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Newton’s method for multicriteria optimization

Newton’s method for multicriteria optimization
Newton’s method for multicriteria optimization
We propose an extension of Newton's method for unconstrained multiobjective optimization (multicriteria optimization). This method does not use a priori chosen weighting factors or any other form of a priori ranking or ordering information for the different objective functions. Newton's direction at each iterate is obtained by minimizing the max-ordering scalarization of the variations on the quadratic approximations of the objective functions. The objective functions are assumed to be twice continuously differentiable and locally strongly convex. Under these hypotheses, the method, as in the classical case, is locally superlinear convergent to optimal points. Again as in the scalar case, if the second derivatives are Lipschitz continuous, the rate of convergence is quadratic. Our convergence analysis uses a Kantorovich-like technique. As a byproduct, existence of optima is obtained under semilocal assumptions.
multicriteria optimization, multiobjective programming, pareto points, newton's method
1052-6234
602-626
Fliege, Joerg
54978787-a271-4f70-8494-3c701c893d98
Drummond, Mauricio G.
f4db0ded-7ad8-4bec-b0c4-21b239aec38a
Svaiter, Benar F.
6260912e-a15f-4ea7-9847-9be3deec3c5b
Fliege, Joerg
54978787-a271-4f70-8494-3c701c893d98
Drummond, Mauricio G.
f4db0ded-7ad8-4bec-b0c4-21b239aec38a
Svaiter, Benar F.
6260912e-a15f-4ea7-9847-9be3deec3c5b

Fliege, Joerg, Drummond, Mauricio G. and Svaiter, Benar F. (2009) Newton’s method for multicriteria optimization. SIAM Journal on Optimization, 20 (2), 602-626. (doi:10.1137/08071692X).

Record type: Article

Abstract

We propose an extension of Newton's method for unconstrained multiobjective optimization (multicriteria optimization). This method does not use a priori chosen weighting factors or any other form of a priori ranking or ordering information for the different objective functions. Newton's direction at each iterate is obtained by minimizing the max-ordering scalarization of the variations on the quadratic approximations of the objective functions. The objective functions are assumed to be twice continuously differentiable and locally strongly convex. Under these hypotheses, the method, as in the classical case, is locally superlinear convergent to optimal points. Again as in the scalar case, if the second derivatives are Lipschitz continuous, the rate of convergence is quadratic. Our convergence analysis uses a Kantorovich-like technique. As a byproduct, existence of optima is obtained under semilocal assumptions.

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More information

Published date: 2009
Keywords: multicriteria optimization, multiobjective programming, pareto points, newton's method
Organisations: Operational Research

Identifiers

Local EPrints ID: 79917
URI: http://eprints.soton.ac.uk/id/eprint/79917
ISSN: 1052-6234
PURE UUID: 63ae02dc-72a9-4c19-9379-a672ba9e0a9d
ORCID for Joerg Fliege: ORCID iD orcid.org/0000-0002-4459-5419

Catalogue record

Date deposited: 22 Mar 2010
Last modified: 28 Apr 2022 01:59

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Contributors

Author: Joerg Fliege ORCID iD
Author: Mauricio G. Drummond
Author: Benar F. Svaiter

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