On Generalizing Lipschitz Global Methods forMultiobjective Optimization

Lovison, Alberto; Hartikainen, Markus

doi:10.1007/978-3-319-15892-1_18

Cited by 5 publications

(9 citation statements)

References 17 publications

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“…Some exact single objective optimization algorithms have inspired or have been used as components in multiobjective optimization algorithms like [9,16,21,24,25]. Nevertheless, the class of exact, global multiobjective optimization algorithms has not been developed to the same extent as its single objective counterpart.…”

Section: Introductionmentioning

confidence: 99%

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On the Extension of the DIRECT Algorithm to Multiple Objectives

Lovison

Miettinen

2020

J Glob Optim

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Deterministic global optimization algorithms like Piyavskii-Shubert, direct, ego and 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 algorithm direct, analyze the available algorithms in the literature that extend direct to 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 in direct. We also discuss its advantages and disadvantages when compared with algorithms existing in the literature.

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Section: Introductionmentioning

confidence: 99%

“…As mentioned above, although global convergence and deterministic characters are desirable features, it is not easy to attain them both in algorithm design. Furthermore, the question of global convergence has not been considered in many papers in the domain of multiobjective optimization [4,12,19,[23][24][25]31,41].…”

Section: Introductionmentioning

confidence: 99%

On the Extension of the DIRECT Algorithm to Multiple Objectives

Lovison

Miettinen

2020

J Glob Optim

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“…At the moment, corresponding exact and global methods for multiobjective optimization have not been developed and employed to the same extent as their single objective counterparts. However, some of these methods have inspired or have been used as a component in a number of multiobjective optimization algorithms [2,3,4,5,6,7]. Nevertheless, most of them have the following characteristics:…”

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confidence: 99%

“…In this framework, we have developed in [4] multishubert, a multiobjective extension of the Piyavskii-Shubert algorithm [14,15], and we have shown the global convergence of the algorithm in the sense explained above. A quite natural generalization of the Piyavskii-Shubert algorithm is direct [16], which outperforms its ancestor in several senses.…”

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confidence: 99%

“…At every iteration the multidirect algorithm will select for subdivision of all those potentially optimal hyperintervals. In the sequence of Figure 3 we present the iterations of the multidirect algorithm on two variables biobjective problem (see [4] and Figure 1(b)). Analogously, it is not difficult to prove the following global convergence result: By taking the sequence S j := x ι f (x ι ) is nondominated by f (x 1 ), .…”

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Exact extension of the DIRECT algorithm to multiple objectives

Lovison¹,

Miettinen²

2019

AIP Conference Proceedings

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The direct algorithm has been recognized as an efficient global optimization method which has few requirements of regularity and has proven to be globally convergent in general cases. direct has been an inspiration or has been used as a component for many multiobjective optimization algorithms. We propose an exact and as genuine as possible extension of the direct method for multiple objectives, providing a proof of global convergence (i.e., a guarantee that in an infinite time the algorithm becomes everywhere dense). We test the efficiency of the algorithm on a nonlinear and nonconvex vector function.

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