Newton's Method for Multiobjective Optimization

Fliege, Jörg; Drummond, L. M. Graña; Svaiter, B. F.

doi:10.1137/08071692x

Cited by 320 publications

(208 citation statements)

References 34 publications

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“…We next provide a necessary condition for Pareto optimality, i.e., a notion of criticality. The criticality definition presented herein is an extension of the criticality definition for unconstrained multiobjective optimization used in [14]. This criticality condition for a given feasible pointx can be seen as the nonexistence of a feasible direction d ∈ R n along which all the objective functions are decreasing.…”

Section: ∀X ∈ Xmentioning

confidence: 99%

“…These problems are available by following the instructions at http://www.mat.uc.pt/dms. We enrich the test problem database with further test problems provided in [8,11,14,24] as well as those available in the MacMOOP database (http://wiki.mcs.anl.gov/leyffer/index.php/MacMOOP). The full test set is presented in Table 1 for bound constrained problems, i.e., for problems where the feasible set is Ω = {x ∈ R n : ≤ x ≤ u} for some ∈ (R ∪ {−∞}) n , u ∈ (R ∪ {∞}) n , and in Table 2 for nonlinearly constrained optimization problems.…”

Section: Test Problemsmentioning

confidence: 99%

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A Method for Constrained Multiobjective Optimization Based on SQP Techniques

Fliege¹,

Vaz²

2016

SIAM J. Optim.

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Abstract. We propose a method for constrained and unconstrained nonlinear multiobjective optimization problems that is based on an SQP-type approach. The proposed algorithm maintains a list of nondominated points that is improved both for spread along the Pareto front and optimality by solving single-objective constrained optimization problems. These single-objective problems are derived as SQP problems based on the given nondominated points. Under appropriate differentiability assumptions we discuss convergence to local optimal Pareto points. We provide numerical results for a set of unconstrained and constrained multiobjective optimization problems in the form of performance and data profiles, where several performance metrics are used. The numerical results confirm the superiority of the proposed algorithm against a state-of-the-art multiobjective solver and a classical scalarization approach, both in the quality of the approximated Pareto front and in the computational effort necessary to compute the approximation.

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Section: ∀X ∈ Xmentioning

confidence: 99%

Section: Test Problemsmentioning

confidence: 99%

A Method for Constrained Multiobjective Optimization Based on SQP Techniques

Fliege¹,

Vaz²

2016

SIAM J. Optim.

View full text Add to dashboard Cite

show abstract

“…9 These sets are approximated by means of simplicial complexes, and by exploiting Newton-type estimates it is possible to prove quadratic convergence in a setwise sense, adopting the Hausdorff measure. Because of this result the present method can be considered a setwise variant of multiobjective Newton methods, as in [18]. 5 Debreu won the Nobel Prize for Economics in 1983 "for having incorporated new analytical methods into economic theory and for his rigorous reformulation of the theory of general equilibrium."…”

Section: Multiobjective Optimization and Pareto Optimality Multiobjementioning

confidence: 99%

Singular Continuation: Generating Piecewise Linear Approximations to Pareto Sets via Global Analysis

Lovison¹

2011

SIAM J. Optim.

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Abstract. We propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale [Global analysis and economics. I. Pareto optimum and a generalization of Morse theory, in Dynamical Systems, Academic Press, New York, 1973, pp. 531-544]. The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. The method distinguishes the hierarchy between singular set, Pareto critical set, and stable Pareto critical set, and it can handle the problem of superposition of local Pareto fronts, occurring in the general nonconvex case. Furthermore, a quadratic convergence result in a suitable setwise sense is proven and tested in a number of numerical examples.

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“…In [168],a directed search (DS) is incorporated as local search method within global indicator based optimization algorithms. In [89], Newton steepest descent method [51] and Hooke &Jeeves [64] have been concurrently integrated within SMS-EMOEA [20] framework to handle ZDT test problems [198].…”

Section: Indicator Based Evolutionary Algorithmmentioning

confidence: 99%

Evolutionary Algorithms Based on Decomposition and Indicator Functions: State-of-the-art Survey

Khan¹,

Salhi²,

Jan³

et al. 2016

ijacsa

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Abstract-In the last two decades, multiobjective optimization has become mainstream because of its wide applicability in a variety of areas such engineering, management, the military and other fields. Multi-Objective Evolutionary Algorithms (MOEAs) play a dominant role in solving problems with multiple conflicting objective functions. They aim at finding a set of representative Pareto optimal solutions in a single run. Classical MOEAs are broadly in three main groups: the Pareto dominance based MOEAs, the Indicator based MOEAs and the decomposition based MOEAs. Those based on decomposition and indicator functions have shown high search abilities as compared to the Pareto dominance based ones. That is possibly due to their firm theoretical background. This paper presents state-of-the-art MOEAs that employ decomposition and indicator functions as fitness evaluation techniques along with other efficient techniques including those which use preference based information, local search optimizers, multiple ensemble search operators together with self-adaptive strategies, metaheuristics, mating restriction approaches, statistical sampling techniques, integration of Fuzzy dominance concepts and many other advanced techniques for dealing with diverse optimization and search problems

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Newton's Method for Multiobjective Optimization

Cited by 320 publications

References 34 publications

A Method for Constrained Multiobjective Optimization Based on SQP Techniques

A Method for Constrained Multiobjective Optimization Based on SQP Techniques

Singular Continuation: Generating Piecewise Linear Approximations to Pareto Sets via Global Analysis

Evolutionary Algorithms Based on Decomposition and Indicator Functions: State-of-the-art Survey

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