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Improving the convergence rate of the direct global optimization algorithm

JOURNAL OF GLOBAL OPTIMIZATION 2017年第4期67卷 851-872页

作者： Liu, Qunfeng Yang, Guang Zhang, Zhongzhi Zeng, Jinping Dongguan Univ Technol Coll Comp Sci & Technol Dongguan 523808 Peoples R China

direct is derivative-free global-search algorithm has been found to perform robustly across a wide variety of low-dimensional test problems. The reason direct's robustness is its lack of algorithmic parameters that need be "tuned" to make the algorithm perform well. In particular, there is no parameter that determines the relative emphasis on global versus local search. Unfortunately, the same algorithmic features that enable direct to perform so robustly have a negative side effect. In particular, direct is usually quick to get close to the global minimum, but very slow to refine the solution to high accuracy. This is what we call direct's "eventually inefficient behavior." In this paper, we outline two root causes for this undesirable behavior and propose modifications to eliminate it. The paper builds upon our previously published "Mrdirect" algorithm, which we can now show only addressed the first root cause of the "eventually inefficient behavior." The key contribution of the current paper is a further enhancement that allows Mrdirect to address the second root cause as well. To demonstrate the effectiveness of the enhanced Mrdirect, we have identified a set of test functions that highlight different situations in which direct has convergence issues. Extensive numerical work with this test suite demonstrates that the enhanced version of Mrdirect does indeed improve the convergence rate of direct.

关键词： Global optimization Multilevel algorithm direct algorithm Mrdirect algorithm Convergence rate

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On the Extension of thedirectalgorithm to Multiple Objectives

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JOURNAL OF GLOBAL OPTIMIZATION 2021年第2期79卷 387-412页

作者： Lovison, Alberto Miettinen, Kaisa Univ Padua Dept Math Tullio Levi Civita Via Trieste 63 I-35121 Padua Italy Univ Jyvaskyla Fac Informat Technol POB 35 Agora FI-40014 Jyvaskyla Finland

Deterministic global optimization algorithms like Piyavskii-Shubert,direct,egoand many more, have a recognized standing, for problems with many local optima. Although many single objective optimization algorithms have been extended to multiple objectives, completely deterministic algorithms for nonlinear problems with guarantees of convergence to global Pareto optimality are still missing. For instance, deterministic algorithms usually make use of some form of scalarization, which may lead to incomplete representations of the Pareto optimal set. Thus, all global Pareto optima may not be obtained, especially in nonconvex cases. On the other hand, algorithms attempting to produce representations of the globally Pareto optimal set are usually based on heuristics. We analyze the concept of global convergence for multiobjective optimization algorithms and propose a convergence criterion based on the Hausdorff distance in the decision space. Under this light, we consider the well-known global optimization algorithmdirect, analyze the available algorithms in the literature that extenddirectto multiple objectives and discuss possible alternatives. In particular, we propose a novel definition for the notion of potential Pareto optimality extending the notion of potential optimality defined indirect. We also discuss its advantages and disadvantages when compared with algorithms existing in the literature.

关键词： Global convergence Multiobjective optimization Multiple criteria optimization direct algorithm Deterministic optimization algorithms

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Solving non-monotone equilibrium problems via a direct-type approach

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JOURNAL OF GLOBAL OPTIMIZATION 2022年第4期83卷 699-725页

作者： Lucidi, Stefano Passacantando, Mauro Rinaldi, Francesco Sapienza Univ Rome Dept Comp Control & Management Engn Antonio Ruber Via Ariosto 25 I-00185 Rome Italy Univ Pisa Dept Comp Sci Largo B Pontecorvo 3 I-56127 Pisa Italy Univ Padua Dept Math Tullio Levi Civita Via Trieste 63 I-35121 Padua Italy

A global optimization approach for solving non-monotone equilibrium problems (EPs) is proposed. The class of (regularized) gap functions is used to reformulate any EP as a constrained global optimization program and some bounds on the Lipschitz constant of such functions are provided. The proposed global optimization approach is a combination of an improved version of the direct algorithm, which exploits local bounds of the Lipschitz constant of the objective function, with local minimizations. Unlike most existing solution methods for EPs, no monotonicity-type condition is assumed in this paper. Preliminary numerical results on several classes of EPs show the effectiveness of the approach.

