A new approach for solving linear bilevel problems using genetic algorithms

Bilevel programming involves two optimization problems where the constraint region of the first level problem is implicitly determined by another optimization problem. This paper develops a genetic algorithm for the linear bilevel problem in which both objective functions are linear and the common constraint region is a polyhedron. Taking into account the existence of an extreme point of the polyhedron which solves the problem, the algorithm aims to combine classical extreme point enumeration techniques with genetic search methods by associating chromosomes with extreme points of the polyhedron. The numerical results show the efficiency of the proposed algorithm. In addition, this genetic algorithm can also be used for solving quasiconcave bilevel problems provided that the second level objective function is linear.     

A genetic algorithm for solving linear fractional bilevel problems

Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. In this paper a genetic algorithm is proposed for the class of bilevel problems in which both level objective functions are linear fractional and the common constraint region is a bounded polyhedron. The algorithm associates chromosomes with extreme points of the polyhedron and searches for a feasible solution close to the optimal solution by proposing efficient crossover and mutation procedures. The computational study shows a good performance of the algorithm, both in terms of solution quality and computational time.     

Solving linear fractional bilevel programs

In this paper, we prove that an optimal solution to the linear fractional bilevel programming problem occurs at a boundary feasible extreme point. Hence, the Kth-best algorithm can be proposed to solve the problem. This property also applies to quasiconcave bilevel problems provided that the first level objective function is explicitly quasimonotonic.     

On the Quasiconcave Bilevel Programming Problem

Bilevel programming involves two optimization problems where the constraint region of the first-level problem is implicitly determined by another optimization problem. In this paper, we consider the case in which both objective functions are quasiconcave and the constraint region common to both levels is a polyhedron. First, it is proved that this problem is equivalent to minimizing a quasiconcave function over a feasible region comprised of connected faces of the polyhedron. Consequently, there is an extreme point of the polyhedron that solves the problem. Finally, it is shown that this model includes the most important case where the objective functions are ratios of concave and convex functions     

Simplex-like sequential methods for a class of generalized fractional programs

A sequential method for a class of generalized fractional programming problems is proposed. The considered objective function is the ratio of powers of affine functions and the feasible region is a polyhedron, not necessarily bounded. Theoretical properties of the optimization problem are first established and the maximal domains of pseudoconcavity are characterized. When the objective function is pseudoconcave in the feasible region, the proposed algorithm takes advantage of the nice optimization properties of pseudoconcave functions; the particular structure of the objective function allows to provide a simplex-like algorithm even when the objective function is not pseudoconcave. Computational results validate the nice performance of the proposed algorithm.     

On the adjustment problem for linear programs

We propose a generalization of the inverse problem which we will call the adjustment problem. For an optimization problem with linear objective function and its restriction defined by a given subset of feasible solutions, the adjustment problem consists in finding the least costly perturbations of the original objective function coefficients, which guarantee that an optimal solution of the perturbed problem is also feasible for the considered restriction. We describe a method of solving the adjustment problem for continuous linear programming problems when variables in the restriction are required to be binary.     

Bilevel problems over polyhedra with extreme point optimal solutions

Bilevel programming involves two optimization problems where the constraint region of the upper level problem is implicitly determined by another optimization problem. In this paper we focus on bilevel problems over polyhedra with upper level constraints involving lower level variables. On the one hand, under the uniqueness of the optimal solution of the lower level problem, we prove that the fact that the objective functions of both levels are quasiconcave characterizes the property of the existence of an extreme point of the polyhedron defined by the whole set of constraints which is an optimal solution of the bilevel problem. An example is used to show that this property is in general violated if the optimal solution of the lower level problem is not unique. On the other hand, if the lower level objective function is not quasiconcave but convex quadratic, assuming the optimistic approach we prove that the optimal solution is attained at an extreme point of an ??enlarged?? polyhedron.     

Parametric computation of a fuzzy set solution to a class of fuzzy linear fractional optimization problems

The class of fuzzy linear fractional optimization problems with fuzzy coefficients in the objective function is considered in this paper. We propose a parametric method for computing the membership values of the extreme points in the fuzzy set solution to such problems. We replace the exhaustive computation of the membership values—found in the literature for solving the same class of problems—by a parametric analysis of the efficiency of the feasible basic solutions to the bi-objective linear fractional programming problem through the optimality test in a related linear programming problem, thus simplifying the computation. An illustrative example from the field of production planning is included in the paper to complete the theoretical presentation of the solving approach, but also to emphasize how many real life problems may be modelled mathematically using fuzzy linear fractional optimization.     

线性半向量二层规划问题的全局优化方法

研究了线性半向量二层规划问题的全局优化方法. 利用下层问题的对偶间隙构造了线性半向量二层规划问题的罚问题, 通过分析原问题的最优解与罚问题可行域顶点之间的关系, 将线性半向量二层规划问题转化为有限个线性规划问题, 从而得到线性半向量二层规划问题的全局最优解. 数值结果表明所设计的全局优化方法对线性半向量二层规划问题是可行的.     

Finite algorithm for generalized linear multiplicative programming

The nonconvex problem of minimizing the sum of a linear function and the product of two linear functions over a convex polyhedron is considered. A finite algorithm is proposed which either finds a global optimum or shows that the objective function is unbounded from below in the feasible region. This is done by means of a sequence of primal and/or dual simplex iterations.The first author gratefully acknowledges the research support received as Visiting Professor of the Dipartimento di Statistica e Matematica Applicata all' Economia, Universitá di Pisa, Pisa, Italy, Spring 1992.     

