The Stationary Point Set Map in General Parametric Optimization Problems
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The Stationary Point Set Map in General Parametric Optimization Problems D. T. K. Huyen1
· J.-C. Yao2 · N. D. Yen3
Received: 21 July 2019 / Accepted: 16 September 2020 / © Springer Nature B.V. 2020
Abstract The present paper shows how the linear independence constraint qualification (LICQ) can be combined with some conditions put on the first-order and second-order derivatives of the objective function and the constraint functions to ensure the Robinson stability and the Lipschitz-like property of the stationary point set map of a general C 2 -smooth parametric constrained optimization problem. So, a part of the results in two preceding papers of the authors [J. Optim. Theory Appl. 180 (2019), 91–116 (Part 1); 117–139 (Part 2)], which were obtained for a problem with just one inequality constraint, now has an adequate extension for problems having finitely many equality and inequality constraints. Our main tool is an estimate of B. S. Mordukhovich and R. T. Rockafellar [SIAM J. Optim. 22 (2012), 953– 986; Theorem 3.3] for a second-order partial subdifferential of a composite function. The obtained results are illustrated by three examples. Keywords C 2 -smooth optimization problem · Stationary point set · Total perturbation · Robinson stability · Lipschitz-like property · Second-order subdifferential · Coderivative Mathematics Subject Classification (2010) 49K40 · 49J53 · 90C31 · 90C20
1 Introduction Mathematical programming problems given by twice continuously differentiable functions (C 2 -smooth functions for brevity) are studied widely in optimization theory. Concerning D. T. K. Huyen
[email protected] J.-C. Yao [email protected] N. D. Yen [email protected] 1
Department of Mathematics, Hanoi Pedagogical University 2, Vinh Phuc, Vietnam
2
Center for General Education, China Medical University, Taichung, 40402, Taiwan
3
Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam
D.T.K. Huyen et al.
optimality conditions and numerical algorithms for such problems, we refer the readers to [4, 38] and [6, 28, 30, 38]. When the data set of the problem is perturbed, the optimal value, the solution set, the local solution set, and stationary point set change. They are functions or multifunctions of the perturbation parameters. In the differential stability theory, various differentiability properties of the optimal value function have been established (see, e.g., [7–9, 21]). The stationary point set mapping can be studied via variational systems (see [18, Chap. 3], [19, Chap. 4], and the references therein). Continuity and Lipschitz continuity properties of the solution map and the local solution map have been studied by Robinson [35], Alt [1, 2], Bonnans and Shapiro [4], and other authors. For instance, Chapter 4 of the book [4] is devoted to stability and sensitivity analysis of parameterized optimization problems of the form Minimize f (x, w) subject to F (x, w) ∈ K,
(1.1)
where w ∈ W is the parameter, f : X × W → R and F : X × W → Y are given maps, and the set K in Y is
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