Algorithm of Barrier Objective Penalty Function

In this paper, an algorithm of barrier objective penalty function for inequality constrained optimization is studied and a conception–the stability of barrier objective penalty function is presented. It is proved that an approximate optimal solution may be obtained by solving a barrier objective penalty function for inequality constrained optimization problem when the barrier objective penalty function is stable. Under some conditions, the stability of barrier objective penalty function is proved for convex programming. Specially, the logarithmic barrier function of convex programming is stable. Based on the barrier objective penalty function, an algorithm is developed for finding an approximate optimal solution to an inequality constrained optimization problem and its convergence is also proved under some conditions. Finally, numerical experiments show that the barrier objective penalty function algorithm has better convergence than the classical barrier function algorithm.     

Network reoptimization procedures for multiobjective network probelems

Network models are attractive because of their computational efficiency. Network applications can involve multiple objective analysis. Multiple objective analysis requires generating nondominated solutions in various forms. Two general methods exist to generate new solutions in continuous optimization: changing objective function weights and inserting objective bounds through constraints. In network flow problems, modifying weights is straightforward, allowing use of efficient network codes. Use of bounds on objective attainment levels can provide a more controlled generation of solutions reflecting tradeoffs among objectives. To constrain objective attainment, however, would require a side constrained network code, sacrificing some computational efficiency for greater model flexibility. We develop reoptimization procedures for the side constrained problem and use them in conjunction with simplex-based techniques. Our approach provides a useful tool for generating solutions allowing greater decision maker control over objective attainments, allowing multiobjective analysis of large-scale problems. Results are compared with solutions obtained from the computationally more attractive weighting technique. Reoptimization procedures are discussed as a means of more efficiently conducting multiple objective network analyses.     

Solving Bilevel Linear Programs Using Multiple Objective Linear Programming

We present an algorithm for solving bilevel linear programs that uses simplex pivots on an expanded tableau. The algorithm uses the relationship between multiple objective linear programs and bilevel linear programs along with results for minimizing a linear objective over the efficient set for a multiple objective problem. Results in multiple objective programming needed are presented. We report computational experience demonstrating that this approach is more effective than a standard branch-and-bound algorithm when the number of leader variables is small.     

Augmented Lagrangian Objective Penalty Function

Augmented Lagrangian function is one of the most important tools used in solving some constrained optimization problems. In this article, we study an augmented Lagrangian objective penalty function and a modified augmented Lagrangian objective penalty function for inequality constrained optimization problems. First, we prove the dual properties of the augmented Lagrangian objective penalty function, which are at least as good as the traditional Lagrangian function's. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker condition. This is especially so when the Karush-Kuhn-Tucker condition holds for convex programming of its saddle point existence. Second, we prove the dual properties of the modified augmented Lagrangian objective penalty function. For a global optimal solution, when the exactness of the modified augmented Lagrangian objective penalty function holds, its saddle point exists. The sufficient and necessary stability conditions used to determine whether the modified augmented Lagrangian objective penalty function is exact for a global solution is proved. Based on the modified augmented Lagrangian objective penalty function, an algorithm is developed to find a global solution to an inequality constrained optimization problem, and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the modified augmented Lagrangian objective penalty function is proved for a local solution. An algorithm is presented in finding a local solution, with its convergence proved under some conditions.     

New Algorithms for Three-Partitioning Containing Kernels

1.IntroductionGivenasetRofmketnelitems(wecallthemkernels),R={ri2o,i=1,2,...,tn}andasetPofn52mnonkernelitems(wecallthemitems),P={Pj2o,j=1,2,...tn}.WewanttopartitionT=RUPintomsubsetssuchthatrjEMjandlMj153.DenoteCj=Zx,callittheloadofMj,Thendefinethemin-maxproblemastominimizethe2eMmaximumloadofthesemsubsets,andthemax-minproblemastomaximizetheminimumloadofthesemsubset.TheseproblemsareNP-complete[1],meantimeaHeuristicalgorithmKLPTisgivenforthemin-maxproblem,andtheworst-caseboundis:-C'thebo…     

自反Banach空间中锥线性优化问题初始——对偶目标函数水平集的几何性质

讨论自反Banach空间中的原——对偶锥线性优化问题的目标函数水平集的几何性质．在自反Banach空间中,证明了原目标函数水平集的最大模与对偶目标函数水平集的最大内切球半径几乎是成反比例的．     

Objective Bayesian probabilistic logic

This paper develops connections between objective Bayesian epistemology—which holds that the strengths of an agent's beliefs should be representable by probabilities, should be calibrated with evidence of empirical probability, and should otherwise be equivocal—and probabilistic logic. After introducing objective Bayesian epistemology over propositional languages, the formalism is extended to handle predicate languages. A rather general probabilistic logic is formulated and then given a natural semantics in terms of objective Bayesian epistemology. The machinery of objective Bayesian nets and objective credal nets is introduced and this machinery is applied to provide a calculus for probabilistic logic that meshes with the objective Bayesian semantics.     

