Proximal Normal Cone Research Articles

Minimizing Curves in Prox-regular Subsets of Riemannian Manifolds

We obtain a characterization of the proximal normal cone to a prox-regular subset of a Riemannian manifold and some properties of Bouligand tangent cones to these sets are presented. Moreover, we show that on an open neighborhood of a prox-regular set, the metric projection is locally Lipschitz and it is directionally differentiable at the boundary points of the set. Finally, a necessary condition for a curve to be a minimizing curve in a prox-regular set is derived.

Criteria for Convexity of Closed Sets in Banach Spaces

Criteria for the convexity of closed sets in general Banach spaces in terms of the Clarke and Bouligand tangent cones are proved. In the case of uniformly convex spaces, these convexity criteria are stated in terms of proximal normal cones. These criteria are used to derive sufficient conditions for the convexity of the images of convex sets under nonlinear mappings and multifunctions.

Approximate proper efficiency for multiobjective optimization problems

This paper is devoted to the study of a new kind of approximate proper efficiency in terms of proximal normal cone and co-radiant set for multiobjective optimization problem. We derive some properties of the new approximate proper efficiency and discuss the relations with the existing approximate concepts, such as approximate efficiency and approximate Benson proper efficiency. At last, we study the linear scalarizations for the new approximate proper efficiency under the generalized convexity assumption and give some examples to illustrate the main results.

On state-dependent sweeping process in Banach spaces

In this paper we prove, in a separable reflexive uniformly smooth Banach space, the existence of solutions of a perturbed first order differential inclusion governed by the proximal normal cone to a moving set depending on the time and on the state. The perturbation is assumed to be separately upper semicontinuous.

A second order differential inclusion with proximal normal cone in Banch spaces

In the present paper we mainly consider the second order evolution inclusion with proximal normal cone: $$ \begin{cases} -\ddot{x}(t)\in N_{K(t)}(\dot{x}(t))+F(t,x(t),\dot{x}(t)), \quad \textmd{a.e.}\\ \dot x(t)\in K(t),\\ x(0)=x_0,\quad\dot x(0)=u_0, \end{cases} \leqno{(*)} $$ where $t\in I=[0,T]$, $E$ is a separable reflexive Banach space, $K(t)$ a ball compact and $r$-prox-regular subset of $E$, $N_{K(t)}(\cdot)$ the proximal normal cone of $K(t)$ and $F$ an u.s.c. set-valued mapping with nonempty closed convex values. First, we prove the existence of solutions of $(*)$. After, we give an other existence result of $(*)$ when $K(t)$ is replaced by $K(x(t))$.

Calculus Rules forV-Proximal Subdifferentials in Smooth Banach Spaces

In 2010, Bounkhel et al. introduced new proximal concepts (analytic proximal subdifferential, geometric proximal subdifferential, and proximal normal cone) in reflexive smooth Banach spaces. They proved, inp-uniformly convex andq-uniformly smooth Banach spaces, the density theorem for the new concepts of proximal subdifferential and various important properties for both proximal subdifferential concepts and the proximal normal cone concept. In this paper, we establish calculus rules (fuzzy sum rule and chain rule) for both proximal subdifferentials and we prove the Bishop-Phelps theorem for the proximal normal cone. The limiting concept for both proximal subdifferentials and for the proximal normal cone is defined and studied. We prove that both limiting constructions coincide with the Mordukhovich constructions under some assumptions on the space. Applications to nonconvex minimisation problems and nonconvex variational inequalities are established.

Sweeping process by prox-regular sets in Riemannian Hilbert manifolds

In this paper, we deal with sweeping processes on (possibly infinite-dimensional) Riemannian Hilbert manifolds. We extend the useful notions (proximal normal cone, prox-regularity) already defined in the setting of a Hilbert space to the framework of such manifolds. Especially we introduce the concept of local prox-regularity of a closed subset in accordance with the geometrical features of the ambient manifold and we check that this regularity implies a property of hypomonotonicity for the proximal normal cone. Moreover we show that the metric projection onto a locally prox-regular set is single-valued in its neighborhood. Then under some assumptions, we prove the well-posedness of perturbed sweeping processes by locally prox-regular sets.

Proximal Normal Cone Analysis on Smooth Banach Spaces and Applications

Unlike the Frechet, Penot, Clarke, and other “order one” notions of normal cones, the notion of proximal normal cone describes variational behavior of “order two,” and most of results on proximal normal cone are established in the Hilbert space framework. A possible reason is that in a general Banach space (such as in the classical Banach spaces $l^p$ and $L^p$ with $1 \leq p < 2$), the proximal normal cone of a “smooth” subset $A$ at every point of $A$ may consist of the zero element only. Nevertheless, on the other hand, we show in this paper that for a fairly big class of smooth Banach spaces (including the classical Banach spaces $l^p$ and $L^p$ with $2 \leq p <\infty$), the consideration of proximal normal cones provides a lot of useful information, especially for leading to satisfactory treatment for constrained optimization problems. Some of our results are new even in the Hilbert space case.

