求解多主多从线性斯坦伯格问题的新方法

在本文中,我们研究斯坦伯格问题,发展了罚函数法。     

确定线性规划全部最优解的方法

使用凸多面体的表示定理 ,导出了标准型线性规划最优解的一般表达式 ,并基于单纯形法 ,给出最优解唯一性条件以及当唯一性条件不满足时求出全部最优解的计算步骤 ,同时附有数值例子 .     

An Interval-Parameter Fuzzy Approach for Multiobjective Linear Programming Under Uncertainty

An interval-parameter fuzzy linear programming method (IFMOLP) is proposed in this study for multiple objective decision-making under uncertainty. As a hybrid of interval-parameter and fuzzy methodologies, the IFMOLP incorporates interval-parameter linear programming and fuzzy multiobjective programming approaches to form an integrated optimization system. The method inherits advantages of interval-parameter programming, and allows uncertainties and decision-makers’ aspirations to be effectively communicated into its programming processes and resulting solutions. Membership functions for both objectives and constraints are formulated to reflect uncertainties in different system components and their interrelationships. An interactive solution procedure has been developed based on solution approaches of the interval-parameter and fuzzy programming techniques, plus necessary measures for handling the multiobjective feature. A didactic example is provided in the paper to illustrate the detailed solution process. Possibilities of further improvements by seeking Pareto optimum and incorporating flexible preference within constraints are also discussed.     

关于线性二层规划分枝定界方法的探讨

对求解线性二层规划的分枝定界方法进行了探讨.给出的一个例子表明,目前的分枝定界方法不能很好地解决上层带有任意线性形式约束的线性二层规划问题,进而在线性二层规划新定义的基础上提出了求解线性二层规划的扩展分枝定界方法.算例表明扩展分枝定界方法可以有效解决原分枝定界方法的不足.     

On the Stability of the Feasible Set in Linear Optimization

This paper deals with the stability of two families of linear optimization problems, each one formed by the dual problems to the members of the other family. We characterize the problems of these families that are stable in the sense that they remain consistent (inconsistent) under sufficiently small arbitrary perturbations of all the data. This characterization is established in terms of the lower semicontinuity property of the feasible set mapping and the boundedness of the optimal set of the corresponding coupled problem. Other continuity properties of the feasible set mapping are also derived. This stability theory extends some well-known theorems of Williams and Robinson on the stability of ordinary linear programming problems to linear optimization problems with infinitely many variables or constraints.     

Differential-Algebraic Approach to Linear Programming

This paper presents a differential-algebraic approach for solving linear programming problems. The paper shows that the differential-algebraic approach is guaranteed to generate optimal solutions to linear programming problems with a superexponential convergence rate. The paper also shows that the path-following interior-point methods for solving linear programming problems can be viewed as a special case of the differential-algebraic approach. The results in this paper demonstrate that the proposed approach provides a promising alternative for solving linear programming problems.     

约束条件下多元线性模型中线性估计的可容许性

本文研究了多元线性模型中未知参数在约束条件:(θ-θ0)′X′NX(θ-θ0)≤U,N≥0下中心点θ0对线性估计的可容许性的影响.研究结果表明对于具有某种结构的θ1和θ2,在约束集(θ-θ1)′X′NX(θ-θ1)≤U,N≥0与(θ-θ2)′X′NX(θ-θ2)≤U,N≥0下的可容许线性估计类是一致的.     

The Concept of Proper Solution in Linear Programming

In this paper, we study the optimal solutions of a dual pair of linear programming problems that correspond to the proper equilibria of their associated matrix game. We give conditions ensuring the existence of such solutions, show that they are especially robust under perturbation of right-hand-side terms, and describe a procedure to obtain them.     

Optimal Prioritized Infeasibility Handling in Model Predictive Control: Parametric Preemptive Multiobjective Linear Programming Approach

All practical implementations of model-based predictive control (MPC) require a means to recover from infeasibility. We propose a strategy designed for linear state-space MPC with prioritized constraints. It relaxes optimally an infeasible MPC optimization problem into a feasible one by solving a single-objective linear program (LP) online in addition to the standard online MPC optimization problem at each sample. By optimal, it is meant that the violation of a lower prioritized constraint cannot be made less without increasing the violation of a higher prioritized constraint. The problem of computing optimal constraint violations is naturally formulated as a parametric preemptive multiobjective LP. By extending well-known results from parametric LP, the preemptive multiobjective LP is reformulated into an equivalent standard single-objective LP. An efficient algorithm for offline design of this LP is given, and the algorithm is illustrated on an example.     