Approximation of convex bodies by multiple objective optimization and an application in reachable sets

In this paper, we focus on approximating convex compact bodies. For a convex body described as the feasible set in objective space of a multiple objective programme, we show that finding it is equivalent to finding the non-dominated set of a multiple objective programme. This equivalence implies that convex bodies can be approximated using multiple objective optimization algorithms. Therefore, we propose a revised outer approximation algorithm for convex multiple objective programming problems to approximate convex bodies. Finally, we apply the algorithm to solve reachable sets of control systems and use numerical examples to show the effectiveness of the algorithm.     

Exactness and algorithm of an objective penalty function

Penalty function is an important tool in solving many constrained optimization problems in areas such as industrial design and management. In this paper, we study exactness and algorithm of an objective penalty function for inequality constrained optimization. In terms of exactness, this objective penalty function is at least as good as traditional exact penalty functions. Especially, in the case of a global solution, the exactness of the proposed objective penalty function shows a significant advantage. The sufficient and necessary stability condition used to determine whether the objective penalty function is exact for a global solution is proved. Based on the objective penalty function, an algorithm is developed for finding a global solution to an inequality constrained optimization problem and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the objective penalty function is proved for a local solution. An algorithm is presented in the paper in finding a local solution, with its convergence proved under some conditions. Finally, numerical experiments show that a satisfactory approximate optimal solution can be obtained by the proposed algorithm.     

关于总误工损失的两个代理单机排序问题

研究了与总误工损失相关的两个代理的单机排序问题。第一个代理以工件的总误工损失为目标函数,第二个代理以工件的总完工时间或总误工工件数为目标函数。目标是寻找一个排序,使得在第二个代理的目标函数不超过给定的上界的条件下,第一个代理的目标函数值最小。对这两个与总误工损失相关的两个代理的单机排序问题,分别给出它们的拟多项式时间的动态规划算法。     

Strictly feasible solutions and strict complementarity in multiple objective linear optimization

Recently, Luc defined a dual program for a multiple objective linear program. The dual problem is also a multiple objective linear problem and the weak duality and strong duality theorems for these primal and dual problems have been established. Here, we use these results to prove some relationships between multiple objective linear primal and dual problems. We extend the available results on single objective linear primal and dual problems to multiple objective linear primal and dual problems. Complementary slackness conditions for efficient solutions, and conditions for the existence of weakly efficient solution sets and existence of strictly primal and dual feasible points are established. We show that primal-dual (weakly) efficient solutions satisfying strictly complementary conditions exist. Furthermore, we consider Isermann’s and Kolumban’s dual problems and establish conditions for the existence of strictly primal and dual feasible points. We show the existence of primal-dual feasible points satisfying strictly complementary conditions for Isermann’s dual problem. Also, we give an alternative proof to establish necessary conditions for weakly efficient solutions of multiple objective programs, assuming the Kuhn–Tucker (KT) constraint qualification. We also provide a new condition to ensure the KT constraint qualification.     

目标属性为真假值函数的多目标决策问题解的存在性与求解方法

本文建立了一类目标属性为真假值函数的多目标决策模型，定义了相应有效解、弱有效解与最优解．并在此基础上，对有效解与弱有效解的存在性、求解等问题进行了探讨．     

Steepest ascent algorithms for nonconical multiple objective programming

A steepest ascent family of algorithms suitable for the direct solution of continuous variable unconstrained nonconical multiple objective programming problems is introduced. Nonconical multiple objective problems, unlike standard (conical) vector optimization problems, cannot be easily solved by examining related single objective problems. The concept of a direction of steepest ascent is generalized to the multiple objective context and the question of algorithmic convergence is treated. A computational example involving a nonconical unanimity order is given.     