Lagrange Multiplier Rules for Weak Approximate Pareto Solutions of Constrained Vector Optimization Problems in Hilbert Spaces

In the Hilbert space case, in terms of proximal normal cone and proximal coderivative, we establish a Lagrange multiplier rule for weak approximate Pareto solutions of constrained vector optimization problems. In this case, our Lagrange multiplier rule improves the main result on vector optimization in Zheng and Ng (SIAM J. Optim. 21: 886---911, 2011). We also introduce a notion of a fuzzy proximal Lagrange point and prove that each Pareto (or weak Pareto) solution is a fuzzy proximal Lagrange point.

Existence of solutions for second-order differential inclusions involving proximal normal cones

In this work, we prove global existence of solutions for second-order differential problems in a general framework. More precisely, we consider second-order differential inclusions involving proximal normal cone to a set-valued map. This set-valued map is supposed to take admissible values (so in particular uniformly prox-regular values, which may be non-smooth and non-convex). Moreover the solution is required to satisfy an impact law, appearing in the description of mechanical systems with inelastic shocks.

EXISTENCE RESULTS FOR NONSMOOTH SECOND-ORDER DIFFERENTIAL INCLUSIONS, CONVERGENCE RESULT FOR A NUMERICAL SCHEME AND APPLICATION TO THE MODELING OF INELASTIC COLLISIONS

We are interested in the existence results for second-order differential inclusions, involving finite number of unilateral constraints in an abstract framework. These constraints are described by a set-valued operator, more precisely a proximal normal cone to a time-dependent set. In order to prove these existence results, we study an extension of the numerical scheme introduced in [10] and prove a convergence result for this scheme.

Prox-regular sets and epigraphs in uniformly convex Banach spaces: Various regularities and other properties

We continue the study of prox-regular sets that we began in a previous work in the setting of uniformly convex Banach spaces endowed with a norm both uniformly smooth and uniformly convex (e.g., L p , W m,p spaces). We prove normal and tangential regularity properties for these sets, and in particular the equality between Mordukhovich and proximal normal cones. We also compare in this setting the proximal normal cone with different Holderian normal cones depending on the power types s, q of moduli of smoothness and convexity of the norm. In the case of sets that are epigraphs of functions, we show that J-primal lower regular functions have prox-regular epigraphs and we compare these functions with Poliquin's primal lower nice functions depending on the power types s, q of the moduli. The preservation of prox-regularity of the intersection of finitely many sets and of the inverse image is obtained under a calmness assumption. A conical derivative formula for the metric projection mapping of prox-regular sets is also established. Among other results of the paper it is proved that the Attouch-Wets convergence preserves the uniform r-prox-regularity property and that the metric projection mapping is in some sense continuous with respect to this convergence for such sets.

Proximal analysis in reflexive smooth Banach spaces

In the present paper, we introduce and study a new proximal normal cone in reflexive Banach spaces in terms of a generalized projection operator. Two new variants of generalized proximal subdifferentials are also introduced in reflexive smooth Banach spaces. The density theorem for both proximal subdifferentials has been proved in p -uniformly convex and q -uniformly smooth Banach spaces. Various important properties and applications of our concepts are also proved.

Generalizing Mutational Equations for Uniqueness of Some Nonlocal 1st-Order Geometric Evolutions

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painleve–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution-like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of ℝ N evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.

Differentiability properties for a class of non-convex functions

Closed sets K ⊂ \(\mathbb R^{n}\) satisfying an external sphere condition with uniform radius (called ϕ-convexity or proximal smoothness) are considered. It is shown that for \(\mathcal H^{n-1}\)-a.e. x ∊ ∂K the proximal normal cone to K at x has dimension one. Moreover if K is the closure of an open set satisfying a (sharp) nondegeneracy condition, then the De Giorgi reduced boundary is equivalent to ∂ K and the unit proximal normal equals \(\mathcal H^{n-1}\)-a.e. the (De Giorgi) external normal. Then lower semicontinuous functions f : \(\mathbb R^{n}\rightarrow \mathbb R\cup\{ +\infty\}\) with ϕ-convex epigraph are shown, among other results, to be locally BV and twice \(\mathcal L^{n}\)-a.e. differentiable; furthermore, the lower dimensional rectifiability of the singular set where f is not differentiable is studied. Finally we show that for \(\mathcal L^{n}\)-a.e. x there exists δ (x) > 0 such that f is semiconvex on B(x,δ(x)). We remark that such functions are neither convex nor locally Lipschitz, in general. Methods of nonsmooth analysis and of geometric measure theory are used.

Proximal proper efficiency for minimisation with respect to normal cones

This paper is devoted to the study of a new kind of proper efficiency in terms of proximal normal cones for vector minimisation. This new notion called proximal proper efficiency is used to obtain a scalar characterisation when a set related to the criterion set is a nonconvex set. Proximal proper efficiency is related with the well known notions of Benson and Borwein proper efficiency which are defined in the literature in terms of tangent cones. The study is further extended to characterise Benson and Borwein proper efficiency in terms of normal cones assuming convexity of a related set.

Weak Sharp Minima: Characterizations and Sufficient Conditions

The problem of identifying weak sharp minima of order m, an important class of (possibly) nonisolated minima, is investigated in this paper. Some sufficient conditions for weak sharp minimality in nonsmooth mathematical programming are presented, and some characterizations of weak sharp minimality are obtained, with special attention given to orders one and two. It is also demonstrated that two of these sufficient conditions guarantee exactness of an l1 penalty function. A key role in this paper is played by two geometric concepts: the limiting proximal normal cone and a generalization of the contingent cone.