Linear multiplicative programming

An algorithm for solving a linear multiplicative programming problem (referred to as LMP) is proposed. LMP minimizes the product of two linear functions subject to general linear constraints. The product of two linear functions is a typical non-convex function, so that it can have multiple local minima. It is shown, however, that LMP can be solved efficiently by the combination of the parametric simplex method and any standard convex minimization procedure. The computational results indicate that the amount of computation is not much different from that of solving linear programs of the same size. In addition, the method proposed for LMP can be extended to a convex multiplicative programming problem (CMP), which minimizes the product of two convex functions under convex constraints.     

Parametric simplex algorithms for solving a special class of nonconvex minimization problems

It is shown that parametric linear programming algorithms work efficiently for a class of nonconvex quadratic programming problems called generalized linear multiplicative programming problems, whose objective function is the sum of a linear function and a product of two linear functions. Also, it is shown that the global minimum of the sum of the two linear fractional functions over a polytope can be obtained by a similar algorithm. Our numerical experiments reveal that these problems can be solved in much the same computational time as that of solving associated linear programs. Furthermore, we will show that the same approach can be extended to a more general class of nonconvex quadratic programming problems.     

Multiplicative Programming Problems: Analysis and Efficient Point Search Heuristic

Multiplicative programming problems are difficult global optimization problems known to be NP-hard. At the same time, these problems have some important applications in engineering, finance, economics, and other fields. This article has two purposes. The first is to present an analysis that shows several relationships between concave multiplicative programs and concave minimization problems, and between concave multiplicative programs and certain multiple-objective mathematical programs. The second purpose is to propose and report computational results for a heuristic efficient-point search algorithm that we have designed for use on linear multiplicative programming problems. To our knowledge, this is the first heuristic algorithm of its type. The theoretical and algorithmic results given in the article offer some potentially important new avenues for analyzing and solving multiplicative programming problems of various types.     

Solving generalized convex multiobjective programming problems by a normal direction method

We use normal directions of the outcome set to develop a method of outer approximation for solving generalized convex multiobjective programming problems. We prove the convergence of the method and report some computational experiments. As an application, we obtain an algorithm to solve an associated multiplicative problem over a convex constraint set.     

Global minimization of a generalized linear multiplicative programming

This article presents a simplicial branch and bound algorithm for globally solving generalized linear multiplicative programming problem (GLMP). Since this problem does not seem to have been studied previously, the algorithm is apparently the first algorithm to be proposed for solving such problem. In this algorithm, a well known simplicial subdivision is used in the branching procedure and the bound estimation is performed by solving certain linear programs. Convergence of this algorithm is established, and some experiments are reported to show the feasibility of the proposed algorithm.     

A practicable branch-and-bound algorithm for globally solving linear multiplicative programming

To globally solve linear multiplicative programming problem (LMP), this paper presents a practicable branch-and-bound method based on the framework of branch-and-bound algorithm. In this method, a new linear relaxation technique is proposed firstly. Then, the branch-and-bound algorithm is developed for solving problem LMP. The proposed algorithm is proven that it is convergent to the global minimum by means of the subsequent solutions of a series of linear programming problems. Some experiments are reported to show the feasibility and efficiency of this algorithm.     

A branch-and-cut algorithm for the equicut problem

We describe an algorithm for solving the equicut problem on complete graphs. The core of the algorithm is a cutting-plane procedure that exploits a subset of the linear inequalities defining the convex hull of the incidence vectors of the edge sets that define an equicut. The cuts are generated by several separation procedures that will be described in the paper. Whenever the cutting-plane procedure does not terminate with an optimal solution, the algorithm uses a branch-and-cut strategy. We also describe the implementation of the algorithm and the interface with the LP solver. Finally, we report on computational results on dense instances with sizes up to 100 nodes.     

An improved sequential quadratic programming algorithm for solving general nonlinear programming problems

In this paper, a class of general nonlinear programming problems with inequality and equality constraints is discussed. Firstly, the original problem is transformed into an associated simpler equivalent problem with only inequality constraints. Then, inspired by the ideals of the sequential quadratic programming (SQP) method and the method of system of linear equations (SLE), a new type of SQP algorithm for solving the original problem is proposed. At each iteration, the search direction is generated by the combination of two directions, which are obtained by solving an always feasible quadratic programming (QP) subproblem and a SLE, respectively. Moreover, in order to overcome the Maratos effect, the higher-order correction direction is obtained by solving another SLE. The two SLEs have the same coefficient matrices, and we only need to solve the one of them after a finite number of iterations. By a new line search technique, the proposed algorithm possesses global and superlinear convergence under some suitable assumptions without the strict complementarity. Finally, some comparative numerical results are reported to show that the proposed algorithm is effective and promising.     

一类线性乘积规划问题的分支定界缩减方法

提出了求解一类线性乘积规划问题的分支定界缩减方法, 并证明了算法的收敛性.在这个方法中, 利用两个变量乘积的凸包络技术, 给出了目标函数与约束函数中乘积的下界, 由此确定原问题的一个松弛凸规划, 从而找到原问题全局最优值的下界和可行解. 为了加快所提算法的收敛速度, 使用了超矩形的缩减策略. 数值结果表明所提出的算法是可行的.     

