On the nonemptiness and compactness of the solution sets for vector variational inequalities

In this article, we investigate conditions for nonemptiness and compactness of the sets of solutions of pseudomonotone vector variational inequalities by using the concept of asymptotical cones. We show that a pseudomonotone vector variational inequality has a nonempty and compact solution set provided that it is strictly feasible. We also obtain some necessary conditions for the set of solutions of a pseudomonotone vector variational inequality to be nonempty and compact.     

基于灰色理论的价格控制问题及其算法

针对排污收费的最优定价问题,提出了基于灰色理论的价格控制问题,并给出了该问题的模型及相关的定理。在约束域为非空紧集的条件下,证明了漂移型价格控制问题的最优解一定可以在约束域的极点达到。针对漂移型价格控制问题,采用价格控制问题的搜索算法的求解技术,把灰参数看做一个新的决策变量,将该问题转化为多个含参数的非线性规划问题。最后,通过一算例验证了模型及求解方法的有效性。     

Optimizing a linear function over an integer efficient set

In this paper, a method for optimizing a linear function over the integer Pareto-optimal set without having to determine all integer efficient solutions is presented. The proposed algorithm is based on a simple selection technique that improves the linear objective value at each iteration. Two types of cuts are performed successively until the optimal value is obtained and the current truncated region contains no integer feasible solution.     

一种具有非线性约束线性规划全局优化算法

本文提出了一种新的适用于处理非线性约束下线性规划问题的全局优化算法。该算法通过构造子问题来寻找优于当前局部最优解的可行解。该子问题可通过模拟退火算法来解决。通过求解一系列的子问题,当前最优解被不断地更新,最终求得全局最优解。最后,本算法应用于几个典型例题,并与罚函数法相比较,数值结果表明该算法是可行的,有效的。     

On boundedness of (quasi-)convex integer optimization problems

In this paper we are concerned with the problem of boundedness and the existence of optimal solutions to the constrained integer optimization problem. We present necessary and sufficient conditions for boundedness of either a faithfully convex or quasi-convex polynomial function over the feasible set contained in , and defined by a system of faithfully convex inequality constraints and/or quasi-convex polynomial inequalities. The conditions for boundedness are provided in the form of an implementable algorithm, terminating after a finite number of iterations, showing that for the considered class of functions, the integer programming problem with nonempty feasible region is unbounded if and only if the associated continuous optimization problem is unbounded. We also prove that for a broad class of objective functions (which in particular includes polynomials with integer coefficients), an optimal solution set of the constrained integer problem is nonempty over any subset of .     

The bilevel linear/linear fractional 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 bilevel linear/linear fractional programming problem in which the objective function of the first level is linear, the objective function of the second level is linear fractional and the feasible region is a polyhedron. For this problem we prove that an optimal solution can be found which is an extreme point of the polyhedron. Moreover, taking into account the relationship between feasible solutions to the problem and bases of the technological coefficient submatrix associated to variables of the second level, an enumerative algorithm is proposed that finds a global optimum to the problem.     

目标控制型线性三级规划的基本性质

本文讨论了一类以下级目标函数最优值为反馈的线性三级递阶优化问题，按照参数规划的方法给出了可行集、最优解等概念，得到了可靠集的弱拟凸性，连通性等性质，为算法设计了基础。     

An Interactive Method as an Aid in Solving Bicriterion Mathematical Programming Problems

In this paper, a man-machine interactive method is presented as an aid in solving the bicriterion mathematical programming problem. It is assumed that the two objective functions are real-valued functions of the decision variables which are themselves constrained to some compact and nonempty set. The overall utility function is assumed to be unknown explicitly to the decision-maker but is assumed to be a real-valued function defined on the pairs of feasible values of the objective functions and monotone non-decreasing in each argument. The decision-maker is required only to provide yes or no answers to questions regarding the desirability of increase or decrease in objective function values of solutions that he will not accept as optimal. Convergence of the method is indicated and a numerical example is presented in order to demonstrate its applicability.     

