Global optimization algorithm for sum of generalized polynomial ratios problem

In this paper, a global optimization algorithm is proposed for solving sum of generalized polynomial ratios problem (P) which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solve the problem (P). For such problems, we present a branch and bound algorithm. In this method, by utilizing exponent transformation and new three-level linear relaxation method, a sequence of linear relaxation programming of the initial nonconvex programming problem (P) are derived which are embedded in a branch and bound algorithm. The proposed method need not introduce new variables and constraints and it is convergent to the global minimum of prime problem by means of the subsequent solutions of a series of linear programming problems. Several numerical examples in the literatures are tested to demonstrate that the proposed algorithm can systematically solve these examples to find the approximate ?-global optimum.     

A branch and bound algorithm for globally solving a class of nonconvex programming problems

A branch and bound algorithm is proposed for globally solving a class of nonconvex programming problems (NP). For minimizing the problem, linear lower bounding functions (LLBFs) of objective function and constraint functions are constructed, then a relaxation linear programming is obtained which is solved by the simplex method and which provides the lower bound of the optimal value. The proposed algorithm is convergent to the global minimum through the successive refinement of linear relaxation of the feasible region and the solutions of a series of linear programming problems. And finally the numerical experiment is reported to show the feasibility and effectiveness of the proposed algorithm.     

A column generation based decomposition algorithm for a parallel machine just-in-time scheduling problem

We propose a column generation based exact decomposition algorithm for the problem of scheduling n jobs with an unrestrictively large common due date on m identical parallel machines to minimize total weighted earliness and tardiness. We first formulate the problem as an integer program, then reformulate it, using Dantzig–Wolfe decomposition, as a set partitioning problem with side constraints. Based on this set partitioning formulation, a branch and bound exact solution algorithm is developed for the problem. In the branch and bound tree, each node is the linear relaxation problem of a set partitioning problem with side constraints. This linear relaxation problem is solved by column generation approach where columns represent partial schedules on single machines and are generated by solving two single machine subproblems. Our computational results show that this decomposition algorithm is capable of solving problems with up to 60 jobs in reasonable cpu time.     

求线性比试和问题的确定性全局优化算法

为求线性比试和问题的全局最优解,本文给出了一个分支定界算法.通过一个等价问题和一个新的线性化松弛技巧,初始的非凸规划问题归结为一系列线性规划问题的求解.借助于这一系列线性规划问题的解,算法可收敛于初始非凸规划问题的最优解.算法的计算量主要是一些线性规划问题的求解.数值算例表明算法是切实可行的.     

Global optimization for sum of geometric fractional functions

This paper presents an efficient branch and bound algorithm for globally solving sum of geometric fractional functions under geometric constraints, which arise in various practical problems. By using an equivalent transformation and a new linear relaxation technique, a linear relaxation programming problem of the equivalent problem is obtained. The proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems. Numerical results are reported to show the feasibility of our algorithm.     

求解非凸区域上凸函数比式和问题的全局优化方法

针对非凸区域上的凸函数比式和问题,给出一种求其全局最优解的确定性方法.该方法基于分支定界框架.首先通过引入变量,将原问题等价转化为d.c.规划问题,然后利用次梯度和凸包络构造松弛线性规划问题,从而将关键的估计下界问题转化为一系列线性规划问题,这些线性规划易于求解而且规模不变,更容易编程实现和应用到实际中;分支采用单纯形对分不但保证其穷举性,而且使得线性规划规模更小.理论分析和数值实验表明所提出的算法可行有效.     

Bound reduction using pairs of linear inequalities

We describe a procedure to reduce variable bounds in mixed integer nonlinear programming (MINLP) as well as mixed integer linear programming (MILP) problems. The procedure works by combining pairs of inequalities of a linear programming (LP) relaxation of the problem. This bound reduction procedure extends the feasibility based bound reduction technique on linear functions, used in MINLP and MILP. However, it can also be seen as a special case of optimality based bound reduction, a method to infer variable bounds from an LP relaxation of the problem. For an LP relaxation with m constraints and n variables, there are O(m ²) pairs of constraints, and a naïve implementation of our bound reduction scheme has complexity O(n ³) for each pair. Therefore, its overall complexity O(m ² n ³) can be prohibitive for relatively large problems. We have developed a more efficient procedure that has complexity O(m ² n ²), and embedded it in two Open-Source solvers: one for MINLP and one for MILP. We provide computational results which substantiate the usefulness of this bound reduction technique for several instances.     

