Maximizing a concave function over the efficient or weakly-efficient set

An important approach in multiple criteria linear programming is the optimization of some function over the efficient or weakly-efficient set. This is a very difficult nonconvex optimization problem, even for the case that the function to be optimized is linear. In this article we consider the problem of maximizing a concave function over the efficient or weakly-efficient set. We show that this problem can essentially be formulated as a special global optimization problem in the space of the extreme criteria of the underlying multiple criteria linear program. An algorithm of branch and bound type is proposed for solving the resulting problem.     

A Bilinear Algorithm for Optimizing a Linear Function over the Efficient Set of a Multiple Objective Linear Programming Problem

The problem Q of optimizing a linear function over the efficient set of a multiple objective linear program serves several useful purposes in multiple criteria decision making. However, Q is in itself a difficult global optimization problem, whose local optima, frequently large in number, need not be globally optimal. Indeed, this is due to the fact that the feasible region of Q is, in general, a nonconvex set. In this paper we present a monotonically increasing algorithm that finds an exact, globally-optimal solution for Q. Our approach does not require any hypothesis on the boundedness of neither the efficient set EP nor the optimal objective value. The proposed algorithm relies on a simplified disjoint bilinear program that can be solved through the use of well-known specifically designed methods within nonconvex optimization. The algorithm has been implemented in C and preliminary numerical results are reported.     

Optimization over the efficient set using an active constraint approach

In this work the problem of maximizing a nonlinear objective over the set of efficient solutions of a multicriteria linear program is considered. This is a nonlinear program with nonconvex constraints. The approach is to develop an active constraint algorithm which utilizes the fact that the efficient structure in decision space can be associated in a natural way with hyperplanes in the space of objective values. Examples and numerical experience are included.     

On solving simple bilevel programs with a nonconvex lower level program

In this paper, we consider a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint and the upper level program has a convex set constraint. By using the value function of the lower level program, we reformulate the bilevel program as a single level optimization problem with a nonsmooth inequality constraint and a convex set constraint. To deal with such a nonsmooth and nonconvex optimization problem, we design a smoothing projected gradient algorithm for a general optimization problem with a nonsmooth inequality constraint and a convex set constraint. We show that, if the sequence of penalty parameters is bounded then any accumulation point is a stationary point of the nonsmooth optimization problem and, if the generated sequence is convergent and the extended Mangasarian-Fromovitz constraint qualification holds at the limit then the limit point is a stationary point of the nonsmooth optimization problem. We apply the smoothing projected gradient algorithm to the bilevel program if a calmness condition holds and to an approximate bilevel program otherwise. Preliminary numerical experiments show that the algorithm is efficient for solving the simple bilevel program.     

求解Nash均衡子问题的一个算法

n人有限博弈的混合策略组合（p1^*,…,pn^*)为Nash均衡,如果其中每一策略pi^*都是参与人i(i=1,2,…,n),对其它n-1个参与人策略组合（p1^*,…,pi 1^*,pi-1^*,…,pn^*)的最优反应,即存在n个概率向量p1^*,…,pn^*使得对i=1,2,…,n及任意k1维概率向量pi恒有vi(p1^*,…,pn^*…)小于vi(pi^*,…,pi-1^*,pi 1^*,…pn^*),其中vi为参与人i的支付函数,pi=(pil,…,piki)）为ki维概率向量,即满足条件,pij大于等于0,∑kij=1pij=1,ki是参与人i的策略空间中策略个数,i=1,2,…,n,由此,Nash均衡的求解可化为下列优化问题：求n个概率向量pi^*,…,pn^8,使得对i=1,2,…,n及任意ki维的概率向量pi满足maxxvi(P1^*,…,pi-1^*,pi,Pi 1^*,…,pn^*)=vi(P1^*,,…,Pn^*)。     

The range of the efficient frontier in multiple objective linear programming

Because a rational decision maker should only select an efficient alternative in multiple criterion decision problems, the efficient frontier defined as the set of all efficient alternatives has become a central solution concept in multiple objective linear programming. Normally this set reduces the set of available alternatives of the underlying problem. There are several methods, mainly based on the simplex method, for computing the efficient frontier. This paper presents a quite different approach which uses a nonlinear parametric program, solved by Wolfe's algorithm, to determine the range of the efficient frontier.     