关键词： Equilibrium problem Gap function Global optimization direct algorithm

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Filter-based direct method for constrained global optimization

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JOURNAL OF GLOBAL OPTIMIZATION 2018年第3期71卷 517-536页

作者： Costa, M. Fernanda P. Rocha, Ana Maria A. C. Fernandes, Edite M. G. P. Univ Minho Dept Math & Applicat Ctr Math Campus Gualtar P-4710057 Braga Portugal Univ Minho Dept Prod & Syst Algoritmi Res Ctr Campus Gualtar P-4710057 Braga Portugal Univ Minho Algoritmi Res Ctr Campus Gualtar P-4710057 Braga Portugal

This paper presents a direct-type method that uses a filter methodology to assure convergence to a feasible and optimal solution of nonsmooth and nonconvex constrained global optimization problems. The filter methodology aims to give priority to the selection of hyperrectangles with feasible center points, followed by those with infeasible and non-dominated center points and finally by those that have infeasible and dominated center points. The convergence properties of the algorithm are analyzed. Preliminary numerical experiments show that the proposed filter-based direct algorithm gives competitive results when compared with other direct-type methods.

关键词： Global optimization direct algorithm Filter method

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A direct-based approach exploiting local minimizations for the solution of large-scale global optimization problems

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2010年第2期45卷 353-375页

作者： Liuzzi, G. Lucidi, S. Piccialli, V. Univ Roma La Sapienza Dipartimento Informat & Sistemist Antonio Ruberti I-00185 Rome Italy CNR IASI A Ruberti I-00185 Rome Italy Univ Roma Tor Vergata Dipartimento Ingn Impresa I-00133 Rome Italy

In this paper we propose a new algorithm for solving difficult large-scale global optimization problems. We draw our inspiration from the well-known direct algorithm which, by exploiting the objective function behavior, produces a set of points that tries to cover the most interesting regions of the feasible set. Unfortunately, it is well-known that this strategy suffers when the dimension of the problem increases. As a first step we define a multi-start algorithm using direct as a deterministic generator of starting points. Then, the new algorithm consists in repeatedly applying the previous multi-start algorithm on suitable modifications of the variable space that exploit the information gained during the optimization process. The efficiency of the new algorithm is pointed out by a consistent numerical experimentation involving both standard test problems and the optimization of Morse potential of molecular clusters.

关键词： Global optimization Nonlinear programming Morse potential direct algorithm

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A direct-based approach exploiting local minimizations for the solution of large-scale global optimization problems

A DIRECT-based approach exploiting local minimizations for t...

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5th Workshop on New Problems and Innovative Methods in Nonlinear Optimization

关键词： Global optimization Nonlinear programming Morse potential direct algorithm

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A Unified Approach Toward Simulating Constant and Varying Amplitude Fatigue Failure Effects of Metals with Fast and Efficient algorithms

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Acta Mechanica Solida Sinica 2021年第1期34卷 53-64页

作者： Si-Yu Wang Lin Zhan Hui-Feng Xi Heng Xiao College of Mechanics and Construction Engineering and MOE Key Lab of Disaster Forecast and Control in Engineering Jinan UniversityGuangzhou510632China

A new finite strain elastoplastic J2-flow model is established with an explicit formulation of work-hardening and softening effects up to eventual failure,in which both a new flow rule free of yielding and an asymptotically vanishing stress limit are *** novelties of this new model are as follows:(i)Fatigue failure effects under repeated loading conditions with either constant or varying amplitudes are automatically characterized as inherent response features;(ii)neither additional damage-like variables nor failure criteria need to be involved;and(iii)both high-and low-cycle fatigue effects may be simultaneously treated.A fast and efficient algorithm of high accuracy is proposed for directly simulating high-and medium-high-cycle fatigue failure effects under repeated loading *** this goal,a direct and explicit relationship between the fatigue life and the stress amplitude is obtained by means of explicit and direct procedures of integrating the coupled elastoplastic rate equations for any given number of loading-unloading cycles with varying stress *** examples suggest that the new algorithm is much more fast and efficient than usual tedious and very time-consuming integration procedures.

