About well-posed optimization problems for functionals in metric spaces
01 Mar 1970-Journal of Optimization Theory and Applications (Kluwer Academic Publishers-Plenum Publishers)-Vol. 5, Iss: 3, pp 225-229
TL;DR: A necessary and sufficient condition of correctness of extremal problems for lower semicontinuous functionals defined in metric spaces is given in this paper, which is a necessary condition for extremal problem correctness.
Abstract: A necessary and sufficient condition of correctness of extremal problems for lower semicontinuous functionals defined in metric spaces is given.
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TL;DR: In this paper, the authors present a survey of work concerning limit-compact operators, measures of noncompactness, and condensing operators, which is a generalization of the theory of completely continuous and contracting operators.
Abstract: The paper contains a survey of investigations concerned with three new concepts: limit-compact operators, measures of non-compactness, and condensing operators. A measure of non-compactness is a function of a set that is invariant under the transition to the closed convex hull of the set. If a certain measure of non-compactness is defined in a space, a condensing operator is defined, roughly speaking, as an operator that decreases the measure of non-compactness of any set whose closure is not compact. The more general concept of a limit-compact operator is defined by means of a property common to all condensing operators; it can be formulated in terms not related to measures of non-compactness. The theory of limit-compact operators can be regarded as a simultaneous generalization of the theory of completely continuous and contracting operators. For non-linear operators the main result is the construction of the theory of the rotation of limit-compact vector fields and, in particular, the proof of a number of new fixed-point principles (Chapter 3 of the present paper). In the theory of linear operators a number of results are obtained that are related to the concept of a Fredholm operator and the Fredholm spectrum of an operator (Chapter 2). The theory of measures of non-compactness and condensing operators has found different applications in general topology, in the theory of ordinary differential equations, functional-differential equations, partial differential equations, the theory of extrema of functionals, etc. The paper contains several examples concerning differential equations in a Banach space and functional-differential equations of neutral type. These examples do not have a special significance but are chosen merely to illustrate the methods. They are therefore investigated with neither maximal generality nor completeness.
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TL;DR: In this article, the well-posedness concept introduced in Ref. 1 for global optimization problems with a unique solution is generalized here to problems with many minimizers, under the name of extended wellposedness.
Abstract: The well-posedness concept introduced in Ref. 1 for global optimization problems with a unique solution is generalized here to problems with many minimizers, under the name of extended well-posedness. It is shown that this new property can be characterized by metric criteria, which parallel to some extent those known about generalized Tikhonov well-posedness.
128 citations
TL;DR: In this article, the authors give a characterization of Tyhonov well-posedness for the problem of minimizing a convex lower-semieontinuous function f on a closed convex set K.
Abstract: We give a characterization of Tyhonov well-posedness for the problem of minimising a convex lower-semieontinuous function f on a closed convex set K. To get this result, we use Ekeland's theorem on the“approximate variational principle”[21; when f is differentiable, the condition is and diam , where . We prove also some results relating the diameter of level sets of a sub- homogeneous function g with a function which characterizes the minimum increase of g above the minimum value. This result allows us to relate our characterization with a former one due to Zolezzi [10] Then we use the condition on diam(T∊) to give a definition of“well-posedness”for variational inequalities and prove some related results.
124 citations
TL;DR: This paper considers Levitin--Polyak-type well-posedness for a general constrained optimization problem and introduces generalized and strongly generalized LevitIn-- Polyak well-posingness.
Abstract: In this paper, we consider Levitin--Polyak-type well-posedness for a general constrained optimization problem. We introduce generalized Levitin--Polyak well-posedness and strongly generalized Levitin--Polyak well-posedness. Necessary and sufficient conditions for these types of well-posedness are given. Relations among these types of well-posedness are investigated. Finally, we consider convergence of a class of penalty methods and a class of augmented Lagrangian methods under the assumption of strongly generalized Levitin--Polyak well-posedness.
106 citations
Cites methods from "About well-posed optimization probl..."
...Next we give a necessary and sufficient condition in the form of Furi and Vignoli [10] to characterize the LP well-posedness in the strongly generalized sense....
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...[10] M. Furi and A. Vignoli, About well-posed minimization problems for functionals in metric spaces, J. Optim....
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TL;DR: In this article, it was shown that well-posedness of convex functions on a Banach space X is the minimal condition that guarantees strong convergence of approximate minima of T-approximating functions to the minimum of f.
Abstract: Let 17(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 f in 17(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 f e r(X) is the minimal condition that guarantees strong convergence of approximate minima of T-approximating functions to the minimum of f . Moreover, we show that most functions in (17(X), Taw ) are well-posed, and that this fails if 17(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.
81 citations
Cites background from "About well-posed optimization probl..."
...We show that when convergence means raw -convergence, this behavior is guaranteed if and only if / is well-posed [40, 21, 29]: / has a unique minimizer x0 , and each minimizing sequence for / converges to x0 ....
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TL;DR: In this paper, the conditions générales d'utilisation (http://www.numdam.math.unipd.org/conditions) of the agreement with the Rendiconti del Seminario Matematico della Università di Padova are discussed.
Abstract: L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
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