On Extensions of the Frank-Wolfe Theorems

In this paper we consider optimization problems defined by a quadratic objective function and a finite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the so-called Frank-Wolfe theorem. In particular, we first prove a general continuity result for the solution set defined by a system of convex quadratic inequalities. This result implies immediately that the optimal solution set of the aforementioned problem is nonempty when all the quadratic functions involved are convex. In the absence of the convexity of the objective function, we give examples showing that the optimal solution set may be empty either when there are two or more convex quadratic constraints, or when the Hessian of the objective function has two or more negative eigenvalues. In the case when there exists only one convex quadratic inequality constraint (together with other linear constraints), or when the constraint functions are all convex quadratic and the objective function is quasi-convex (thus allowing one negative eigenvalue in its Hessian matrix), we prove that the optimal solution set is nonempty.     

Convex-concave fractional programming with each variable occurring in a single constraint

The problem considered is that of maximizing the ratio of a concave and a convex function under the assumption that each variable occurs in exactly one component constraint. Such problems occur in the allocation of resources to activities. It is demonstrated that the problem is separable and that componentwise optimization can be applied to determine a solution. A method is given that can be used to evaluate the quality of any feasible solution in terms of an associated upper bound of the optimal value of the objective function: optimal and almost optimal solutions can be recognized. A fast incremental method of generating feasible solutions is described.     

An interactive method as an aid in solving multi-objective mathematical programming problems

In this paper, an interactive method is presented as an aid in solving multi-objective programming problems. It is assumed that the m objective functions are real-valued functions of the decision variables which are themselves constrained to some compact and nonempty set. The overall utility function is assumed to be unknown explicitly to the decision-maker but is assumed to be a real-valued, unimodal function defined on the m-tuples of feasible values of the objective functions and monotone nondecreasing in each argument. The decision-maker is required only to provide yes or no answers to questions regarding the desirability of increase or decrease in objective function values of solutions that he will not accept as optimal. Convergence of the method is indicated and a numerical example is presented in order to demonstrate its applicability.     

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.     

A variation on Karmarkar’s algorithm for solving linear programming problems

The algorithm described here is a variation on Karmarkar’s algorithm for linear programming. It has several advantages over Karmarkar’s original algorithm. In the first place, it applies to the standard form of a linear programming problem and produces a monotone decreasing sequence of values of the objective function. The minimum value of the objective function does not have to be known in advance. Secondly, in the absence of degeneracy, the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero. This makes it possible to identify an optimal basis before the algorithm converges.     

AN EFFECTIVE SEQUENTIAL QUADRATIC PROGRAMMING ALGORITHM FOR NONLINEAR OPTIMIZATION PROBLEMS

In this paper,a new globally convergent algorithm for nonlinear optimization prablems with equality and inequality constraints is presented. The new algorithm is of SQP type which determines a search direction by solving a quadratic programming subproblem per itera-tion. Some revisions on the quadratic programming subproblem have been made in such a way that the associated constraint region is nonempty for each point x generated by the algorithm, i. e. , the subproblems always have optimal solutions. The new algorithm has two important properties. The computation of revision parameter for guaranteeing the consistency of quadratic sub-problem and the computation of the second order correction step for superlinear convergence use the same inverse of a matrix per iteration, so the computation amount of the new algorithm will not be increased much more than other SQP type algorithms; Another is that the new algorithm can give automatically a feasible point as a starting point for the quadratic subproblems pe     