关键词： Metals Finite strain Elastoplasticity Fatigue failure direct algorithm

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NUMERICAL STUDIES OF SOME GENERALIZED CONTROLLED RANDOM SEARCH algorithmS

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ASIA-PACIFIC JOURNAL OF OPERATIONAL RESEARCH 2012年第2期29卷 1250016-1-1250016-24页

作者： Kaelo, P. Ali, M. M. Univ Witwatersrand Sch Computat & Appl Math ZA-2050 Johannesburg South Africa Univ Botswana Dept Math Gaborone Botswana

This paper presents motivations and algorithmic details of some generalized controlled random search (CRS) algorithms for global optimization. It also carries out an extensive numerical study of the generalized CRS algorithms to demonstrate their superiorities over their original counterparts. The numerical study is carried out using a set of 50 test problems many of which are inspired by practical applications. Numerical experiments indicate that the generalized algorithms are considerably better than the previous versions. The algorithms are also compared with the direct algorithm (Jones et al., 1993). The comparison shows that the generalized CRS algorithms are better than the direct algorithm in high dimensional problems. Thus, they offer a reasonable alternative to many currently available stochastic algorithms, especially for problems requiring "direct search type" methods.

关键词： Global optimization direct algorithm controlled random search linear interpolation probabilistic adaptation

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Global optimization by multilevel partition

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JOURNAL OF GLOBAL OPTIMIZATION 2015年第1期61卷 47-69页

作者： Liu, Qunfeng Zeng, Jinping Dongguan Univ Technol Sch Comp Dongguan 523808 Peoples R China

Partition-based algorithms, such as the direct algorithm, are popular algorithms for solving global optimization problems. However, these algorithms often have an eventually inefficient behavior due to much more costs requirement to obtain a solution with higher accuracy. In this paper, we present an algorithm framework for bound constrained global optimization problems based on a multilevel partition strategy. This multilevel partition strategy can be regarded as a combination of the basic partition strategy and the multigrid algorithm, which is one of the best algorithms to solve partial differential equation. Our basic idea is to combine the multigrid algorithm with the partition-based algorithm to improve the eventually inefficient behavior of the partition-based algorithm. First, we provide a general framework of the partition-based algorithms which include the direct algorithm as a special case. Then we present a strategy to build the subproblem at the coarse level. This strategy is easy to implement and brings no more computational costs. Under mild conditions, we show that the sequence generated by the proposed global optimization algorithm framework converges to the global optimum. Finally, we employ the original direct algorithm to build a specific global optimization algorithm based on multilevel partition and compare it with the original direct algorithm. Our numerical results show that obtained algorithm improves significantly the eventually inefficient behavior of the original direct algorithm when the required accuracy is high.

关键词： Global optimization Partition-based algorithm Multigrid algorithm direct algorithm

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Multiple spheres detection problem-Center based clustering approach

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PATTERN RECOGNITION LETTERS 2023年 176卷 34-41页

作者： Sabo, Kristian Scitovski, Rudolf Ungar, Sime Univ Osijek Dept Math Trg Lj Gaja 6 Osijek 31000 Croatia Univ Zagreb Dept Math Bijenicka 30 Zagreb 10000 Croatia

In this paper, we propose an adaptation of the well-known k-means algorithm for solving the multiple spheres detection problem when data points are homogeneously scattered around several spheres. We call this adaptation the k-closest spheres algorithm. In order to choose good initial spheres, we use a few iterations of the global optimizing algorithm direct, resulting in the high efficiency of the proposed k-closest spheres algorithm. We present illustrative examples for the case of non-intersecting and for the case of intersecting spheres. We also show a real-world application in analyzing earthquake depths.

关键词： Multiple spheres detection k-means algorithm k-closest spheres algorithm direct algorithm

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