Solving geometric programs using GRG: Results and comparisons

This paper describes the performance of a general-purpose GRG code for nonlinear programming in solving geometric programs. The main conclusions drawn from the experiments reported are: (i) GRG competes well with special-purpose geometric programming codes in solving geometric programs; and (ii) standard time, as defined by Colville, is an inadequate means of compensating for different computing environments while comparing optimization algorithms.This research was partially supported by the Office of Naval Research under Contracts Nos. N00014-75-C-0267 and N00014-75-C-0865, the US Energy Research and Development Administration, Contract No. E(04-3)-326 PA-18, and the National Science Foundation, Grant No. DCR75-04544 at Stanford University; and by the Office of Naval Research under Contract No. N00014-75-C-0240, and the National Science Foundation, Grant No. SOC74-23808, at Case Western Reserve University.     

Global optimization for generalized geometric programming problems with discrete variables

Generalized geometric programming (GGP) problems occur frequently in engineering design and management, but most existing methods for solving GGP actually only consider continuous variables. This article presents a new branch-and-bound algorithm for globally solving GGP problems with discrete variables. For minimizing the problem, an equivalent monotonic optimization problem (P) with discrete variables is presented by exploiting the special structure of GGP. In the algorithm, the lower bounds are computed by solving ordinary linear programming problems that are derived via a linearization technique. In contrast to pure branch-and-bound methods, the algorithm can perform a domain reduction cut per iteration by using the monotonicity of problem (P), which can suppress the rapid growth of branching tree in the branch-and-bound search so that the performance of the algorithm is further improved. Computational results for several sample examples and small randomly generated problems are reported to vindicate our conclusions.     

A global optimization using linear relaxation for generalized geometric programming

Many local optimal solution methods have been developed for solving generalized geometric programming (GGP). But up to now, less work has been devoted to solving global optimization of (GGP) problem due to the inherent difficulty. This paper considers the global minimum of (GGP) problems. By utilizing an exponential variable transformation and the inherent property of the exponential function and some other techniques the initial nonlinear and nonconvex (GGP) problem is reduced to a sequence of linear programming problems. The proposed algorithm is proven that it is convergent to the global minimum through the solutions of a series of linear programming problems. Test results indicate that the proposed algorithm is extremely robust and can be used successfully to solve the global minimum of (GGP) on a microcomputer.     

Solving Posynomial Geometric Programming Problems via Generalized Linear Programming

This paper revisits an efficient procedure for solving posynomial geometric programming (GP) problems, which was initially developed by Avriel et al. The procedure, which used the concept of condensation, was embedded within an algorithm for the more general (signomial) GP problem. It is shown here that a computationally equivalent dual-based algorithm may be independently derived based on some more recent work where the GP primal-dual pair was reformulated as a set of inexact linear programs. The constraint structure of the reformulation provides insight into why the algorithm is successful in avoiding all of the computational problems traditionally associated with dual-based algorithms. Test results indicate that the algorithm can be used to successfully solve large-scale geometric programming problems on a desktop computer.     

一个新的线性规划无人工变量算法

本文研究了线性规划的求解问题.利用对偶转化的方法,获得了一个计算效率高的新的无人工变量通用算法.该新算法比最近提出的无人工变量算法push-to-pull算法效率更高.     

On generalized geometric programming problems with non-positive variables

Generalized geometric programming (GGP) problems occur frequently in engineering design and management. Some exponential-based decomposition methods have been developed for solving global optimization of GGP problems. However, the use of logarithmic/exponential transformations restricts these methods to handle the problems with strictly positive variables. This paper proposes a technique for treating non-positive variables with integer powers in GGP problems. By means of variable transformation, the GGP problem with non-positive variables can be equivalently solved with another one having positive variables. In addition, we present some computationally efficient convexification rules for signomial terms to enhance the efficiency of the optimization approach. Numerical examples are presented to demonstrate the usefulness of the proposed method in GGP problems with non-positive variables.     

