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Journal ArticleDOI

Stability of Locally Optimal Solutions

01 Jun 1999-Siam Journal on Optimization (Society for Industrial and Applied Mathematics)-Vol. 10, Iss: 2, pp 580-604
TL;DR: Property of prox-regularity of the essential objective function and positive definiteness of its coderivative Hessian are the keys to the Lipschitzian stability of local solutions to finite-dimensional parameterized optimization problems in a very general setting.
Abstract: Necessary and sufficient conditions are obtained for the Lipschitzian stability of local solutions to finite-dimensional parameterized optimization problems in a very general setting. Properties of prox-regularity of the essential objective function and positive definiteness of its coderivative Hessian are the keys to these results. A previous characterization of tilt stability arises as a special case.
Citations
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01 Jan 2002

230 citations

Journal ArticleDOI
TL;DR: In this paper, the second-order generalized differentiation theory of variational analysis is applied to some problems of constrained optimization in finite-dimensional spaces, such as nonlinear programming and extended nonlinear programs described in composite terms.
Abstract: This paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finite- dimensional spaces. The main focus is the so-called (full and partial) second-order subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of first-order subdifferential mappings. We develop an extended second-order subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal second-order chain rule for strongly and fully amenable compositions. We also calculate the second- order subdifferentials for some major classes of piecewise linear-quadratic functions. These results are applied to the study of tilt stability of local minimizers for important classes of problems in constrained optimization that include, in particular, problems of nonlinear programming and certain classes of extended nonlinear programs described in composite terms.

129 citations

Journal ArticleDOI
TL;DR: The author proves the local convergence of multiplier methods for nonmonotone variational inequalities and nonconvex nonlinear programming by combining the new convergence results with an abstract duality framework for variational inclusions.
Abstract: This paper studies the convergence of the classical proximal point algorithm without assuming monotonicity of the underlying mapping. Practical conditions are given that guarantee the local convergence of the iterates to a solution ofT( x) ? 0, whereT is an arbitrary set-valued mapping from a Hilbert space to itself. In particular, when the problem is "nonsingular" in the sense thatT-1 has a Lipschitz localization around one of the solutions, local linear convergence is obtained. This kind of regularity property of variational inclusions has been extensively studied in the literature under the name ofstrong regularity, and it can be viewed as a natural generalization of classical constraint qualifications in nonlinear programming to more general problem classes. Combining the new convergence results with an abstract duality framework for variational inclusions, the author proves the local convergence of multiplier methods for a very general class of problems. This gives as special cases new convergence results for multiplier methods for nonmonotone variational inequalities and nonconvex nonlinear programming.

121 citations


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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Journal ArticleDOI
TL;DR: Applications of the obtained results to Lipschitzian stability of parametric variational and hemivariational inequalities and efficiently express the derived conditions in terms of the initial data for selected problems of continuum mechanics are provided.
Abstract: We study second-order subdifferentials of nonsmooth functions that are particularly important for applications to sensitivity analysis in optimization and related problems First we develop various calculus rules for these subdifferentials in rather general settings Then we obtain exact formulas for computing the second-order subdifferentials for a class of separable piecewise smooth functions Functions of this class arise, in particular, in equilibrium models related to some practical problems of continuum mechanics Finally we provide applications of the obtained results to Lipschitzian stability of parametric variational and hemivariational inequalities and efficiently express the derived conditions in terms of the initial data for selected problems of continuum mechanics

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
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Journal ArticleDOI
TL;DR: A regularity condition is introduced for generalized equations and it is shown to be in a certain sense the weakest possible condition under which the stated properties will hold.
Abstract: This paper considers generalized equations, which are convenient tools for formulating problems in complementarity and in mathematical programming, as well as variational inequalities. We introduce a regularity condition for such problems and, with its help, prove existence, uniqueness and Lipschitz continuity of solutions to generalized equations with parametric data. Applications to nonlinear programming and to other areas are discussed, and for important classes of such applications the regularity condition given here is shown to be in a certain sense the weakest possible condition under which the stated properties will hold.

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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Journal ArticleDOI
TL;DR: The Lipschitz dependence of the set of solutions of a convex minimization problem and its Lagrange multipliers upon the natural parameters from an inverse function theorem for set-valued maps is derived.
Abstract: We derive the Lipschitz dependence of the set of solutions of a convex minimization problem and its Lagrange multipliers upon the natural parameters from an inverse function theorem for set-valued maps. This requires the use of contingent and Clarke derivatives of set-valued maps, as well as generalized second derivatives of convex functions.

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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Journal ArticleDOI
TL;DR: In this article, the authors consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior.
Abstract: We consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior. These properties have many important applications to various problems in nonlinear analysis, optimization, control theory, etc., especially for studying sensitivity and stability questions with respect to perturbations of initial data and parameters. We establish interrelations between these properties and prove effective criteria for their fulfillment stated in terms of robust generalized derivatives for multifunctions and nonsmooth mappings

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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Journal ArticleDOI
TL;DR: The class of prox-regular functions covers all lsc, proper, convex functions, lower-C2 functions and strongly amenable functions, hence a large core of functions of interest in variational analysis and optimization as mentioned in this paper.
Abstract: The class of prox-regular functions covers all lsc, proper, convex functions, lower-C2 functions and strongly amenable functions, hence a large core of functions of interest in variational analysis and optimization The subgradient mappings associated with prox-regular functions have unusually rich properties, which are brought to light here through the study of the associated Moreau envelope functions and proximal mappings Connections are made between second-order epi-derivatives of the functions and proto-derivatives of their subdifferentials Conditions are identified under which the Moreau envelope functions are convex or strongly convex, even if the given functions are not

337 citations