On the Stability of the Boundary of the Feasible Set in Linear Optimization

This paper analizes the relationship between the stability properties of the closed convex sets in finite dimensions and the stability properties of their corresponding boundaries. We consider a given closed convex set represented by a certain linear inequality system whose coefficients can be arbitrarily perturbed, and we measure the size of these perturbations by means of the pseudometric of the uniform convergence. It is shown that the feasible set mapping is Berge lower semicontinuous at if and only if the boundary mapping satisfies the same property. Moreover, if the boundary mapping is semicontinuous in any sense (lower or upper; Berge or Hausdorff) at , then it is also closed at . All the mentioned stability properties are equivalent when the feasible set is a convex body.     

有无穷多最优解线性规划问题

本文给出了线性规划有无穷多最优解的判别条件及其求出所有最优解的具体方法．     

An Algorithm for Approximate Multiparametric Linear Programming

Multiparametric programming considers optimization problems where the data are functions of a parameter vector and describes the optimal value and an optimizer as explicit functions of the parameters. In this paper, we consider a linear program where the right-hand side is an affine function of a parameter vector; we propose an algorithm for approximating its solution. Given a full-dimensional simplex in the parameter space and an optimizer for each simplex vertex, the algorithm formulates the linear interpolation of the given solutions as an explicit function of the parameters, giving a primal feasible approximation of an optimizer inside the simplex. If the resulting absolute error in the objective exceeds a prescribed tolerance, then the algorithm subdivides the simplex into smaller simplices where it applies recursively. We propose both a basic version and a refined version of the algorithm. The basic version is polynomial in the output size, provided a polynomial LP solver is used; the refined version may give a smaller output. A global error bound for the optimizer is derived and some computational tests are discussed.     

Geometric Algorithm for Multiparametric Linear Programming

We propose a novel algorithm for solving multiparametric linear programming problems. Rather than visiting different bases of the associated LP tableau, we follow a geometric approach based on the direct exploration of the parameter space. The resulting algorithm has computational advantages, namely the simplicity of its implementation in a recursive form and an efficient handling of primal and dual degeneracy. Illustrative examples describe the approach throughout the paper. The algorithm is used to solve finite-time constrained optimal control problems for discrete-time linear dynamical systems.     

线性分式规划最优解集的求法

本文使用多面集的表示定理，导出了线性分式规划最优解集的结构，并给出确定全部最优解的计算步骤。     

Stackelberg Solutions to Multiobjective Two-Level Linear Programming Problems

In this paper, we consider a multiobjective two-level linear programming problem in which the decision maker at each level has multiple-objective functions conflicting with each other. The decision maker at the upper level must take account of multiple or infinite rational responses of the decision maker at the lower level in the problem. We examine three kinds of situations based on anticipation of the decision maker at the upper level: optimistic anticipation, pessimistic anticipation, and anticipation arising from the past behavior of the decision maker at the lower level. Mathematical programming problems for obtaining the Stackelberg solutions based on the three kinds of anticipation are formulated and algorithms for solving the problems are presented. Illustrative numerical examples are provided to understand the geometrical properties of the solutions and demonstrate the feasibility of the proposed methods.     

线性规划问题的一种求解算法

本通过初等变换，并剔除常变量和零变量而对所给的线性规划问题进行预处理，得到的等价问题不仅易找初始可行基且初始可行解较优，易差别无可行解情形，而且可能使所含方程个数与变量个数减少，从而减少了求解问题的计算量和迭代次数。     

线性规划的基解算法

提出了求解线性规划问题的一种新方法-基解算法,它是一个不需引入人工变量,不必预先求出一个可行基的直接求解算法。     

Outcome Space Partition of the Weight Set in Multiobjective Linear Programming

Approaches for generating the set of efficient extreme points of the decision set of a multiple-objective linear program (P) that are based upon decompositions of the weight set W⁰ suffer from one of two special drawbacks. Either the required computations are redundant, or not all of the efficient extreme point set is found. This article shows that the weight set for problem (P) can be decomposed into a partition based upon the outcome set Y of the problem, where the elements of the partition are in one-to-one correspondence with the efficient extreme points of Y. As a result, the drawbacks of the decompositions of W⁰ based upon the decision set of problem (P) disappear. The article explains also how this new partition offers the potential to construct algorithms for solving large-scale applications of problem (P) in the outcome space, rather than in the decision space.     

区间数线性规划问题的解法

基于区间数与实数之间的关系,提出了区间数线性规划的激进最优解,保守最优解的定义.利用约束集之间以及目标函数值之间的关系,在原有区间数线性规划的基础之上,给出了两个求解激进最优解、保守最优解的方法.数值例子验证了该方法的有效性和可行性.     

The Dual Active Set Algorithm and Its Application to Linear Programming

The Dual Active Set Algorithm (DASA), presented in Hager, Advances in Optimization and Parallel Computing, P.M. Pardalos (Ed.), North Holland: Amsterdam, 1992, pp. 137–142, for strictly convex optimization problems, is extended to handle linear programming problems. Line search versions of both the DASA and the LPDASA are given.