求广义几何规划全局最优解的线性化方法

对广义几何规划问题(GGP)提出了一个确定型全局优化算法，这类优化问题能广泛应用于工程设计和非线性系统的鲁棒稳定性分析等实际问题中，使用指数变换及对目标函数和约束函数的线性下界估计，建立了GGP的松弛线性规划(RLP)，通过对RLP可行域的细分以及一系列RLP的求解过程，从理论上证明了算法能收敛到GGP的全局最优解，对一个化学工程设计问题应用本文算法，数值实验表明本文方法是可行的。     

广义几何规划的全局优化算法

对许多工程设计中常用的广义几何规划问题(GGP)提出一种确定性全局优化算法,该算法利用目标和约束函数的线性下界估计,建立GGP的松弛线性规划(RLP),从而将原来非凸问题(GGP)的求解过程转化为求解一系列线性规划问题(RLP).通过可行域的连续细分以及一系列线性规划的解,提出的分枝定界算法收敛到GGP的全局最优解,且数值例子表明了算法的可行性.     

线性规划基线算法群部分算法计算实验

本文简要介绍了基线算法的构思原理 ,对其中部分算法的具体实现形式进行了测试 ,并与单纯形法进行了比较 .理论和数值结果表明基线算法是一种可靠、有效的算法 .作者还给出了一些对其它算法在计算实践中的看法     

Treating Free Variables in Generalized Geometric Global Optimization Programs

Generalized geometric programming (GGP) problems occur frequently in engineering design and management. Recently, some exponential-based decomposition methods [Maranas and Floudas, 1997,Computers and Chemical Engineering 21(4), 351–370; Floudas et al., 1999 , Handbook of Test Problems in Local and Global Optimization, Kluwer Academic Publishers, Boston, pp. 5–105; Floudas, 2000 Deterministic Global Optimizaion: Theory, Methods and Application, Kluwer Academic Publishers, Boston, pp. 257–306] have been developed for GGP problems. These methods can only handle problems with positive variables, and are incapable of solving more general GGP problems. This study proposes a technique for treating free (i.e., positive, zero or negative) variables in GGP problems. Computationally effective convexification rules are also provided for signomial terms with three variables.     

广义几何规划的压缩共轭梯度路径非单调信赖域算法(英文)

In this paper,on the basis of making full use of the characteristics of unconstrained generalized geometric programming（GGP）,we establish a nonmonotonic trust region algorithm via the conjugate path for solving unconstrained GGP problem.A new type of condensation problem is presented,then a particular conjugate path is constructed for the problem,along which we get the approximate solution of the problem by nonmonotonic trust region algorithm,and further prove that the algorithm has global convergence and quadratic convergence properties.     

A numerical investigation of rank-two ellipsoid algorithms for nonlinear programming

We study the performance of some rank-two ellipsoid algorithms when used to solve nonlinear programming problems. Experiments are reported which show that the rank-two algorithms studied are slightly less efficient than the usual rank-one (center-cut) algorithm. Some results are also presented concerning the growth of ellipsoid asphericity in rank-one and rank-two algorithms.     

求解中大规模复杂凸二次整数规划问题的新型分枝定界算法

针对现有分枝定界算法在求解高维复杂二次整数规划问题时所存在的诸多不足，本文通过充分挖掘二次整数规划问题的结构特性来设计选择分枝变量与分枝方向的新方法，并将HNF算法与原问题松弛问题的求解相结合来寻求较好的初始整数可行解，由此导出可用于有效求解中大规模复杂二次整数规划问题的改进型分枝定界算法．数值试验结果表明所给算法大大改进了已有相关的分枝定界算法，并具有较好的稳定性与广泛的适用性．     

一种基于进化算法的非线性两层规划的快速全局优化方法

由于非线性两层规划具有非凸性、NP-难等计算困难，高效的算法并不多见。本文设计了一种新的进化算法，基于此进化算法提出了求解带有一重或多重下层的非线性两层规划的高效算法。该算法充分利用两层规划的结构特点。最后，给出了六个不同类型的算例，数值结果表明，本算法是快速和有效的。     

An iterative approach to solve multiobjective linear fractional programming problems

This paper suggests an iterative parametric approach for solving multiobjective linear fractional programming (MOLFP) problems which only uses linear programming to obtain efficient solutions and always converges to an efficient solution. A numerical example shows that this approach performs better than some existing algorithms. Randomly generated MOLFP problems are also solved to demonstrate the performance of new introduced algorithm.     

