Abstract

We extend the scalar concept of strong convex function of order k due to G. H. Lin and M. Fukushima ["Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints", J. Optim. Theory Appl. 118 (2003) 67–80] to a vector-valued function, by considering a partial order given by a convex cone. We analyze some properties of higher order strong convex functions, and we give two characterizations of this kind of strong convexity for locally Lipschitz functions, one of them, through a new property of the Clarke generalized Jacobian, called strong monotonicity of order k. Similar results are obtained for Fréchet differentiable functions. In the second part, we study connections between strong convexity of order k and global strict minimizers of order k, and we establish sufficient optimality conditions for this class of minimizers in multiobjective optimization problems involving strong cone-convex functions.

Author information

Contact details are reproduced from the original publication and may be historical.

César Gutiérrez

Dep. de Matemática Aplicada, Universidad de Valladolid, Edificio Tec. Inf. Telecomunicaciones, Campus Miguel Delibes, 47011 Valladolid, Spain

cesargv@mat.uva.es

Bienvenido Jiménez

Dep. de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, Ciudad Universitaria, 28040 Madrid, Spain

bjimenez@ind.uned.es

Vicente Novo

Dep. de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, Ciudad Universitaria, 28040 Madrid, Spain

vnovo@ind.uned.es

Keywords

  • Vector optimization
  • higher order strong convexity
  • optimality conditions
  • strict minimizers
  • generalized Jacobian
  • higher order strong monotonicity
  • partial-quasiconvexity

Mathematics Subject Classification

52A41, 90C29, 90C46, 49K27, 49J52

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Suggested citation

C. Gutiérrez, B. Jiménez, V. Novo. “Higher Order Strong Convexity and Global Strict Minimizers in Multiobjective Optimization.” Journal of Convex Analysis 18 (2011), No. 1, 85–103. https://doi.org/10.68381/jca18005

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.