scholarly journals Global Optimization for a Class of Nonlinear Sum of Ratios Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Li Jin ◽  
Xue-Ping Hou
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Important Application ◽  
Global Optimization Algorithm ◽  
Numerical Examples ◽  
Sum Of Ratios ◽  
Financial Optimization ◽  
Decision Variables ◽  
Branch And Bound Search ◽  
Affine Functions

We present a branch and bound algorithm for globally solving the sum of ratios problem. In this problem, each term in the objective function is a ratio of two functions which are the sums of the absolute values of affine functions with coefficients. This problem has an important application in financial optimization, but the global optimization algorithm for this problem is still rare in the literature so far. In the algorithm we presented, 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 this problem belong. Convergence of the algorithm is shown. At last, some numerical examples are given to vindicate our conclusions.

scholarly journals A Global Optimization Algorithm for Sum of Linear Ratios Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yuelin Gao ◽  
Siqiao Jin
Keyword(s):  
Global Optimization ◽  
Lower Bound ◽  
Programming Problem ◽  
Branch And Bound ◽  
Numerical Experiments ◽  
Original Problem ◽  
Bilinear Programming ◽  
Global Optimization Algorithm ◽  
Product Function ◽  
Optimal Value

We equivalently transform the sum of linear ratios programming problem into bilinear programming problem, then by using the linear characteristics of convex envelope and concave envelope of double variables product function, linear relaxation programming of the bilinear programming problem is given, which can determine the lower bound of the optimal value of original problem. Therefore, a branch and bound algorithm for solving sum of linear ratios programming problem is put forward, and the convergence of the algorithm is proved. Numerical experiments are reported to show the effectiveness of the proposed algorithm.

scholarly journals Global Optimization for Solving Linear Multiplicative Programming Based on a New Linearization Method

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Chun-Feng Wang ◽  
Yan-Qin Bai
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Optimization Algorithm ◽  
Linearization Method ◽  
Branch And Bound Algorithm ◽  
Linear Relaxation ◽  
Global Optimization Algorithm ◽  
Multiplicative Programming ◽  
Linear Multiplicative Programming ◽  
Pruning Techniques

This paper presents a new global optimization algorithm for solving a class of linear multiplicative programming (LMP) problem. First, a new linear relaxation technique is proposed. Then, to improve the convergence speed of our algorithm, two pruning techniques are presented. Finally, a branch and bound algorithm is developed for solving the LMP problem. The convergence of this algorithm is proved, and some experiments are reported to illustrate the feasibility and efficiency of this algorithm.

GLOBAL OPTIMIZATION USING THE BRANCH‐AND‐BOUND ALGORITHM WITH A COMBINATION OF LIPSCHITZ BOUNDS OVER SIMPLICES

2009 ◽  
Vol 15 (2) ◽  
pp. 310-325 ◽  
Author(s):  
Remigijus Paulavičius ◽  
Julius Žilinskas
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Optimization Problems ◽  
The Other ◽  
Branch And Bound Algorithm ◽  
Global Optimization Algorithm ◽  
Lipschitz Optimization ◽  
Branch And Bound Algorithms ◽  
Simple Bounds ◽  
Branch And Bound Techniques

Many problems in economy may be formulated as global optimization problems. Most numerically promising methods for solution of multivariate unconstrained Lipschitz optimization problems of dimension greater than 2 use rectangular or simplicial branch‐and‐bound techniques with computationally cheap, but rather crude lower bounds. The proposed branch‐and‐bound algorithm with simplicial partitions for global optimization uses a combination of 2 types of Lipschitz bounds. One is an improved Lipschitz bound with the first norm. The other is a combination of simple bounds with different norms. The efficiency of the proposed global optimization algorithm is evaluated experimentally and compared with the results of other well‐known algorithms. The proposed algorithm often outperforms the comparable branch‐and‐bound algorithms. Santrauka Daug įvairių ekonomikos uždavinių yra formuluojami kaip globaliojo optimizavimo uždaviniai. Didžioji dalis Lipšico globaliojo optimizavimo metodų, tinkamų spręsti didesnės dimensijos, t. y. n > 2, uždavinius, naudoja stačiakampį arba simpleksinį šakų ir rėžių metodus bei paprastesnius rėžius. Šiame darbe pasiūlytas simpleksinis šakų ir rėžių algoritmas, naudojantis dviejų tipų viršutinių rėžių junginį. Pirmasis yra pagerintas rėžis su pirmąja norma, kitas – trijų paprastesnių rėžių su skirtingomis normomis junginys. Gautieji eksperimentiniai pasiūlyto algoritmo rezultatai yra palyginti su kitų gerai žinomų Lipšico optimizavimo algoritmų rezultatais.

IMPROVED LIPSCHITZ BOUNDS WITH THE FIRST NORM FOR FUNCTION VALUES OVER MULTIDIMENSIONAL SIMPLEX

2008 ◽  
Vol 13 (4) ◽  
pp. 553-563 ◽  
Author(s):  
Remigijus Paulavičius ◽  
Julius Žilinskas
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Lipschitz Condition ◽  
Optimization Algorithm ◽  
Upper Bound ◽  
Linear Equations ◽  
Branch And Bound Algorithm ◽  
Maximum Point ◽  
System Of Linear Equations ◽  
Global Optimization Algorithm

A branch and bound algorithm for global optimization is proposed, where the maximum of an upper bounding function based on Lipschitz condition and the first norm over a simplex is used as the upper bound of function. In this case the graph of bounding function is intersection of n‐dimensional pyramids and its maximum point is found solving a system of linear equations. The efficiency of the proposed global optimization algorithm is evaluated experimentally.

Optimization Algorithm for Solving a Kind of Mathematical Problems

2010 ◽  
Vol 44-47 ◽  
pp. 3423-3426 ◽  
Author(s):  
Hong Wei Jiao ◽  
Kun Li
Keyword(s):  
Global Optimization ◽  
Optimization Algorithm ◽  
Original Problem ◽  
Mathematical Problem ◽  
Relaxation Method ◽  
Linear Relaxation ◽  
Global Optimization Algorithm ◽  
Equivalent Problem ◽  
Numerical Examples ◽  
Mathematical Problems

In this paper, we develop an algorithm to globally solve a kind of mathematical problem. Firstly, by utilizing equivalent problem and linear relaxation method, a linear relaxation programming of original problem is established. Secondly, by using branch and bound technique, a determined global optimization algorithm is proposed for solving equivalent problem. Finally, the convergence of the proposed algorithm is proven and numerical examples showed that the presented algorithm is feasible to solve the kind of mathematical problems.

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