Multiobjective bilevel optimization

In this work nonlinear non-convex multiobjective bilevel optimization problems are discussed using an optimistic approach. It is shown that the set of feasible points of the upper level function, the so-called induced set, can be expressed as the set of minimal solutions of a multiobjective optimization problem. This artificial problem is solved by using a scalarization approach by Pascoletti and Serafini combined with an adaptive parameter control based on sensitivity results for this problem. The bilevel optimization problem is then solved by an iterative process using again sensitivity theorems for exploring the induced set and the whole efficient set is approximated. For the case of bicriteria optimization problems on both levels and for a one dimensional upper level variable, an algorithm is presented for the first time and applied to two problems: a theoretical example and a problem arising in applications.     

Utility Function Programs and Optimization over the Efficient Set in Multiple-Objective Decision Making

Natural basic concepts in multiple-objective optimization lead to difficult multiextremal global optimization problems. Examples include detection of efficient points when nonconvexities occur, and optimization of a linear function over the efficient set in the convex (even linear) case. Assuming that a utility function exists allows one to replace in general the multiple-objective program by a single, nonconvex optimization problem, which amounts to a minimization over the efficient set when the utility function is increasing. A new algorithm is discussed for this utility function program which, under natural mild conditions, converges to an -approximate global solution in a finite number of iterations. Applications include linear, convex, indefinite quadratic, Lipschitz, and d.c. objectives and constraints.     

An all-linear programming relaxation algorithm for optimizing over the efficient set

The problem (P) of optimizing a linear function over the efficient set of a multiple objective linear program has many important applications in multiple criteria decision making. Since the efficient set is in general a nonconvex set, problem (P) can be classified as a global optimization problem. Perhaps due to its inherent difficulty, it appears that no precisely-delineated implementable algorithm exists for solving problem (P) globally. In this paper a relaxation algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact optimal solution to the problem after a finite number of iterations. A detailed discussion is included of how to implement the algorithm using only linear programming methods. Convergence of the algorithm is proven, and a sample problem is solved.Research supported by a grant from the College of Business Administration, University of Florida, Gainesville, Florida, U.S.A.     

On the Solution of Generalized Multiplicative Extremum Problems

The paper addresses the problem of maximizing a sum of products of positive and concave real-valued functions over a convex feasible set. A reformulation based on the image of the feasible set through the vector-valued function which describes the problem, combined with an adequate application of convex analysis results, lead to an equivalent indefinite quadratic extremum problem with infinitely many linear constraints. Special properties of this later problem allow to solve it by an efficient relaxation algorithm. Some numerical tests illustrate the approach proposed.     

A smoothing augmented Lagrangian method for solving simple bilevel programs

In this paper, we design a numerical algorithm for solving a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint. We propose to solve a combined problem where the first order condition and the value function are both present in the constraints. Since the value function is in general nonsmooth, the combined problem is in general a nonsmooth and nonconvex optimization problem. We propose a smoothing augmented Lagrangian method for solving a general class of nonsmooth and nonconvex constrained optimization problems. We show that, if the sequence of penalty parameters is bounded, then any accumulation point is a Karush-Kuch-Tucker (KKT) point of the nonsmooth optimization problem. The smoothing augmented Lagrangian method is used to solve the combined problem. Numerical experiments show that the algorithm is efficient for solving the simple bilevel program.     

Approximate solutions and scalarization in set-valued optimization

Abstract

Certain notions of approximate weak efficient solutions are considered for a set-valued optimization problem based on vector and set criteria approaches. For approximate solutions based on the vector approach, a characterization is provided in terms of an extended Gerstewitz’s function. For the set approach case, two notions of approximate weak efficient solutions are introduced using a lower and an upper quasi order relations for sets and further compactness and stability aspects are discussed for these approximate solutions. Existence and scalarization using a generalized Gerstewitz’s function are also established for approximate solutions, based on the lower set order relation.     

An efficient deterministic optimization approach for rectangular packing problems

The rectangular packing problem aims to seek the best way of placing a given set of rectangular pieces within a large rectangle of minimal area. Such a problem is often constructed as a quadratic mixed-integer program. To find the global optimum of a rectangular packing problem, this study transforms the original problem as a mixed-integer linear programming problem by logarithmic transformations and an efficient piecewise linearization approach that uses a number of binary variables and constraints logarithmic in the number of piecewise line segments. The reformulated problem can be solved to obtain an optimal solution within a tolerable error. Numerical examples demonstrate the computational efficiency of the proposed method in globally solving rectangular packing problems.     

