Stochastic Method for the Solution of Unconstrained Vector Optimization Problems

Schäffler, Stefan; Schultz, Reinhart; Weinzierl, К.

doi:10.1023/a:1015472306888

Cited by 114 publications

(72 citation statements)

References 9 publications

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“…Finally, we consider an example where the Pareto set is disconnected. This is an example from production and was introduced in [SSW02]. We want to minimize the failure of a product which consists of n components.…”

Section: Resultsmentioning

confidence: 99%

“…One way to compute a descent direction satisfying (2) is to solve the following auxiliary optimization problem [SSW02]:…”

Section: Multiobjective Optimizationmentioning

confidence: 99%

“…Finally, several authors also develop gradientbased methods for MOPs directly. In [FS00,SSW02,Des12], algorithms are developed where a single descent direction is computed in which all objectives decrease. In [Bos12], a method is presented by which the entire set of descent directions can be determined, an extension of Newton's method to MOPs with quadratic convergence is presented in [FDS09].…”

Section: Introductionmentioning

confidence: 99%

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Gradient-Based Multiobjective Optimization with Uncertainties

Peitz

Dellnitz

2017

Studies in Computational Intelligence

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In this article we develop a gradient-based algorithm for the solution of multiobjective optimization problems with uncertainties. To this end, an additional condition is derived for the descent direction in order to account for inaccuracies in the gradients and then incorporated in a subdivison algorithm for the computation of global solutions to multiobjective optimization problems. Convergence to a superset of the Pareto set is proved and an upper bound for the maximal distance to the set of substationary points is given. Besides the applicability to problems with uncertainties, the algorithm is developed with the intention to use it in combination with model order reduction techniques in order to efficiently solve PDE-constrained multiobjective optimization problems.

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Section: Resultsmentioning

confidence: 99%

“…One way to compute a descent direction satisfying (2) is to solve the following auxiliary optimization problem [SSW02]:…”

Section: Multiobjective Optimizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gradient-Based Multiobjective Optimization with Uncertainties

Peitz

Dellnitz

2017

Studies in Computational Intelligence

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“…This direction is referred to by the authors as the multi-objective gradient. A similar derivation of a single direction of descent is originally given by Mukai [17] and additionally by Schäffler, Schultz and Weinzierl [18]. If the objectives have different ranges, the largest direction of simultaneous descent will be biased towards the objective with the largest range.…”

Section: B Gradient Information Directly Based On the Objectivesmentioning

confidence: 87%

On Gradients and Hybrid Evolutionary Algorithms for Real-Valued Multiobjective Optimization

Bosman

2012

IEEE Trans. Evol. Computat.

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Abstract-Algorithms that make use of the gradient, i.e. the direction of maximum improvement, to search for the optimum of a single objective function have been around for decades. They are commonly accepted to be important and have been applied to tackle single-objective optimization problems in many fields. For multi-objective optimization problems, much less is known about the gradient and its algorithmic use. In this article we aim to contribute to the understanding of gradients for numerical, i.e. real-valued, multi-objective optimization. Specifically, we provide an analytical parametric description of the set of all nondominated, i.e. most promising, directions in which a solution can be moved such that the objective values either improve or remain the same. This result completes previous work where this set is described only for one particular case, namely when some of the non-dominated directions have positive, i.e. non-improving, components and the final set can be computed by taking the subset of directions that are all non-positive. In addition we use our result to assess the utility of using gradient information for multi-objective optimization where the goal is to obtain a Pareto set of solutions that approximates the optimal Pareto set. To this end, we design and consider various multi-objective gradientbased optimization algorithms. One of these algorithms uses the description of the multi-objective gradient provided here. Also, we hybridize an existing multi-objective evolutionary algorithm (MOEA) with the various multi-objective gradient-based optimization algorithms. During optimization, the performance of the gradient-based optimization algorithms is monitored and the available computational resources are redistributed to allow the (currently) most effective algorithm to spend the most resources. We perform an experimental analysis using a few well-known benchmark problems to compare the performance of different optimization methods. The results underline that the use of a population of solutions that is characteristic of MOEAs is a powerful concept if the goal is to obtain a good Pareto set, i.e. instead of only a single solution. This makes it hard to increase convergence speed in the initial phase using gradient information to improve any single solution. However, in the longer run, the use of gradient information does ultimately allow for better finetuning of the results and thus better overall convergence.

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“…This cone is determined by means of the local gradient directions of the objectives. A similar line of research is given by methods that generate non-dominated points by linear combinations of the negative gradients with positive weights [12,19]. For small step-sizes this yields non-dominated or dominating solutions.…”

Section: Related Workmentioning

confidence: 99%

Gradient-Based/Evolutionary Relay Hybrid for Computing Pareto Front Approximations Maximizing the S-Metric

Emmerich

Deutz

Beume

Lecture Notes in Computer Science

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Abstract. The computation of a good approximation set of the Pareto front of a multiobjective optimization problem can be recasted as the maximization of its S-metric value. A high-precision method for computing approximation sets of a Pareto front with maximal S-Metric is presented in this paper as a high-level relay hybrid of an evolutionary algorithm and a gradient method, both guided by the S-metric. First, an evolutionary multiobjective optimizer moves the initial population close to the Pareto front. The gradient-based method takes this population as its starting point for computing a local maximal approximation set with respect to the S-metric. As opposed to existing work on gradient-based multicriteria optimization in the new gradient approach we compute gradients based on a set of points rather than for single points. We will term this approach indicatorbased gradient method, and exemplify it for the S-metric. We derive expressions for computing the gradient of a set of points with respect to its S-metric based on gradients of the objective functions. To deal with the problem of vanishing gradient components in case of dominated points in an approximation set, a penalty approach is introduced. We present a gradient based method for the aforementioned hybridization scheme and report on first results on artificial test problems.

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Stochastic Method for the Solution of Unconstrained Vector Optimization Problems

Cited by 114 publications

References 9 publications

Gradient-Based Multiobjective Optimization with Uncertainties

Gradient-Based Multiobjective Optimization with Uncertainties

On Gradients and Hybrid Evolutionary Algorithms for Real-Valued Multiobjective Optimization

Gradient-Based/Evolutionary Relay Hybrid for Computing Pareto Front Approximations Maximizing the S-Metric

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