一类多乘积规划的加速算法

针对一类多乘积规划问题(MP),给出一个加速算法.首先导出一个与(MP)等价的逆凸问题(RCP),然后构造问题(RCP)的线性松弛化问题.算法的主要特点是提出了两个加速技巧,这些技巧可以用于改善算法的收敛速度.数值算例表明算法是可行的.     

Implementing the simplex method as a cutting-plane method, with a view to regularization

We show that the simplex method can be interpreted as a cutting-plane method, assuming that a special pricing rule is used. This approach is motivated by the recent success of the cutting-plane method in the solution of special stochastic programming problems. We focus on the special linear programming problem of finding the largest ball that fits into a given polyhedron. In a computational study we demonstrate that ball-fitting problems have such special characteristics which indicate their utility in regularization schemes.     

稀疏线性规划研究

稀疏线性规划在金融计算、工业生产、装配调度等领域应用十分广泛.本文首先给出稀疏线性规划问题的一般模型并证明问题是NP困难问题;其次采用交替方向乘子法(ADMM)求解该问题;最后证明了算法在近似问题上的收敛性.数值实验表明,算法在大规模数值算例上的表现优于已有的混合遗传算法;同时通过对金融实例的计算验证了算法及模型在稀疏投资组合问题上的有效性.     

Two-variable linear programming in parallel

Two-variable linear programming is a fundamental problem in computational geometry. Sequentially, this problem was solved optimally in linear time by Megiddo and Dyer using the elegant prune-and-search technique. In parallel, the previously best known deterministic algorithm on the EREW PRAM for this problem takes O(lognloglogn) time and O(n) work. In this paper, we present a faster parallel deterministic two-variable linear programming algorithm, which takes O(lognlog^*n) time and O(n) work on the EREW PRAM. Our algorithm is based on an interesting parallel prune-and-search technique, and makes use of new geometric observations which can be viewed as generalizations of those used by Megiddo and Dyer's sequential algorithms. Our parallel prune-and-search technique also leads to efficient EREW PRAM algorithm for the weighted selection problem, and is likely to be useful in solving other problems.     

Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming problems

Many global optimization approaches for solving signomial geometric programming problems are based on transformation techniques and piecewise linear approximations of the inverse transformations. Since using numerous break points in the linearization process leads to a significant increase in the computational burden for solving the reformulated problem, this study integrates the range reduction techniques in a global optimization algorithm for signomial geometric programming to improve computational efficiency. In the proposed algorithm, the non-convex geometric programming problem is first converted into a convex mixed-integer nonlinear programming problem by convexification and piecewise linearization techniques. Then, an optimization-based approach is used to reduce the range of each variable. Tightening variable bounds iteratively allows the proposed method to reach an approximate solution within an acceptable error by using fewer break points in the linearization process, therefore decreasing the required CPU time. Several numerical experiments are presented to demonstrate the advantages of the proposed method in terms of both computational efficiency and solution quality.     

求解具有确定性线性不等式约束的有补偿随机规划问题的近似ε——容许方向法

1引言形如线性不等式确定性约束的有补偿随机规划问题     

The zero-one knapsack problem with equality constraint

The zero-one knapsack problem is a linear zero-one programming problem with a single inequality constraint. This problem has been extensively studied and many applications and efficient algorithms have been published. In this paper we consider a similar problem, one with an equality instead of the inequality constraint. By replacing the equality by two inequalities one of which is placed in the economic function, a Lagrangean relaxation of the problem is obtained. The relation between the relaxed problem and the original problem is examined and it is shown how the optimal value of the relaxed problem varies with increasing values of the Lagrangean multiplier. Using these results an algorithm for solving the problem is proposed.The paper concludes with a discussion of computational experience.     

Connectivity-and-hop-constrained design of electricity distribution networks

This paper addresses the problem of designing the configuration of an interconnected electricity distribution network, so as to maximize the minimum power margin over the feeders. In addition to the limitation of feeder power capacity, the distance (as hop count) between any customer and its allocated feeder is also limited for preventing power losses and voltage drops. Feasibility conditions are studied and a complexity analysis is performed before introducing a heuristic algorithm and two integer linear programming formulations for addressing the problem. A cutting-plane algorithm relying on the generation of two classes of cuts for enforcing connectivity and distance requirements respectively is proposed for solving the second integer linear programming formulation. All the approaches are then compared on a set of 190 instances before discussing their performances.     

线性分式多乘积规划问题的完全多项式时间近似算法

本文针对线性分式多乘积规划问题,通过Charnes-Cooper转化将原问题转化为一个等价问题,借助此等价问题提出一个获得原问题全局近似最优解的算法,最终证明了算法的收敛性,且提供了算法运算时间的理论分析.     

求一类非凸规划问题全局解的确定性算法

本文对一类非凸规划问题(NP)给出一确定性全局优化算法.这类问题包括:在非凸的可行域上极小化有限个带指数的线性函数乘积的和与差,广义线性多乘积规划,多项式规划等.通过利用等价问题和线性化技巧提出的算法收敛到问题(NP)的全局极小.