An Ellipsoidal Branch and Bound Algorithm for Global Optimization

2009 ◽  
Vol 20 (2) ◽  
pp. 740-758 ◽  
Author(s):  
William W. Hager ◽  
Dzung T. Phan
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Branch And Bound Algorithm

An interval branch and bound algorithm for global optimization of a multiperiod pricing model

1995 ◽  
Vol 22 (7) ◽  
pp. 681-687 ◽  
Author(s):  
M.A. Venkataramanan ◽  
A.V. Cabot ◽  
W.L. Winston
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Branch And Bound Algorithm ◽  
Pricing Model ◽  
Interval Branch And Bound

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.

scholarly journals A branch and bound algorithm for the global optimization of Hessian Lipschitz continuous functions

2012 ◽  
Vol 56 (4) ◽  
pp. 1791-1815 ◽  
Author(s):  
Jaroslav M. Fowkes ◽  
Nicholas I. M. Gould ◽  
Chris L. Farmer
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Continuous Functions ◽  
Branch And Bound Algorithm ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous

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.

scholarly journals INFLUENCE OF LIPSCHITZ BOUNDS ON THE SPEED OF GLOBAL OPTIMIZATION

2012 ◽  
Vol 18 (1) ◽  
pp. 54-66 ◽  
Author(s):  
Remigijus Paulavičius ◽  
Julius Žilinskas
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Lipschitz Function ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Branch And Bound Algorithm ◽  
Test Problems ◽  
Global Optimization Methods

Global optimization methods based on Lipschitz bounds have been analyzed and applied widely to solve various optimization problems. In this paper a bound for Lipschitz function is proposed, which is computed using function values at the vertices of a simplex and the radius of the circumscribed sphere. The efficiency of a branch and bound algorithm with proposed bound and combinations of bounds is evaluated experimentally while solving a number of multidimensional test problems for global optimization. The influence of different bounds on the performance of a branch and bound algorithm has been investigated.

scholarly journals ANALYSIS OF DIFFERENT NORMS AND CORRESPONDING LIPSCHITZ CONSTANTS FOR GLOBAL OPTIMIZATION

2006 ◽  
Vol 12 (4) ◽  
pp. 301-306 ◽  
Author(s):  
Remigijus Paulavičius ◽  
Julius Žilinskas
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Lipschitz Constant ◽  
Branch And Bound Algorithm ◽  
Test Functions ◽  
Lipschitz Optimization ◽  
Lipschitz Constants

The paper discusses how the used norm and corresponding Lipschitz constant influence the speed of algorithms for global optimization. For this reason Lipschitz constants corresponding to different norms were estimated. Different test functions for global optimization were solved using branch‐and‐bound algorithm for Lipschitz optimization with different norms. Experiments have shown that the best results are achieved when combination of extreme (infinite and first) and sometimes Euclidean norms is used.

Combination of two underestimators for univariate global optimization

2018 ◽  
Vol 52 (1) ◽  
pp. 177-186
Author(s):  
Mohand Ouanes ◽  
Mohammed Chebbah ◽  
Ahmed Zidna
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Optimization Problems ◽  
Branch And Bound Algorithm ◽  
Test Problems

In this work, we propose a new underestimator in branch and bound algorithm for solving univariate global optimization problems. The new underestimator is a combination of two underestimators, the classical one used in αBB method (see Androulakis et al. [J. Glob. Optim. 7 (1995) 337–3637]) and the quadratic underestimator developed in Hoai An and Ouanes [RAIRO: OR 40 (2006) 285–302]. We show that the new underestimator is tighter than the two underestimators. A convex/concave test is used to accelerate the convergence of the proposed algorithm. The convergence of our algorithm is shown and a set of test problems given in Casado et al. [J. Glob. Optim. 25 (2003) 345–362] are solved efficiently.

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