# On gap functions for nonsmooth multiobjective optimization problems

On gap functions for nonsmooth multiobjective optimization problems A set-valued gap function, $$\phi$$ ϕ , existing in the literature for smooth and nonsmooth multiobjective optimization problems is dealt with. It is known that $$0\in \phi (x^*)$$ 0 ∈ ϕ ( x ∗ ) is a sufficient condition for efficiency of a feasible solution $$x^*$$ x ∗ , while the converse does not hold. In the current work, the converse of this assertion is proved for properly efficient solutions. Afterwards, to avoid the complexities of set-valued maps some new single-valued gap functions, for nonsmooth multiobjective optimization problems with locally Lipschitz data are introduced. Important properties of the new gap functions are established. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Optimization Letters Springer Journals

# On gap functions for nonsmooth multiobjective optimization problems

Optimization Letters, Volume 12 (2) – Feb 2, 2017
14 pages

/lp/springer_journal/on-gap-functions-for-nonsmooth-multiobjective-optimization-problems-t2Xb55jMyF
Publisher
Springer Journals
Subject
Mathematics; Optimization; Operations Research/Decision Theory; Computational Intelligence; Numerical and Computational Physics, Simulation
ISSN
1862-4472
eISSN
1862-4480
D.O.I.
10.1007/s11590-017-1110-4
Publisher site
See Article on Publisher Site

### Abstract

A set-valued gap function, $$\phi$$ ϕ , existing in the literature for smooth and nonsmooth multiobjective optimization problems is dealt with. It is known that $$0\in \phi (x^*)$$ 0 ∈ ϕ ( x ∗ ) is a sufficient condition for efficiency of a feasible solution $$x^*$$ x ∗ , while the converse does not hold. In the current work, the converse of this assertion is proved for properly efficient solutions. Afterwards, to avoid the complexities of set-valued maps some new single-valued gap functions, for nonsmooth multiobjective optimization problems with locally Lipschitz data are introduced. Important properties of the new gap functions are established.

### Journal

Published: Feb 2, 2017

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