Posynomial geometric programming as a special case of semi-infinite linear programming

This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual problem is equivalent to a generalized linear program. Furthermore, the duality results that are available for the traditionally defined primal-dual pair are readily obtained from the duality theory for semi-infinite linear programs. It is also shown that two efficient algorithms (one primal based and the other dual based) for geometric programming actually operate on the semi-infinite linear program and its dual.     

An algorithm of sequential systems of linear equations for nonlinear optimization problems with arbitrary initial point

For current sequential quadratic programming (SQP) type algorithms, there exist two problems: (i) in order to obtain a search direction, one must solve one or more quadratic programming subproblems per iteration, and the computation amount of this algorithm is very large. So they are not suitable for the large-scale problems; (ii) the SQP algorithms require that the related quadratic programming subproblems be solvable per iteration, but it is difficult to be satisfied. By using ε-active set procedure with a special penalty function as the merit function, a new algorithm of sequential systems of linear equations for general nonlinear optimization problems with arbitrary initial point is presented. This new algorithm only needs to solve three systems of linear equations having the same coefficient matrix per iteration, and has global convergence and local superlinear convergence. To some extent, the new algorithm can overcome the shortcomings of the SQP algorithms mentioned above. Project partly supported by the National Natural Science Foundation of China and Tianyuan Foundation of China.     

Two-storage inventory model with lot-size dependent fuzzy lead-time under possibility constraints via genetic algorithm

Multi-item inventory models with stock dependent demand and two storage facilities are developed in a fuzzy environment where processing time of each unit is fuzzy and the processing time of a lot is correlated with its size. These are order-quantity reorder-point models with back-ordering if required. Here possibility and crisp constraints on investment and capacity of the small storehouse respectively are considered. The models are formulated as fuzzy chance constrained programming problem and is solved via generalized reduced gradient (GRG) technique when crisp equivalent of the constraints are available. A genetic algorithm (GA) is developed based on fuzzy simulation and entropy where region of search space gradually decreases to a small neighborhood of the optima and it is used to solve the models whenever the equivalent crisp form of the constraint is not available. The models are illustrated with some numerical examples and some sensitivity analyses have been done. For some particular cases results observed via GRG and GA are compared.     

A penalty treatment of equality constraints in generalized geometric programming

A technique is described for solving generalized geometric programs whose constraints include one or more strict equalities. The algorithm solves a sequence of penalized geometric programs; the penalty functions are derived from the arithmetic-geometric inequality as condensed posynomials. Two examples serve to illustrate the idea.The authors appreciate the use of the program GGP provided by Professor R. S. Dembo.     

Progressive hedging and tabu search applied to mixed integer (0,1) multistage stochastic programming

Many problems faced by decision makers are characterized by a multistage decision process with uncertainty about the future and some decisions constrained to take on values of either zero or one (for example, either open a facility at a location or do not open it). Although some mathematical theory exists concerning such problems, no general-purpose algorithms have been available to address them. In this article, we introduce the first implementation of general purpose methods for finding good solutions to multistage, stochastic mixed-integer (0, 1) programming problems. The solution method makes use of Rockafellar and Wets' progressive hedging algorithm that averages solutions rather than data. Solutions to the induced quadratic (0,1) mixed-integer subproblems are obtained using a tabu search algorithm. We introduce the notion of integer convergence for progressive hedging. Computational experiments verify that the method is effective. The software that we have developed reads standard (SMPS) data files.