Complexity of gradient descent for multiobjective optimization

A number of first-order methods have been proposed for smooth multiobjective optimization for which some form of convergence to first-order criticality has been proved. Such convergence is global in the sense of being independent of the starting point. In this paper, we analyse the rate of convergen...

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Published inOptimization methods & software Vol. 34; no. 5; pp. 949 - 959
Main Authors Fliege, J., Vaz, A. I. F., Vicente, L. N.
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 03.09.2019
Taylor and Francis
Taylor & Francis Ltd
Subjects
Complexity
Convergence
Global rates
Gradient descent
Mathematical programming
Multiobjective optimization
Multiple objective analysis
Optimization
Pareto optimum
Science & Technology
Steepest descent
worst-case complexity
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Abstract A number of first-order methods have been proposed for smooth multiobjective optimization for which some form of convergence to first-order criticality has been proved. Such convergence is global in the sense of being independent of the starting point. In this paper, we analyse the rate of convergence of gradient descent for smooth unconstrained multiobjective optimization, and we do it for non-convex, convex, and strongly convex vector functions. These global rates are shown to be the same as for gradient descent in single-objective optimization and correspond to appropriate worst-case complexity bounds. In the convex cases, the rates are given for implicit scalarizations of the problem vector function.
AbstractList Published online: 29 Aug 2018 A number of first-order methods have been proposed for smooth multiobjective optimization for which some form of convergence to first-order criticality has been proved. Such convergence is global in the sense of being independent of the starting point. In this paper, we analyse the rate of convergence of gradient descent for smooth unconstrained multiobjective optimization, and we do it for non-convex, convex, and strongly convex vector functions. These global rates are shown to be the same as for gradient descent in single-objective optimization and correspond to appropriate worstcase complexity bounds. In the convex cases, the rates are given for implicit scalarizations of the problem vector function. Support for A.I.F. Vaz was partially provided by FCT [grant number COMPETE:POCI-01- 0145-FEDER-007043], [grant number UID/CEC/00319/2013], and support for L.N. Vicente was partially provided by FCT [grant number UID/MAT/00324/2013], [grant number P2020 SAICTPAC/0011/2015.]
A number of first-order methods have been proposed for smooth multiobjective optimization for which some form of convergence to first-order criticality has been proved. Such convergence is global in the sense of being independent of the starting point. In this paper, we analyse the rate of convergence of gradient descent for smooth unconstrained multiobjective optimization, and we do it for non-convex, convex, and strongly convex vector functions. These global rates are shown to be the same as for gradient descent in single-objective optimization and correspond to appropriate worst-case complexity bounds. In the convex cases, the rates are given for implicit scalarizations of the problem vector function.
Author Vaz, A. I. F.
Vicente, L. N.
Fliege, J.
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  fullname: Vaz, A. I. F.
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  givenname: L. N.
  surname: Vicente
  fullname: Vicente, L. N.
  email: lnv@mat.uc.pt
  organization: CMUC, Department of Mathematics, University of Coimbra
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Snippet A number of first-order methods have been proposed for smooth multiobjective optimization for which some form of convergence to first-order criticality has...
Published online: 29 Aug 2018 A number of first-order methods have been proposed for smooth multiobjective optimization for which some form of convergence to...
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SubjectTerms Complexity
Convergence
Global rates
Gradient descent
Mathematical programming
Multiobjective optimization
Multiple objective analysis
Optimization
Pareto optimum
Science & Technology
Steepest descent
worst-case complexity
Title Complexity of gradient descent for multiobjective optimization
URI https://www.tandfonline.com/doi/abs/10.1080/10556788.2018.1510928
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Volume 34
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