Stability of Locally Optimal Solutions
Citations
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Cites background from "Stability of Locally Optimal Soluti..."
...…in the literature, and there are available specific conditions guaranteeing it for many classes of problems; see, for example, Robinson (1980), Klatte and Tammer (1990), Dontchev and Rockafellar (1996, 1998, 2001), Klatte and Kummer (1999), Levy et al. (2000), and the references therein....
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94 citations
Cites background from "Stability of Locally Optimal Soluti..."
...tional inequalities over convex polyhedra in [4], to second-order characterizations of stable optimal solutions to nonsmooth optimization problems in [20] and [6], and to necessary optimality conditions obtained in [17], [18], [19], [24], [25], [26], and [27] for various problems of hierarchical optimization unified under the name of mathematical programs with equilibrium constraints [7]....
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References
975 citations
"Stability of Locally Optimal Soluti..." refers background in this paper
...S. M. Robinson, “Strongly regular generalized equations,” Math....
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...The pioneering contribution of Robinson [1] put the focus on single-valued Lipschitzian behavior of optimal solutions....
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377 citations
"Stability of Locally Optimal Soluti..." refers background in this paper
...The Aubin property of G at ū for (x̄, v̄) entails the Aubin property at u for (x, v) whenever (u, x, v) is near enough to (ū, x̄, v̄) in gphG....
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...J.-P. Aubin, “Lipschitz behavior of solutions to convex minimization problems,” Math....
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...Crucial among them is the form of localized Lipschitz continuity for set-valued mappings that was defined by Aubin [4] and the criterion for it that was derived by Mordukhovich [5] in terms of his coderivative mappings....
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...With convenient adjustments of notation to fit the epigraphical setting, the Aubin property in question can be identified with the existence of neighborhoods X1 of x̄ and U1 of ū along with ε > 0 and κ ≥ 0 such that, for all u, u′ ∈ U1, one has [epi fu] ∩ ( [X1 × [ᾱ− ε, ᾱ+ ε] ) ⊂ [epi fu′ ] + κ|u′ − u| ( IB × [−1, 1] ) , or in other words the implication x ∈ X1 α ≥ f(x, u) |α− ᾱ| ≤ ε =⇒ ∃(x′, α′) with f(x′, u′) ≤ α′, |x′ − x| ≤ κ|u′ − u|, |α′ − α| ≤ κ|u′ − u|....
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...With such closedness, G has the Aubin property at ū for (x̄, v̄) if and only if the Mordukhovich criterion is satisfied, namely that u′ ∈ D∗G(ū | x̄, v̄)(0, 0) only for u′ = 0....
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345 citations
"Stability of Locally Optimal Soluti..." refers background in this paper
...criterion has been found by Mordukhovich [5], [12], [13]: As long as gphS is closed relative to a neighborhood of (z̄, w̄), the Aubin property holds if and only if...
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337 citations