order necessary conditions for a local weakly efficient solution. Two second-order sufficient conditions are also presented.
The linearization method, for solving the general problem of nonlinear programming and its various modifications, is considered. On the basic ideas of the linearization method, the algorithms for solving the various p...
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The linearization method, for solving the general problem of nonlinear programming and its various modifications, is considered. On the basic ideas of the linearization method, the algorithms for solving the various problems of mathematical programming are constructed for (a) solving systems of equalities and inequalities, (b) multiobjective programming and (c) complementary problem.
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