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 Estimation of the numerical efficiency of global optimization methods

2021 ◽  
Vol 3 (134) ◽  
pp. 31-39
Author(s):  
Anatolii Kosolap
Keyword(s):  
Global Optimization ◽  
Unconstrained Optimization ◽  
Regularization Method ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Test Problems ◽  
Numerical Efficiency ◽  
Global Optimization Methods ◽  
Search Program ◽  
Conditional Optimization

Currently, test problems are used to test the effectiveness of new global optimization methods. In this article, we analyze test global optimization problems to test the numerical efficiency of methods for their solution. At present, about 200 test problems of unconditional optimization and more than 1000 problems of conditional optimization have been developed. We can find these test problems on the Internet. However, most of these test problems are not informative for testing the effectiveness of global optimization methods. The solution of test problems of conditional optimization, as a rule, has trivial solutions. This allows the parameters of the algorithms to be tuned before these solutions are obtained. In test problems of conditional optimization, the accuracy of the fulfillment of constraints is important. Often, small errors in the constraints lead to a significant change in the value of an objective function. Construction of a new package of test problems to test the numerical efficiency of global optimization methods and compare the exact quadratic regularization method with existing methods.The author suggests limiting oneself to test problems of unconstrained optimization with unknown solutions. A package of test problems of unconstrained optimization is pro-posed, which includes known test problems with unknown solutions and modifications of some test problems proposed by the author. We also propose to include in this package J. Nie polynomial functions with unknown solutions. This package of test problems will simplify the verification of the numerical effectiveness of methods. The more effective methods will be those that provide the best solutions. The paper compares existing global optimization methods with the exact quadratic regularization method proposed by the author. This method has shown the best results in solving most of the test problems. This paper presents some of the results of the author's numerical experiments. In particular, the best solutions were obtained for test problems with unknown solutions. This method allows solving multimodal problems of large dimensions and only a local search program is required for its implementation.

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.

A New Global Optimization Method for Simultaneous Computation on Expensive Black-Box Functions

2003 ◽  
Author(s):  
Liqun Wang ◽  
Songqing Shan ◽  
G. Gary Wang
Keyword(s):  
Global Optimization ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Search Space ◽  
Optimization Method ◽  
Global Optimum ◽  
Black Box ◽  
Efficient Global Optimization ◽  
Global Optimization Method ◽  
Global Optimization Methods

The presence of black-box functions in engineering design, which are usually computation-intensive, demands efficient global optimization methods. This work proposes a new global optimization method for black-box functions. The global optimization method is based on a novel mode-pursuing sampling (MPS) method which systematically generates more sample points in the neighborhood of the function mode while statistically covers the entire search space. Quadratic regression is performed to detect the region containing the global optimum. The sampling and detection process iterates until the global optimum is obtained. Through intensive testing, this method is found to be effective, efficient, robust, and applicable to both continuous and discontinuous functions. It supports simultaneous computation and applies to both unconstrained and constrained optimization problems. Because it does not call any existing global optimization tool, it can be used as a standalone global optimization method for inexpensive problems as well. Limitation of the method is also identified and discussed.

AN IMPLEMENTATION OF A PARALLEL GENERALIZED BRANCH AND BOUND TEMPLATE

2007 ◽  
Vol 12 (3) ◽  
pp. 277-289 ◽  
Author(s):  
Milda Baravykaitė ◽  
Raimondas Čiegis
Keyword(s):  
Global Optimization ◽  
Load Balancing ◽  
Branch And Bound ◽  
Optimization Problems ◽  
Test Problems ◽  
Search Spaces ◽  
Parallel Version ◽  
Derivative Free ◽  
Template Library ◽  
Selection Of

Branch and bound (BnB) is a general algorithm to solve optimization problems. We present a template implementation of the BnB paradigm. A BnB template is implemented using C++ object oriented paradigm. MPI is used for underlying communications. A paradigm of domain decomposition (data parallelization) is used to construct a parallel algorithm. To obtain a better load balancing, the BnB template has the load balancing module that allows the redistribution of search spaces among the processors at run time. A parallel version of user's algorithm is obtained automatically. A new derivative-free global optimization algorithm is proposed for solving nonlinear global optimization problems. It is based on the BnB algorithm and its implementation is done by using the developed BnB algorithm template library. The robustness of the new algorithm is demonstrated by solving a selection of test problems.