双目标规划问题的像集与求解

经济管理的决策目标往往与成本、收益相关,双目标规划在经济管理中具有广泛应用.然而,尚缺乏成熟的算法确定双目标规划问题的全部解.给出双目标规划问题像集的一般性确定法,以求其解,为研究目的所在.具体而言,构造一个带等式约束的单目标规划问题,以确定双目标规划问题像集之部分边界,并借助拉格朗日乘子符号判断其单调性,据此确定原问题的帕累托解与弱帕累托解.这相当于提供了一个求解双目标规划问题的一般性框架.     

Constructing a quasi-concave quadratic objective function from interviewing a decision maker

A model for constructing quadratic objective functions (=utility functions) from interviewing a decision maker is considered. The interview is designed to guarantee a unique non-trivial output of the model and to enable estimating both cardinal and ordinal utility, depending on interview scenarios. The model is provided with operational restrictions for the monotonicity of the objective function (=either only growth, or only decrease in every variable) and its quasi-concavity (=convexity of the associated preference). Thereby constructing a monotonic quasi-concave quadratic objective function is reduced to a problem of non-linear programming. To support interactive editing of a quadratic objective function, the stability of the model (the continuous dependence of the output ordinal preference on the input data) is proved. In illustration, we construct a quadratic objective function of ski station customers. Then it is used to adjust prices of 10 ski stations in the south of Stuttgart.     

Building models of values

To achieve a set of objectives, whether they are made explicit or not, is the entire intent of decisionmaking. When they are explicitly stated, the objectives are often quantified with an objective function. Because of its critical role for decisionmaking, the objective function should be developed from first principles, sound logic, reasoned judgments, and carefully acquired consistent data. Unfortunately, in practice many objective functions are hastily chosen froma process that can at best be described as arbitrary. This paper presents a better alternative for developing the objective function, namely to construct it from a quality modelling effort.     

基于相对目标接近度的多目标决策方法及其应用

对多目标决策问题 ,引进多目标决策问题的理想点、多目标决策问题的负理想点和任意可行解对应的的目标向量的概念 ;然后将多目标决策问题的理想点、多目标决策问题的负理想点和任意可行解对应的的目标向量标准化 ;再利用 AHP法计算目标函数的权重向量 ;考虑权重后 ,定义任意可行解对应的目标向量的标准化向量到理想点的标准化向量 (和负理想点的标准化向量 )的加权距离 ,从而引进目标向量与理想点的相对目标接近度概念 ,进而提出了一种基于相对目标接近度的多目标决策方法 .并应用该方法对投资组合问题进行决策     

A computer graphics-based Decision Support System for multiple objective linear programming

In this paper we propose a computer-graphics based Decision Support System for multiple objective linear programming that builds on the VIG system (Visual Interactive Goal programming). The essential part of the VIG system is Pareto Race, a dynamic and visual approach for exploring the efficient frontier of a multiple objective linear programming problem. Our objective is to extend Pareto Race to large-scale multiple objective linear programming. The approach works with any efficient solutions that are in general not extreme point solutions. Interactive use of computer graphics plays a central role. The approach, the underlying theory, and an illustrative example are described.     

Pareto quantity flexibility contracts for a supply chain under multiple objectives

We analyse a decentralized supply chain consisting of a supplier and a retailer. The terms of trade between the two agents are specified by a quantity flexibility (QF) contract. We first identify the Pareto QF contracts for the supply chain where each agent adopts a satisficing objective, that is, to maximize the probability of achieving his/her predetermined target profit. It is shown that to coordinate such a supply chain, QF contracts have to degenerate into wholesale price (WP) contracts. This provides an additional justification for the popularity of WP contracts besides their simplicities and lower administration costs. Next, we consider the supply chain where each agent adopts multiple objectives, namely the satisficing objective and the objective of expected profit maximization (EPM). It is shown that there always exist QF contracts that coordinate the supply chain under the objective of EPM and are simultaneously Pareto optimal for the satisficing objective.     

A computational study of an objective hyperplane search heuristic for the general integer linear programming problem

The paper describes an objective function hyperplane search heuristic for solving the general all-integer linear programming problem (ILP). The algorithm searches a series of objective function hyperplanes and the search over any given hyperplane is formulated as a bounded knapsack problem. Theory developed for combinations of the objective function and problem constraints is used to guide the search. We evaluate the algorithm's performance on a class of ILP problems to assess the areas of effectiveness.