On the Convergence of Sequences of Convex Sets in Finite Dimensions

Salinetti, Gabriella; Wets, Roger J.-B.

doi:10.1137/1021002

Cited by 153 publications

(66 citation statements)

References 10 publications

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“…Such set convergence can be characterized in many ways, which we will not attempt to review (see Salinetti and Wets [39], for instance.) Epi-convergence of rp t to rp can itself be shown to be equivalent to the relation…”

Section: Epi-derivatives Of Convex Functionsmentioning

confidence: 99%

Generalized second derivatives of convex functions and saddle functions

Rockafellar¹

1990

Trans. Amer. Math. Soc.

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ABSTRACT. The theory of second-order epi-derivatives of extended-real-valued functions is applied to convex functions on ]K" and shown to be closely tied to proto-differentiation of the corresponding subgradient muitifunctions, as well as to second-order epi-differentiation of conjugate functions. An extension is then made to saddle functions, which by definition are convex in one argument and concave in another. For this case a concept of epi-hypo-differentiability is introduced. The saddle function results provide a foundation for the sensitivity analysis of primal and dual optimal solutions to general finite-dimensional problems in convex optimization, since such solutions are characterized as saddlepoints of a convex-concave Lagrangian function, or equivalently as subgradients of the saddle function conjugate to the Lagrangian.

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Section: Epi-derivatives Of Convex Functionsmentioning

confidence: 99%

Generalized second derivatives of convex functions and saddle functions

Rockafellar¹

1990

Trans. Amer. Math. Soc.

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“…In fact, this equivalence holds assuming only that the limit set is bounded, a fact noted by Salinetti and Wets in finite dimensions [38]. As a result, the topology xaw reduces to the usually stronger Hausdorff metric topology when restricted to the bounded elements of W(X).…”

Section: Some Tool Theorems For Xaw -Convergencementioning

confidence: 91%

Convex optimization and the epi-distance topology

Beer¹,

Lucchetti²

1991

Trans. Amer. Math. Soc.

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Abstract. Let T{X) denote the proper, lower semicontinuous, convex functions on a Banach space X , equipped with the completely metrizable topology t of uniform convergence of distance functions on bounded sets. A function / in T(X) is called well-posed provided it has a unique minimizer, and each minimizing sequence converges to this minimizer. We show that well-posedness of / e T(X) is the minimal condition that guarantees strong convergence of approximate minima of r-approximating functions to the minimum of /.Moreover, we show that most functions in (T(X), Taw ) are well-posed, and that this fails if T(X) is topologized by the weaker topology of Mosco convergence, whenever X is infinite dimensional. Applications to metric projections are also given, including a fundamental characterization of approximative compactness.

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“…Consider any sequence {X k } of non-empty, compact, convex and increasing subsets of X. The sequence then has a set limit (e.g., Mosco [65], and Salinetti and Wets [85]), say X ⊆ X, which is also non-empty, compact and convex. Further, let, for all k, x k ∈ X * k .…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…Thus, it has a set limit (e.g., Mosco [65] and Salinetti and Wets [85]), say X ⊆ X, which is also non-empty, compact and convex.…”

Section: Remark 2 (Initiation)mentioning

confidence: 99%

A generic column generation principle: derivation and convergence analysis

2015

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Given a non-empty, compact and convex set, and an a priori defined condition which each element either satisfies or not, we want to find an element belonging to the former category. This is a fundamental problem of mathematical programming which encompasses nonlinear programs, variational inequalities, and saddle-point problems.We present a conceptual column generation scheme, which alternates between solving a restriction of the original problem and a column generation phase which is used to augment the restricted problems. We establish the general applicability of the conceptual method, as well as to the three problem classes mentioned. We also establish a version of the conceptual method in which the restricted and column generation problems are allowed to be solved approximately, and of a version allowing for the dropping of columns.We show that some solution methods (e.g., Dantzig-Wolfe decomposition and simplicial decomposition) are special instances, and present new convergent column generation methods in nonlinear programming, such as a sequential linear programming (SLP) type method. Along the way, we also relate our quite general scheme in nonlinear programming presented in this paper with several other classic, and more recent, iterative methods in nonlinear optimization.

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On the Convergence of Sequences of Convex Sets in Finite Dimensions

Cited by 153 publications

References 10 publications

Generalized second derivatives of convex functions and saddle functions

Generalized second derivatives of convex functions and saddle functions

Convex optimization and the epi-distance topology

A generic column generation principle: derivation and convergence analysis

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