Global optimization for the sum of generalized polynomial fractional functions

In this paper, a branch and bound approach is proposed for global optimization problem (P) of the sum of generalized polynomial fractional functions under generalized polynomial constraints, which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solving this problem. By utilizing an equivalent problem and some linear underestimating approximations, a linear relaxation programming problem of the equivalent form is obtained. Consequently, the initial non-convex nonlinear problem (P) is reduced to a sequence of linear programming problems through successively refining the feasible region of linear relaxation problem. The proposed algorithm is convergent to the global minimum of the primal problem by means of the solutions to a series of linear programming problems. Numerical results show that the proposed algorithm is feasible and can successfully be used to solve the present problem (P).     

带非凸二次约束的二次比式和问题的全局优化算法(英文)

对带非凸二次约束的二次比式和问题(P)给出分枝定界算法,首先将问题(P)转化为其等价问题(Q),然后利用线性化技术,建立了(Q)松弛线性规划问题(RLP),通过对(RLP)可行域的细分及求解一系列线性规划问题,不断更新(Q)的上下界,从理论上证明了算法的收敛性,数值实验表明了算法的可行性和有效性.     

解带有二次约束非凸二次规划问题的一个分枝缩减方法

在这篇论文里,有机地把外逼近方法与分枝定界技术结合起来,提出了解带有二次约束非凸二次规划问题的一个分枝缩减方法;给出了原问题的一个新的线性规划松弛,以便确定它在超矩形上全局最优值的一个下界;利用超矩形的一个深度二级剖分方法,以及超矩形的缩减和删除技术,提高算法的收敛速度;证明了在知道原问题可行点的条件下,该算法在有限步里就可以获得原问题的一个全局最优化解,并且用一个例子说明了该算法是有效的.     

An algorithm for solving the problem of convex programming with several objective functions

This work aims to establish an algorithm for solving the problem of convex programming with several objective-functions, with linear constraints. Starting from the idea of Rosen’s algorithm for solving the problem of convex programming with linear constraints, and taking into account the solution concept from multidimensional programming, represented by a program which reaches ”the best compromise”, we are extending this method in the case of multidimensional programming. The concept of direction of minimization is introduced, and a necessary and sufficient condition is given for as ∈R ⁿ direction to be a direction of minimization, according to the values of a criteria ensemble in a given point. The algorithm is interactive, and the intervention of the decident is minimal. The two numerical examples presented at the end validate the algorithm.     

Solving large single allocation p-hub problems with two or three hubs

In this paper we present an efficient approach for solving single allocation p-hub problems with two or three hubs. Two different variants of the problem are considered: the uncapacitated single allocation p-hub median problem and the p-hub allocation problem. We solve these problems using new mixed integer linear programming formulations that require fewer variables than those formerly used in the literature. The problems that we solve here are the largest single allocation problems ever solved. The numerical results presented here will demonstrate the superior performance of our mixed integer linear programs over traditional approaches for large problems. Finally we present the first mixed integer linear program for solving single allocation hub location problems that requires only O(n²) variables and O(n²) constraints that is valid for any number of hubs.     

线性比式和规划问题的输出空间分支定界算法

本文旨在针对线性比式和规划这一NP-Hard非线性规划问题提出新的全局优化算法.首先,通过引入p个辅助变量把原问题等价的转化为一个非线性规划问题,这个非线性规划问题的目标函数是乘积和的形式并给原问题增加了p个新的非线性约束,再通过构造凸凹包络的技巧对等价问题的目标函数和约束条件进行相应的线性放缩,构成等价问题的一个下界线性松弛规划问题,从而提出了一个求解原问题的分支定界算法,并证明了算法的收敛性.最后,通过数值结果比较表明所提出的算法是可行有效的.     

