Fixed point theory via nonsmooth analysis

Clarke, Frank H.; Ledyaev, Yu. S.; Stern, Ronald J.

doi:10.1090/conm/204/02624

Cited by 136 publications

(236 citation statements)

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“…However, many of the basic ideas driving this development extend to nonconvex settings, a central example being the normal cone and its continuity properties. Nonconvex variational analysis has grown into a complete, powerful and elegant theory over the past several decades: fine expositions may be found in [27,7,34]. Extending the results described in this paper to nonconvex settings is the topic of ongoing research.…”

Section: Introductionmentioning

confidence: 87%

Generic Optimality Conditions for Semialgebraic Convex Programs

Bolte¹,

Daniilidis

Lewis

2011

Mathematics of OR

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We consider linear optimization over a nonempty convex semialgebraic feasible region F . Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique "active" manifold, around which F is "partly smooth", and the second-order sufficient conditions hold. Perturbing the objective results in smooth variation of the optimal solution. The active manifold consists, locally, of these perturbed optimal solutions; it is independent of the representation of F , and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes. These results extend to unbounded sets F .

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Section: Introductionmentioning

confidence: 87%

Generic Optimality Conditions for Semialgebraic Convex Programs

Bolte¹,

Daniilidis

Lewis

2011

Mathematics of OR

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“…The extension to higher dimension is however far more involved than the extension of classical differential calculus to higher dimensions. The domain-theoretic derivative is closely related to the so-called generalized (or Clarke's) gradient, which is a key tool in nonsmooth analysis, control theory and optimization theory [3,4]. For any locally Lipschitz function, the domain-theoretic derivative at a point gives the smallest hyper-rectangle, with sides parallel to the coordinate planes, which contains the Clarke's gradient.…”

Section: Related Workmentioning

confidence: 99%

A Computational Model for Multi-variable Differential Calculus

Edalat

Lieutier

Pattinson

2005

Foundations of Software Science and Computational Structures

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Abstract. We introduce a domain-theoretic computational model for multivariable differential calculus, which for the first time gives rise to data types for differentiable functions. The model, a continuous Scott domain for differentiable functions of n variables, is built as a sub-domain of the product of n + 1 copies of the function space on the domain of intervals by tupling together consistent information about locally Lipschitz (piecewise differentiable) functions and their differential properties (partial derivatives). The main result of the paper is to show, in two stages, that consistency is decidable on basis elements, which implies that the domain can be given an effective structure. First, a domain-theoretic notion of line integral is used to extend Green's theorem to interval-valued vector fields and show that integrability of the derivative information is decidable. Then, we use techniques from the theory of minimal surfaces to construct the least and the greatest piecewise linear functions that can be obtained from a tuple of n+1 rational step functions, assuming the integrability of the n-tuple of the derivative part. This provides an algorithm to check consistency on the rational basis elements of the domain, giving an effective framework for multi-variable differential calculus.

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“…This later notion is particularly relevant for the investigation regarding variational behavior of "order two" (cf [11,12,30,35]). We recall that x * ∈ N P (A, a) if and only if there exist σ, δ ∈ (0, +∞) such that…”

Section: Preliminariesmentioning

confidence: 99%

Metric subregularity for proximal generalized equations in Hilbert spaces

Zheng

2012

Nonlinear Analysis: Theory, Methods & Applications

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In this paper, we introduce and consider the concept of the prox-regularity of a multifunction. We mainly study the metric subregularity of a generalized equation defined by a proximal closed multifunction between two Hilbert spaces. Using proximal analysis techniques, we provide sufficient and/or necessary conditions for such a generalized equation to have the metric subregularity in Hilbert spaces. We also establish results of Robinson-Ursescu theorem type for prox-regular multifunctions.

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Fixed point theory via nonsmooth analysis

Cited by 136 publications

References 0 publications

Generic Optimality Conditions for Semialgebraic Convex Programs

Generic Optimality Conditions for Semialgebraic Convex Programs

A Computational Model for Multi-variable Differential Calculus

Metric subregularity for proximal generalized equations in Hilbert spaces

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