<scp>L</scp>ipschitz Global Optimization

Sergeyev, Yaroslav D.; Kvasov, Dmitri E.

doi:10.1002/9780470400531.eorms1006

Cited by 14 publications

(6 citation statements)

References 145 publications

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“…DIRECT (DIvide RECTangles) algorithm developed by Jones [14,12] is one of the most widely used partitioning-based algorithms due to its simplicity, and it only needs one algorithmic parameter ([8, 7, 4, 5, 3, 1]. The algorithm is an extension of classical Lipschitz optimization (see, e.g., [29,2,32,38,41]), where the need to know the Lipschitz constant is eliminated. However, DIRECT algorithm may converge slowly if it gets close to the optimum, requiring it to divide incessantly near the location of this optimum.…”

Section: Introductionmentioning

confidence: 99%

“…This strategy is typical for diagonal-based algorithms, which produce many unnecessary sampling points of the objective function. Every vertex where the function has been evaluated can belong to up to 2 n hyper-rectangles [35,38,18]. Especially the algorithm takes significantly longer than usual to find a solution close to a global optimum.…”

Section: Introductionmentioning

confidence: 99%

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Diagonal Partitioning Strategy Using Bisection of Rectangles and a Novel Sampling Scheme

Guessoum,

Chiter

2023

mendel

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In this paper, we consider a global optimization problem where the objective function is assumed to be Lipschitz-continuous with an unknown Lipschitz constant. Building upon the recently introduced BIRECT (BIsection of RECTangles) algorithm, we propose a new diagonal partitioning and sampling scheme. Our framework, named BIRECT-V (V for vertices), combines bisection with the sampling of two points. In the initial hyper-rectangle, these points are located at 1/3 and 1 along the main diagonal. Unlike most DIRECT-type algorithms, where evaluating the objective function at vertices is not suitable for bisection, our strategy, when combined with bisection, provides more comprehensive information about the objective function. However, the creation of new sampling points may coincide with existing ones at shared vertices, resulting in additional evaluations of the objective function and increasing the number of function evaluations per iteration. To overcome this issue, we propose modifying the original optimization domain to obtain a good approximation of the global solution. Experimental investigations demonstrate that this modification positively impacts the performance of the BIRECT-V algorithm. Our proposal shows promise as a global optimization algorithm compared to the original BIRECT and two popular DIRECT-type algorithms on a set of test problems. It particularly excels at high-dimensional problems

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Diagonal Partitioning Strategy Using Bisection of Rectangles and a Novel Sampling Scheme

Guessoum,

Chiter

2023

mendel

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“…The DIRECT algorithm developed by Jones [7] is a popular and widely used deterministic solution technique for various real-world optimization problems [8][9][10][11][12]. The proposed algorithm is an extension of the classical Lipschitz optimization [13][14][15], which no longer requires the knowledge of the Lipschitz constant. The DIRECT algorithm [7] seeks global optima by dividing the most promising hyper-rectangles and evaluating the objective function at their centers.…”

Section: Introductionmentioning

confidence: 99%

Experimental Study of Excessive Local Refinement Reduction Techniques for Global Optimization DIRECT-Type Algorithms

Stripinis

Paulavičius

2022

Mathematics

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This article considers a box-constrained global optimization problem for Lipschitz continuous functions with an unknown Lipschitz constant. The well-known derivative-free global search algorithm DIRECT (DIvide RECTangle) is a promising approach for such problems. Several studies have shown that recent two-step (global and local) Pareto selection-based algorithms are very efficient among all DIRECT-type approaches. However, despite its encouraging performance, it was also observed that the candidate selection procedure has two possible shortcomings. First, there is no limit on how small the size of selected candidates can be. Secondly, a balancing strategy between global and local candidate selection is missing. Therefore, it may waste function evaluations by over-exploring the current local minimum and delaying finding the global one. This paper reviews and employs different strategies in a two-step Pareto selection framework (1-DTC-GL) to overcome these limitations. A detailed experimental study has revealed that existing strategies do not always improve and sometimes even worsen results. Since 1-DTC-GL is a DIRECT-type algorithm, the results of this paper provide general guidance for all DIRECT-type algorithms on how to deal with excessive local refinement more efficiently.

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“…Two of these global solvers (TOMLAB/GLCCLUSTER and MCS [23,31]) are based on the DIRECT (DIviding RECTangle) algorithm [24]. The DIRECT algorithm is a modification of standard Lipschitzian approaches [10,11,19,20,22,27,[34][35][36][38][39][40]43,44] eliminating the need to specify a Lipschitz constant. Due to its simplicity, DIRECT has been used to solve many engineering optimization problems [1][2][3]6,9,17,18,28].…”

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confidence: 99%

Advantages of simplicial partitioning for Lipschitz optimization problems with linear constraints

Paulavičius

Žilinskas

2014

Optim Lett

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The well known DIRECT (DIviding RECTangles) algorithm for global optimization requires bound constraints on variables and does not naturally address additional linear or nonlinear constraints. A feasible region defined by linear constraints may be covered by simplices, therefore simplicial partitioning may tackle linear constraints in a very subtle way. In this paper we demonstrate this advantage of simplicial partitioning by applying a recently proposed deterministic simplicial partitions based DISIMPL algorithm for optimization problems defined by general linear constraints (Lc-DISIMPL). An extensive experimental investigation reveals advantages of this approach to such problems comparing with different constraint-handling methods, proposed for use with DIRECT. Furthermore the Lc-DISIMPL algorithm gives very competitive results compared to a derivative-free particle swarm algorithm (PSwarm) which was previously shown to give very promising results. Moreover, DISIMPL guarantees the convergence to the global solution, whereas the PSwarm algorithm sometimes fails to converge to the global minimum.

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Lipschitz Global Optimization

Cited by 14 publications

References 145 publications

Diagonal Partitioning Strategy Using Bisection of Rectangles and a Novel Sampling Scheme

Diagonal Partitioning Strategy Using Bisection of Rectangles and a Novel Sampling Scheme

Experimental Study of Excessive Local Refinement Reduction Techniques for Global Optimization DIRECT-Type Algorithms

Advantages of simplicial partitioning for Lipschitz optimization problems with linear constraints

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