Absolute Value Equation Solution Via Linear Programming

By utilizing a dual complementarity property, we propose a new linear programming method for solving the NP-hard absolute value equation (AVE): Ax?|x|=b, where A is an n×n square matrix. The algorithm makes no assumptions on the AVE other than solvability and consists of solving a few linear programs, typically less than four. The algorithm was tested on 500 consecutively generated random solvable instances of the AVE with n=10, 50, 100, 500 and 1000. The algorithm solved 100 % of the test problems to an accuracy of 10^?8 by solving an average of 3.3 linear programs per AVE problem.     

Solving Convex MINLP Optimization Problems Using a Sequential Cutting Plane Algorithm

In this article we look at a new algorithm for solving convex mixed integer nonlinear programming problems. The algorithm uses an integrated approach, where a branch and bound strategy is mixed with solving nonlinear programming problems at each node of the tree. The nonlinear programming problems, at each node, are not solved to optimality, rather one iteration step is taken at each node and then branching is applied. A Sequential Cutting Plane (SCP) algorithm is used for solving the nonlinear programming problems by solving a sequence of linear programming problems. The proposed algorithm generates explicit lower bounds for the nodes in the branch and bound tree, which is a significant improvement over previous algorithms based on QP techniques. Initial numerical results indicate that the described algorithm is a competitive alternative to other existing algorithms for these types of problems.     

An O(n2) simplex algorithm for a class of linear programs with tree structure

In connection with the optimal design of centralized circuit-free networks linear 0–1 programming problems arise which are related to rooted trees. For each problem the variables correspond to the edges of a given rooted tree T. Each path from a leaf to the root of T, together with edge weights, defines a linear constraint, and a global linear objective is to be maximized. We consider relaxations of such problems where the variables are not restricted to 0 or 1 but are allowed to vary continouosly between these bounds. The values of the optimal solutions of such relaxations may be used in a branch and bound procedure for the original 0–1 problem. While in [10] a primal algorithm for these relaxations is discussed, in this paper, we deal with the dual linear program and present a version of the simplex algorithm for its solution which can be implemented to run in time O(n²). For balanced trees T this time can be reduced to O(n log n).     

Lagrangean relaxation for a lower bound to a set partitioning problem with side constraints: properties and algorithms

The problem (P) addressed here is a special set partitioning problem with two additional non-trivial constraints. A Lagrangean Relaxation (LR_u) is proposed to provide a lower bound to the optimal solution to this problem. This Lagrangean relaxation is accomplished by a subgradient optimization procedure which solves at each iteration a special 0–1 knapsack problem (KP-k). We give two procedures to solve (KP-k), namely an implicity enumeration algorithm and a column generation method. The approach is promising for it provides feasible integer solutions to the side constraints that will hopefully be optimal to most of the instances of the problem (P). Properties of the feasible solutions to (KP-k) are highlighted and it is shown that the linear programming relaxation to this problem has a worst case time bound of order O(n³).     

On solving a D.C. programming problem by a sequence of linear programs

We are dealing with a numerical method for solving the problem of minimizing a difference of two convex functions (a d.c. function) over a closed convex set in ⁿ . This algorithm combines a new prismatic branch and bound technique with polyhedral outer approximation in such a way that only linear programming problems have to be solved.Parts of this research were accomplished while the third author was visiting the University of Trier, Germany, as a fellow of the Alexander von Humboldt foundation.     

A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem

In this article, we present and validate a simplicial branch and bound duality-bounds algorithm for globally solving the linear sum-of-ratios fractional program. The algorithm computes the lower bounds called for during the branch and bound search by solving ordinary linear programming problems. These problems are derived by using Lagrangian duality theory. The algorithm applies to a wide class of linear sum-of-ratios fractional programs. Two sample problems are solved, and the potential practical and computational advantages of the algorithm are indicated.     

A Branch and Bound Algorithm for Solving Low Rank Linear Multiplicative and Fractional Programming Problems

This paper is concerned with a practical algorithm for solving low rank linear multiplicative programming problems and low rank linear fractional programming problems. The former is the minimization of the sum of the product of two linear functions while the latter is the minimization of the sum of linear fractional functions over a polytope. Both of these problems are nonconvex minimization problems with a lot of practical applications. We will show that these problems can be solved in an efficient manner by adapting a branch and bound algorithm proposed by Androulakis–Maranas–Floudas for nonconvex problems containing products of two variables. Computational experiments show that this algorithm performs much better than other reported algorithms for these class of problems.