Simplicial Lipschitz optimization without the Lipschitz constant

In this paper we propose a new simplicial partition-based deterministic algorithm for global optimization of Lipschitz-continuous functions without requiring any knowledge of the Lipschitz constant. Our algorithm is motivated by the well-known Direct algorithm which evaluates the objective function on a set of points that tries to cover the most promising subregions of the feasible region. Almost all previous modifications of Direct algorithm use hyper-rectangular partitions. However, other types of partitions may be more suitable for some optimization problems. Simplicial partitions may be preferable when the initial feasible region is either already a simplex or may be covered by one or a manageable number of simplices. Therefore in this paper we propose and investigate simplicial versions of the partition-based algorithm. In the case of simplicial partitions, definition of potentially optimal subregion cannot be the same as in the rectangular version. In this paper we propose and investigate two definitions of potentially optimal simplices: one involves function values at the vertices of the simplex and another uses function value at the centroid of the simplex. We use experimental investigation to compare performance of the algorithms with different definitions of potentially optimal partitions. The experimental investigation shows, that proposed simplicial algorithm gives very competitive results to Direct algorithm using standard test problems and performs particularly well when the search space and the numbers of local and global optimizers may be reduced by taking into account symmetries of the objective function.     

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.     

Conic approximation to quadratic optimization with linear complementarity constraints

This paper proposes a conic approximation algorithm for solving quadratic optimization problems with linear complementarity constraints.We provide a conic reformulation and its dual for the original problem such that these three problems share the same optimal objective value. Moreover, we show that the conic reformulation problem is attainable when the original problem has a nonempty and bounded feasible domain. Since the conic reformulation is in general a hard problem, some conic relaxations are further considered. We offer a condition under which both the semidefinite relaxation and its dual problem become strictly feasible for finding a lower bound in polynomial time. For more general cases, by adaptively refining the outer approximation of the feasible set, we propose a conic approximation algorithm to identify an optimal solution or an \(\epsilon \)-optimal solution of the original problem. A convergence proof is given under simple assumptions. Some computational results are included to illustrate the effectiveness of the proposed algorithm.     

Balinski—Tucker simplex tableaus: Dimensions, degeneracy degrees, and interior points of optimal faces

This paper shows the relationship between degeneracy degrees and multiple solutions in linear programming (LP) models. The usual definition of degeneracy is restricted to vertices of a polyhedron. We introduce degeneracy for nonempty subsets of polyhedra and show that for LP-models for which the feasible region contains at least one vertex it holds that the dimension of the optimal face is equal to the degeneracy degree of the optimal face of the corresponding dual model. This result is obtained by means of the so-called Balinski—Tucker (B—T) Simplex Tableaus. Furthermore, we give a strong polynomial algorithm for constructing such a B—T Simplex Tableau when a solution in the relative interior of the optimal face is known. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems

In this paper, a notion of Levitin–Polyak (LP in short) well-posedness is introduced for a vector optimization problem in terms of minimizing sequences and efficient solutions. Sufficient conditions for the LP well-posedness are studied under the assumptions of compactness of the feasible set, closedness of the set of minimal solutions and continuity of the objective function. The continuity assumption is then weakened to cone lower semicontinuity for vector-valued functions. A notion of LP minimizing sequence of sets is studied to establish another set of sufficient conditions for the LP well-posedness of the vector problem. For a quasiconvex vector optimization problem, sufficient conditions are obtained by weakening the compactness of the feasible set to a certain level-boundedness condition. This in turn leads to the equivalence of LP well-posedness and compactness of the set of efficient solutions. Some characterizations of LP well-posedness are given in terms of the upper Hausdorff convergence of the sequence of sets of approximate efficient solutions and the upper semicontinuity of an approximate efficient map by assuming the compactness of the set of efficient solutions, even when the objective function is not necessarily quasiconvex. Finally, a characterization of LP well-posedness in terms of the closedness of the approximate efficient map is provided by assuming the compactness of the feasible set.     

求标准线性规划问题的一种截解法

本提出了求解线性规划问题的一种新思路，就是通过平行移动目标函数等值面，即改变目标函数作为参数的取值来截取基本可行解，甚至最优解。值得注意的是，本算法可能会克服由退化引起的迭代循环。