On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming

• Published in: Mathematical programming Vol. 123; no. 1; pp. 199 - 223
• Main Authors: Gutiérrez, C., Jiménez, B., Novo, V.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-05-2010; Springer; Springer Nature B.V
• Subjects: Applied sciences; Calculus of variations and optimal control; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Decision theory. Utility theory; Exact sciences and technology; Full Length Paper; Inequality; Lagrange multiplier; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Optimization; Parabolas; Sciences and techniques of general use; Studies; Theoretical; Optimality conditions; Second order Clarke subdifferential; Parabolic second order derivative; 90C46; Lagrange multipliers; Multiobjective optimization; 90C29; 49K27; Multivalued function; Ordered set; Program optimization; Subdifferential; Multiobjective programming; Non linear programming; Lagrange multiplier; Nonsmooth analysis; Inequality constraint; Variational calculus; Mathematical programming; Second derivative; Set constraint; Directional derivative; Equality constraint; Optimality condition; Optimality criterion
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-009-0318-1

We study a multiobjective optimization program with a feasible set defined by equality constraints and a generalized inequality constraint. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point considered. We provide necessary second order optimality conditions and also sufficient conditions via a Fritz John type Lagrange multiplier rule and a set-valued second order directional derivative, in such a way that our sufficient conditions are close to the necessary conditions. Some consequences are obtained for parabolic directionally differentiable functions and
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1,1
functions, in this last case, expressed by means of the second order Clarke subdifferential. Some illustrative examples are also given.