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 Numerical verification of the efficiency of global optimization methods

2020 ◽  
Vol 8 (2) ◽  
pp. 15-23
Author(s):  
A.I. Kosolap ◽  
Keyword(s):  
Global Optimization ◽  
Regularization Method ◽  
Convex Relaxation ◽  
Optimization Methods ◽  
Test Problems ◽  
Numerical Verification ◽  
Test Functions ◽  
Estimates Of Solutions ◽  
Global Optimization Methods ◽  
Arbitrary Dimensions

In this paper, new difficult test problems are proposed to test the numerical efficiency of global optimization methods. These are problems of unconstrained optimization with unknown solutions. The proposed test problems are inseparable and have arbitrary dimensions. The author also proposes to include the test functions by J. Nie in the list of test functions for numerical verification of the effectiveness of methods. These functions are also inseparable functions of arbitrary dimensions with unknown solutions. The proposed test problems have many local extrema. Testing the effectiveness of global optimization methods for such functions is simplified. If the method allows improving the found solutions to test problems, then it will be more effective. The existing global optimization methods are compared with the exact quadratic regularization method developed by the author. This method is compared with known software packages that implement modern methods of global optimization. These packages include several methods. The best of them use convex relaxation of the problem to obtain estimates of solutions with subsequent use of local optimization programs. But even such powerful packages have difficulties in solving the considered test problems. Some test problems, for example, with the Rana or Egg Holder function, have been solved by different methods for over 20 years. During this time, no method has allowed obtaining results that are obtained by the method of exact quadratic regularization. For almost all complex test problems with unknown solutions, this method yielded better solutions. Sometimes the advantage of this method was significant, as is the case with the Rana test function. The essence of the exact quadratic regularization method is to transform any global optimization problem to maximize the square of the Euclidean norm of a vector on a convex set. This problem is computationally much simpler. Often, with such a transformation, the multimodal problem becomes unimodal, which is easy to solve. Keywords: test problems, global optimization, unimodal problems, multimodal problems, numerical methods.

scholarly journals SOLVING THE NONLINEAR TRANSPORTATION PROBLEM BY GLOBAL OPTIMIZATION

Transport ◽  
2010 ◽  
Vol 25 (3) ◽  
pp. 314-324 ◽  
Author(s):  
Uroš Klanšek ◽  
Mirko Pšunder
Keyword(s):  
Global Optimization ◽  
Transportation Problem ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Branch And Cut ◽  
Cut Method ◽  
Global Optimization Methods ◽  
Optimum Solution ◽  
Global And Local ◽  
Branch And Cut Method

The aim of this paper is to present the suitability of three different global optimization methods for specifically the exact optimum solution of the nonlinear transportation problem (NTP). The evaluated global optimization methods include the branch and reduce method, the branch and cut method and the combination of global and local search strategies. The considered global optimization methods were applied to solve NTPs with reference to literature. NTPs were formulated as nonlinear programming (NLP) optimization problems. The obtained optimal results were compared with those got from literature. A comparative evaluation of global optimization methods is presented at the end of the paper to show their suitability for solving NTPs.

scholarly journals TEMPLATE REALIZATION OF GENERALIZED BRANCH AND BOUND ALGORITHM

2005 ◽  
Vol 10 (3) ◽  
pp. 217-236 ◽  
Author(s):  
M. Baravykaite ◽  
R. Čiegis ◽  
J. Žilinskas
Keyword(s):  
Global Optimization ◽  
Combinatorial Optimization ◽  
Branch And Bound ◽  
Optimization Algorithms ◽  
Optimization Methods ◽  
Branch And Bound Algorithm ◽  
Computational Experiments ◽  
Branch And Bound Algorithms ◽  
Parallel Branch And Bound

In this work we consider a template for implementation of parallel branch and bound algorithms. The main aim of this package to ease implementation of covering and combinatorial optimization methods for global optimization. Standard parts of global optimization algorithms are implemented in the package and only method specific rules should be implemented by the user. The parallelization part of the tool is described in details. Results of computational experiments are presented and discussed. Straipsnyje pristatyta apibendrinto šaku ir režiu algoritmo šablono realizacija. Irankis skirtas palengvinti nuosekliuju ir lygiagrečiuju optimizacijos uždaviniu programu kūrima. Nuo uždavinio nepriklausančios algoritmo dalys yra idiegtos šablone ir vartotojui reikia sukurti tik nuo uždavinio priklausančiu daliu realizacija. Šablone idiegti keli lygiagretieji algoritmai, paremti tyrimo srities padalinimu tarp procesoriu. Pateikiami skaičiavimo eksperimentu rezultatai.

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