Parametric linear fractional programming for an unbounded feasible region

Parametric analysis in linear fractional programming is significantly more complicated in case of an unbounded feasible region. We propose procedures which are based on a modified version of Martos' algorithm or a modification of Charnes-Cooper's algorithm, applying each to problems where either the objective function or the right-hand side is parametrized.     

Solving convex quadratic bilevel programming problems using an enumeration sequential quadratic programming algorithm

In this paper, we present an original method to solve convex bilevel programming problems in an optimistic approach. Both upper and lower level objective functions are convex and the feasible region is a polyhedron. The enumeration sequential linear programming algorithm uses primal and dual monotonicity properties of the primal and dual lower level objective functions and constraints within an enumeration frame work. New optimality conditions are given, expressed in terms of tightness of the constraints of lower level problem. These optimality conditions are used at each step of our algorithm to compute an improving rational solution within some indexes of lower level primal-dual variables and monotonicity networks as well. Some preliminary computational results are reported.     

Bilevel multiplicative problems: A penalty approach to optimality and a cutting plane based algorithm

Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterised by the existence of two optimisation problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimisation problem. In this paper we focus on the class of bilevel problems in which the upper level objective function is linear multiplicative, the lower level one is linear and the common constraint region is a bounded polyhedron. After replacing the lower level problem by its Karush–Kuhn–Tucker conditions, the existence of an extreme point which solves the problem is proved by using a penalty function approach. Besides, an algorithm based on the successive introduction of valid cutting planes is developed obtaining a global optimal solution. Finally, we generalise the problem by including upper level constraints which involve both level variables.     

一类模糊线性规划模型的模糊最优区间值

讨论一类既有模糊不等式约束又有模糊等式约束的全模糊系数线性规划问题。在给定的模糊隶属度水平下 ,将模型转化为区间数线性规划模型 ,通过确定区间模型的最佳目标函数和最大可行域以及最劣目标函数和最小可行域 ,求出目标函数的模糊最优区间值 ,从而为决策者提供更多的决策信息。最后给出一个数值例子。     

Weighting method for bi-level linear fractional programming problems

In this paper we consider the solution of a bi-level linear fractional programming problem (BLLFPP) by weighting method. A non-dominated solution set is obtained by this method. In this article decision makers (DMs) provide their preference bounds to the decision variables that is the upper and lower bounds to the decision variables they control. We convert the hierarchical system into scalar optimization problem (SOP) by finding proper weights using the analytic hierarchy process (AHP) so that objective functions of both levels can be combined into one objective function. Here the relative weights represent the relative importance of the objective functions.     

求解多层线性规划的模糊规划法

本文用模糊集理论中的隶属函数描述多层线性规划的各层目标,在第一层给定最小满意水平下,通过求解相应层次的模糊规划来确定各层的最小满意度,从而最终得到问题的一个满意解。提出的方法只需求解一系列线性规划问题,具有较好的计算复杂性和可行性,最后的算例进一步验证了方法的有效性。     

Generalized linear multiplicative and fractional programming

This paper addresses itself to the algorithm for minimizing the sum of a convex function and a product of two linear functions over a polytope. It is shown that this nonconvex minimization problem can be solved by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in higher dimension and apply a parametric programming (path following) approach. Also it is shown that the same idea can be applied to a generalized linear fractional programming problem whose objective function is the sum of a convex function and a linear fractional function.     

On the transformation of lexicographic nonlinear multiobjective programs to single objective programs

This paper deals with multiobjective optimization programs in which the objective functions are ordered by their degree of priority. A number of approaches have been proposed (and several implemented) for the solution of lexicographic (preemptive priority) multiobjective optimization programs. These approaches may be divided into two classes. The first encompasses the development of algorithms specifically designed to deal directly with the initial model. Considered only for linear multiobjective programs and multiobjective programs with a finite discrete feasible region, the second one attempts to transform, efficiently, the lexicographic multiobjective model into an equvivalent model, i.e. a single objective programming problem. In this paper, we deal with the second approach for lexicographic nonlinear multiobjective programs.     

Linear bilevel programs with multiple objectives at the upper level

Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. Focus of the paper is on general bilevel optimization problems with multiple objectives at the upper level of decision making. When all objective functions are linear and constraints at both levels define polyhedra, it is proved that the set of efficient solutions is non-empty. Taking into account the properties of the feasible region of the bilevel problem, some methods of computing efficient solutions are given based on both weighted sum scalarization and scalarization techniques. All the methods result in solving linear bilevel problems with a single objective function at each level.     

Optimal value range in interval linear programming

We deal with the linear programming problem in which input data can vary in some given real compact intervals. The aim is to compute the exact range of the optimal value function. We present a general approach to the situation the feasible set is described by an arbitrary linear interval system. Moreover, certain dependencies between the constraint matrix coefficients can be involved. As long as we are able to characterize the primal and dual solution set (the set of all possible primal and dual feasible solutions, respectively), the bounds of the objective function result from two nonlinear programming problems. We demonstrate our approach on various cases of the interval linear programming problem (with and without dependencies).