Conditions on Twice Continuous Codifferentiability of Quasidifferentiable Functions

I. M. Prudnikov () and N. S. Duxbury ()
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I. M. Prudnikov: Scientific Center of Smolensk Federal Medical University
N. S. Duxbury: Space Science Institute

Journal of Optimization Theory and Applications, 2024, vol. 203, issue 3, No 26, 2848-2869

Abstract: Abstract The authors study the twice codifferentiable functions introduced by Professor V.F. Demyanov and how to calculate their second codifferentials. We proved that a twice hypodifferentiable positively homogeneous (p.h.) function of the second order is the maximum of quadratic forms over some set of matrices from a convex compact set, which coincides with the subdifferential of the second order introduced by the authors. This paper is the logical continuation of previous papers. The subdifferentials of the first and second orders are used to calculate the second codifferential at a point up to equivalence. The proved theorems give the rules for calculation of the second codifferentials. The theorem deals with twice continuous codifferentiability of a locally Lipschitz quasidifferentiable function whose second subdifferential and second superdifferential are upper semicontinuous (USC).

Keywords: Positively homogeneous functions; Quasidifferentiable functions; Generalized gradients; The Clarke subdifferential; Codifferentiable functions; Subdifferential of the first and second orders; Second codifferential; Generalized matrices; Necessary and sufficient conditions of optimality; Twice continuous codifferentiability; 49J52; 90C30; 90C31 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02549-5

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