A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems
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A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems Alexey F. Izmailov · Alexey S. Kurennoy · Mikhail V. Solodov
Received: 2 May 2011 / Accepted: 11 August 2012 / Published online: 2 September 2012 © Springer and Mathematical Optimization Society 2012
Abstract We prove a new local upper Lipschitz stability result and the associated local error bound for solutions of parametric Karush–Kuhn–Tucker systems corresponding to variational problems with Lipschitzian base mappings and constraints possessing Lipschitzian derivatives, and without any constraint qualifications. This property is equivalent to the appropriately extended to this nonsmooth setting notion of noncriticality of the Lagrange multiplier associated to the primal solution, which is weaker than second-order sufficiency. All this extends several results previously known only for optimization problems with twice differentiable data, or assuming some constraint qualifications. In addition, our results are obtained in the more general variational setting. Keywords KKT system · Error bound · Second-order sufficiency · Critical multipliers
Research of the first two authors is supported by the Russian Foundation for Basic Research Grant 10-01-00251. The third author is supported in part by CNPq Grant 302637/2011-7, by PRONEX-Optimization, and by FAPERJ. A. F. Izmailov, A. S. Kurennoy Uchebniy Korpus 2, OR Department, VMK Faculty, Moscow State University, MSU, Leninskiye Gory, 119991 Moscow, Russia e-mail: [email protected] A. S. Kurennoy e-mail: [email protected] M. V. Solodov (B) IMPA-Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil e-mail: [email protected]
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Mathematics Subject Classification (2000)
90C30 · 90C55 · 65K05
1 Introduction Given the base mapping Φ : Rn → Rn and the constraints mappings h : Rn → Rl and g : Rn → Rm , consider the variational problem x ∈ D, Φ(x), ξ ≥ 0 ∀ ξ ∈ TD (x),
(1)
where D = {x ∈ Rn | h(x) = 0, g(x) ≤ 0}, and TD (x) is the usual tangent (contingent) cone to D at x ∈ D. This problem setting is fairly general. In particular, it contains optimality conditions: for a given smooth function f : Rn → R, any local solution of the optimization problem minimize f (x) subject to h(x) = 0, g(x) ≤ 0,
(2)
necessarily satisfies (1) with the base mapping defined by Φ(x) = f (x), x ∈ Rn .
(3)
Assuming that the constraints mappings h and g are smooth, for any local solution x¯ of problem (1) under the appropriate constraint qualifications there exists a multiplier (λ, μ) ∈ Rl × Rm satisfying the Karush–Kuhn–Tucker (KKT) system Φ(x) + (h (x))T λ + (g (x))T μ = 0, h(x) = 0, μ ≥ 0, g(x) ≤ 0, μ, g(x) = 0
(4)
for x = x. ¯ In particular, if (3) holds then (4) is the KKT system of optimization problem (2). Let M(x) ¯ stand for the set of such multipliers associated with x. ¯ Define the mapping Ψ : Rn × Rl × Rm → Rn by Ψ (x, λ, μ) = Φ(x) + (h (x))T λ + (g (x))T μ.
(5)
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