Global Maximization of a Generalized Concave Multiplicative Function

This article presents a branch-and-bound algorithm for globally solving the problem (P) of maximizing a generalized concave multiplicative function over a compact convex set. Since problem (P) does not seem to have been studied previously, the algorithm is apparently the first algorithm to be proposed for solving this problem. It works by globally solving a problem (P1) equivalent to problem (P). The branch-and-bound search undertaken by the algorithm uses rectangular partitioning and takes place in a space which typically has a much smaller dimension than the space to which the decision variables of problem (P) belong. Convergence of the algorithm is shown; computational considerations and benefits for users of the algorithm are given. A sample problem is also solved.     

A branch and bound network approach to the canonical constrained entropy problem

In this paper, we present a branch and bound algorithm for solving the constrained entropy mathematical programming problem. Unlike other methods for solving this problem, our method solves more general problems with inequality constraints. The advantage of the proposed technique is that the relaxed problem solved at each node is a singly constrained network problem. The disadvantage is that the relaxed problem has twice as many variables as the original problem. An application to regional planning is given, and an example problem is solved.     

一类特殊DC规划的对偶性

根据共轭函数和DC规划的性质，给出一类特殊DC规划的共轭对偶并讨论其对偶规划的特殊性质，然后利用该性质，把对这类特殊DC规划的求解转化为对一个凸规划的求解。     

Sensitivity method for basis inverse representation in multistage stochastic linear programming problems

A version of the simplex method for solving stochastic linear control problems is presented. The method uses a compact basis inverse representation that extensively exploits the original problem data and takes advantage of the supersparse structure of the problem. Computational experience indicates that the method is capable of solving large problems.This research was supported by Programs CPBP02.15 and RPI.02.     

Interactive TOPSIS algorithms for solving multi-level non-linear multi-objective decision-making problems

This paper extended the concept of the technique for order preference by similarity to ideal solution (TOPSIS) to develop a methodology for solving multi-level non-linear multi-objective decision-making (MLN-MODM) problems of maximization-type. Also, two new interactive algorithms are presented for the proposed TOPSIS approach for solving these types of mathematical programming problems. The first proposed interactive TOPSIS algorithm includes the membership functions of the decision variables for each level except the lower level of the multi-level problem. These satisfactory decisions are evaluated separately by solving the corresponding single-level MODM problems. The second proposed interactive TOPSIS algorithm lexicographically solves the MODM problems of the MLN-MOLP problem by taking into consideration the decisions of the MODM problems for the upper levels. To demonstrate the proposed algorithms, a numerical example is solved and compared the solutions of proposed algorithms with the solution of the interactive algorithm of Osman et al. (2003) [4]. Also, an example of an application is presented to clarify the applicability of the proposed TOPSIS algorithms in solving real world multi-level multi-objective decision-making problems.     

An MOLP based procedure for finding efficient units in DEA models

In this paper a multiple objective linear programming (MOLP) problem whose feasible region is the production possibility set with variable returns to scale is proposed. By solving this MOLP problem by multicriterion simplex method, the extreme efficient Pareto points can be obtained. Then the extreme efficient units in data envelopment analysis (DEA) with variable returns to scale, considering the specified theorems and conditions, can be obtained. Therefore, by solving the proposed MOLP problem, the non-dominant units in DEA can be found. Finally, a numerical example is provided.     

A dynamic programming algorithm for the bilevel knapsack problem

We propose an efficient dynamic programming algorithm for solving a bilevel program where the leader controls the capacity of a knapsack, and the follower solves the resulting knapsack problem. We propose new recursive rules and show how to solve the problem as a sequence of two standard knapsack problems.     

Simple solution methods for separable mixed linear and quadratic knapsack problem

Quadratic knapsack problem has a central role in integer and nonlinear optimization, which has been intensively studied due to its immediate applications in many fields and theoretical reasons. Although quadratic knapsack problem can be solved using traditional nonlinear optimization methods, specialized algorithms are much faster and more reliable than the nonlinear programming solvers. In this paper, we study a mixed linear and quadratic knapsack with a convex separable objective function subject to a single linear constraint and box constraints. We investigate the structural properties of the studied problem, and develop a simple method for solving the continuous version of the problem based on bi-section search, and then we present heuristics for solving the integer version of the problem. Numerical experiments are conducted to show the effectiveness of the proposed solution methods by comparing our methods with some state of the art linear and quadratic convex solvers.     

一类多商品设施选址问题的基于线性松弛解的启发式方法

多商品设施选址问题是众多设施选址问题中一类重要而困难的问题.在这一问题中,顾客的需求可能包含不止一种商品.对于大规模问题,成熟的商业求解器往往不能在满意的时间内找到高质量的可行解.研究了无容量限制的单货源多商品设施选址问题的一般形式,并给出了应用于此类问题的两个启发式方法.这两个方法基于原选址问题的线性规划松弛问题的最优解,分别通过求解紧问题和邻域搜索的方式给出了原问题的一个可行上界.理论分析指出所提方法可以实施于任意可行问题的实例.数值结果表明所提方法可以显著地提高求解器求解此类设施选址问题的求解效率.     