Hyperbolic set covering problems with competing ground-set elements

Motivated by a challenging problem arising in wireless network design, we investigate a new nonlinear variant of the set covering problem with hyperbolic objective function. Each ground-set element (user) competes with all its neighbors (interfering users) for a shared resource (the network access time), and the goal is to maximize the sum of the resource share assigned to each ground-set element (the network efficiency) while covering all of them. The hyperbolic objective function privileges covers with limited overlaps among selected subsets. In a sense, this variant lies in between the set partitioning problem, where overlaps are forbidden, and the standard set covering problem, where overlaps are not an issue at all. We study the complexity and approximability of generic and Euclidean versions of the problem, present an efficient Lagrangean relaxation approach to tackle medium-to-large-scale instances, and compare the computational results with those obtained by linearizations.     

Reverse Convex Programming Approach in the Space of Extreme Criteria for Optimization over Efficient Sets

The problem of minimizing a convex function over the difference of two convex sets is called ‘reverse convex program’. This is a typical problem in global optimization, in which local optima are in general different from global optima. Another typical example in global optimization is the optimization problem over the efficient set of a multiple criteria programming problem. In this article, we investigate some special cases of optimization problems over the efficient set, which can be transformed equivalently into reverse convex programs in the space of so-called extreme criteria of multiple criteria programming problems under consideration. A suitable algorithm of branch and bound type is then established for globally solving resulting problems. Preliminary computational results with the proposed algorithm are reported.     

Existence of efficient solutions for vector maximization problems

The vector maximization problem arises when more than one objective function is to be maximized over a given feasibility region. The concept of efficiency has played a useful role in analyzing this problem. In order to exclude efficient solutions of a certain anomalous type, the concept of proper efficiency has also been utilized. In this paper, an examination of the existence of efficient and properly efficient solutions for the vector maximization problem is undertaken. Given a feasible solution for the vector maximization problem, a related single-objective mathematical programming problem is investigated. Any optimal solution to this program, if one exists, yields an efficient solution for the vector maximization problem. In many cases, the unboundedness of this problem shows that no properly efficient solutions exist. Conditions are pointed out under which the latter conclusion implies that the set of efficient solutions is null. As a byproduct of our results, conditions are derived which guarantee that the outcome of any improperly efficient point is the limit of the outcomes of some sequence of properly efficient points. Examples are provided to illustrate these results.The author would like to thank Professor T. L. Morin for his helpful comments. Thanks also go to an anonymous reviewer for his useful comments concerning an earlier version of this paper.The author would like to acknowledge a useful discussion with Professor G. Bitran which helped in motivating Example 4.1.     

Inverse multi-objective combinatorial optimization

Inverse multi-objective combinatorial optimization consists of finding a minimal adjustment of the objective functions coefficients such that a given set of feasible solutions becomes efficient. An algorithm is proposed for rendering a given feasible solution into an efficient one. This is a simplified version of the inverse problem when the cardinality of the set is equal to one. The adjustment is measured by the Chebyshev distance. It is shown how to build an optimal adjustment in linear time based on this distance, and why it is right to perform a binary search for determining the optimal distance. These results led us to develop an approach based on the resolution of mixed-integer linear programs. A second approach based on a branch-and-bound is proposed to handle any distance function that can be linearized. Finally, the initial inverse problem is solved by a cutting plane algorithm.     

Linear programming based decomposition methods for inventory distribution systems

We consider an inventory distribution system consisting of one warehouse and multiple retailers. The retailers face random demand and are supplied by the warehouse. The warehouse replenishes its stock from an external supplier. The objective is to minimize the total expected replenishment, holding and backlogging cost over a finite planning horizon. The problem can be formulated as a dynamic program, but this dynamic program is difficult to solve due to its high dimensional state variable. It has been observed in the earlier literature that if the warehouse is allowed to ship negative quantities to the retailers, then the problem decomposes by the locations. One way to exploit this observation is to relax the constraints that ensure the nonnegativity of the shipments to the retailers by associating Lagrange multipliers with them, which naturally raises the question of how to choose a good set of Lagrange multipliers. In this paper, we propose efficient methods that choose a good set of Lagrange multipliers by solving linear programming approximations to the inventory distribution problem. Computational experiments indicate that the inventory replenishment policies obtained by our approach can outperform several standard benchmarks by significant margins.