Higher-order set-valued Hadamard directional derivatives: calculus rules and sensitivity analysis of equilibrium problems and generalized equations

Nguyen Minh Tung () and Nguyen Xuan Duy Bao ()
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Nguyen Minh Tung: Banking University of Ho Chi Minh City
Nguyen Xuan Duy Bao: University of Science

Journal of Global Optimization, 2022, vol. 83, issue 2, No 10, 377-402

Abstract: Abstract In this paper, we propose a notion of higher-order directional derivatives in the sense of Hadamard for set-valued maps, which is a natural extension of the classical directional derivatives. Some of the usual calculus rules, for unions, intersections, products, sums, and compositions are given under directional metric subregularity conditions. The Hadamard differentiability of the efficient value mapping and a formula to compute its derivative are also obtained. Then, we apply these derivatives to establish an implicit set-valued map theorem and employ it to higher-order sensitivity analysis of the solution mapping for a parametric vector equilibrium problem. Sensitivity for solutions to a parametric generalized equation is also investigated. Many examples are provided for analyzing and illustrating the obtained results.

Keywords: Sensitivity analysis; Generalized equation; Equilibrium problem; Robinson directional metric subregularity; Hadamard derivative; Studniarski’s derivative; 49Q12; 54C60; 90C31; 90C33 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10898-021-01090-3

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