Integrating SQP and Branch-and-Bound for Mixed Integer Nonlinear Programming

This paper considers the solution of Mixed Integer Nonlinear Programming (MINLP) problems. Classical methods for the solution of MINLP problems decompose the problem by separating the nonlinear part from the integer part. This approach is largely due to the existence of packaged software for solving Nonlinear Programming (NLP) and Mixed Integer Linear Programming problems.In contrast, an integrated approach to solving MINLP problems is considered here. This new algorithm is based on branch-and-bound, but does not require the NLP problem at each node to be solved to optimality. Instead, branching is allowed after each iteration of the NLP solver. In this way, the nonlinear part of the MINLP problem is solved whilst searching the tree. The nonlinear solver that is considered in this paper is a Sequential Quadratic Programming solver.A numerical comparison of the new method with nonlinear branch-and-bound is presented and a factor of up to 3 improvement over branch-and-bound is observed.     

The bilevel knapsack problem with stochastic right-hand sides

We introduce the bilevel knapsack problem with stochastic right-hand sides, and provide necessary and sufficient conditions for the existence of an optimal solution. When the leader’s decisions can take only integer values, we present an equivalent two-stage stochastic programming reformulation with binary recourse. We develop a branch-and-cut algorithm for solving this reformulation, and a branch-and-backtrack algorithm for solving the scenario subproblems. Computational experiments indicate that our approach can solve large instances in a reasonable amount of time.     

Outcome-Space Cutting-Plane Algorithm for Linear Multiplicative Programming

This article presents an outcome-space pure cutting-plane algorithm for globally solving the linear multiplicative programming problem. The framework of the algorithm is taken from a pure cutting-plane decision set-based method developed by Horst and Tuy for solving concave minimization problems. By adapting this method to an outcome-space reformulation of the linear multiplicative programming problem, rather than applying directly the method to the original decision-set formulation, it is expected that considerable computational savings can be obtained. Also, we show how additional computational benefits might be obtained by implementing the new algorithm appropriately. To illustrate the new algorithm, we apply it to the solution of a sample problem.     

A relaxed cutting plane algorithm for solving the Vasicek-type forward interest rate model

This work considers the solution of the Vasicek-type forward interest rate model. A deterministic process is adopted to model the random behavior of interest rate variation as a deterministic perturbation. It shows that the solution of the Vasicek-type forward interest rate model can be obtained by solving a nonlinear semi-infinite programming problem. A relaxed cutting plane algorithm is then proposed for solving the resulting optimization problem. The features of the proposed method are tested using a set of real data and compared with some commonly used spline fitting methods.     

Exploiting equalities in polynomial programming

We propose a novel approach for solving polynomial programs over compact domains with equality constraints. By means of a generic transformation, we show that existing solution schemes for the, typically simpler, problem without equalities can be used to address the problem with equalities.     

Multi-objective integer programming: A general approach for generating all non-dominated solutions

In this paper we develop a general approach to generate all non-dominated solutions of the multi-objective integer programming (MOIP) Problem. Our approach, which is based on the identification of objective efficiency ranges, is an improvement over classical ε-constraint method. Objective efficiency ranges are identified by solving simpler MOIP problems with fewer objectives. We first provide the classical ε-constraint method on the bi-objective integer programming problem for the sake of completeness and comment on its efficiency. Then present our method on tri-objective integer programming problem and then extend it to the general MOIP problem with k objectives. A numerical example considering tri-objective assignment problem is also provided.     

并行技术在约束凸规划化问题的对偶算法中的应用

用 Rosen(196 1)的投影梯度的方法求解约束凸规划化问题的对偶问题 ,在计算投影梯度方向时 ,涉及求关于原始变量的最小化问题的最优解 .我们用并行梯度分布算法 (PGD)计算出这一极小化问题的近似解 ,证明近似解可以达到任何给定的精度 ,并说明当精度选取合适时 ,Rosen方法仍然是收敛的     

Dynamic Programming Method for Constrained Discrete-Time Optimal Control

A dynamic programming method is presented for solving constrained, discrete-time, optimal control problems. The method is based on an efficient algorithm for solving the subproblems of sequential quadratic programming. By using an interior-point method to accommodate inequality constraints, a modification of an existing algorithm for equality constrained problems can be used iteratively to solve the subproblems. Two test problems and two application problems are presented. The application examples include a rest-to-rest maneuver of a flexible structure and a constrained brachistochrone problem.     

An Efficient Algorithm for Min-Max Convex Semi-Infinite Programming Problems

In this article, we consider the convex min-max problem with infinite constraints. We propose an exchange method to solve the problem by using efficient inactive constraint dropping rules. There is no need to solve the maximization problem over the metric space, as the algorithm has merely to find some points in the metric space such that a certain criterion is satisfied at each iteration. Under some mild assumptions, the proposed algorithm is shown to terminate in a finite number of iterations and to provide an approximate solution to the original problem. Preliminary numerical results with the algorithm are promising. To our knowledge, this article is the first one conceived to apply explicit exchange methods for solving nonlinear semi-infinite convex min-max problems.     

求解二阶锥规划问题的VU分解方法

给出解决二阶锥规划(SOCP)问题的VU-分解方法.问题首先被转化为非线性规划,并给出相应的精确罚函数的Clarke次微分结构及VU-空间分解．在某种条件下,可以计算出一个二阶连续可微的轨道,进而得到目标函数f在其上的二阶展开．最后给出一个具有超线性收敛速度的概念型算法．     

解可分离约束双凹规划问题的一种外逼近方法[英文]

在本文中，我们提出了双凹规划问题和更一般的广义凹规划问题。我们给出了双凹规划问题的整体最优性条件，并构造了一个有限终止外逼近算法。