Clarke Normal Cone Research Articles

Value Functions and Optimality Conditions for Nonconvex Variational Problems with an Infinite Horizon in Banach Spaces

We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. First, we provide an upper estimate of its Dini–Hadamard subdifferential in terms of the Clarke subdifferential of the Lipschitz continuous integrand and the Clarke normal cone to the graph of the set-valued mapping describing dynamics. Second, we derive a necessary condition for optimality in the form of an adjoint inclusion that grasps a connection between the Euler–Lagrange condition and the maximum principle. The main results are applied to the derivation of the necessary optimality condition of the spatial Ramsey growth model.

State dependent nonconvex sweeping processes in smooth Banach spaces

In the setting of 2-uniformly smooth and q -uniformly convex Banach spaces, we prove the existence of solutions of the following multivalued differential equation: -\frac{d}{dt} J(u(t)) \in N^C(C(t,u(t));u(t)) \text{ a.e. in } [0,T]. \:\:\: \mathrm{(SDNSP)} This inclusion is called State Dependent Nonconvex Sweeping Process (SDNSP). Here N^C(C(t,u(t)); u(t)) stands for the Clarke normal cone. The perturbed (SDNSPP) is also considered. Our results extend recent existing results from the setting of Hilbert spaces to the setting of Banach spaces. In our proofs we use some new results on V -uniformly generalized prox-regular sets in Banach spaces.

Pontryagin’s Direct Method for Optimization Problems with Differential Inclusion

We develop Pontryagin’s direct variational method, which allows us to obtain necessary conditions in the Mayer extremal problem on a fixed interval under constraints on the trajectories given by a differential inclusion with generally unbounded right-hand side. The established necessary optimality conditions contain the Euler—Lagrange differential inclusion. The results are proved under maximally weak conditions, and very strong statements compared with the known ones are obtained; moreover, admissible velocity sets may be unbounded and nonconvex under a general hypothesis that the right-hand side of the differential inclusion is pseudo-Lipschitz. In the statements, we refine conditions on the Euler—Lagrange differential inclusion, in which neither the Clarke normal cone nor the limiting normal cone is used, as is common in the works of other authors. We also give an example demonstrating the efficiency of the results obtained.

The Limiting Normal Cone to Pointwise Defined Sets in Lebesgue Spaces

We consider subsets of Lebesgue spaces which are defined by pointwise constraints. We provide formulas for corresponding variational objects (tangent and normal cones). Our main result shows that the limiting normal cone is always dense in the Clarke normal cone and contains the convex hull of the pointwise limiting normal cone. A crucial assumption for this result is that the underlying measure is non-atomic, and this is satisfied in many important applications (Lebesgue measure on subsets of \(\mathbb {R}^{d}\) or the surface measure on hypersurfaces in \(\mathbb {R}^{d}\)). Finally, we apply our findings to an optimization problem with complementarity constraints in Lebesgue spaces.

Weak Necessary and Sufficient Stochastic Maximum Principle for Markovian Regime-Switching Diffusion Models

In this paper we prove a weak necessary and sufficient maximum principle for Markovian regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that $$ 0 $$ 0 belongs to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle.

On subdifferentials of a minimal time function in Hausdorff topological vector spaces

In a general Hausdorff topological vector space , we study the minimal time function associated with a nonempty closed (both cases: convex and nonconvex) set and a bounded closed convex set defined by . We prove and extend various important properties on directional derivatives and subdifferentials of at points in . These results are used to prove various new characterizations of the convex tangent cone, the convex normal cone, the Clarke tangent cone, the Bouligand tangent cone, and Clarke normal cone to in terms of the minimal time function for points inside , in Hausdorff topological vector spaces. Those characterizations are used to scalarize the tangential regularity of sets, in Hausdorff topological vector spaces, as the directional regularity of . Our results extend various existing results, in convex and nonconvex cases, from Banach spaces and normed vector spaces to Hausdorff topological vector spaces. Even in Banach spaces and in normed spaces our results are new, and they extend various existing results on the distance function to closed sets by taking to be the closed unit ball. Applications to normal and subidfferential regularities in normed vector spaces are also given in the last section of the paper.

Subdifferential of Optimal Value Functions in Nonlinear Infinite Programming

This paper presents an exact formula for computing the normal cones of the constraint set mapping including the Clarke normal cone and the Mordukhovich normal cone in infinite programming under the extended Mangasarian-Fromovitz constraint qualification condition. Then, we derive an upper estimate as well as an exact formula for the limiting subdifferential of the marginal/optimal value function in a general Banach space setting.

Super-efficiency of vector optimization in Banach spaces

Using variational analysis, in terms of the Clarke normal cone, we consider super-efficiency of vector optimization in Banach spaces. We establish some characterizations for super-efficiency. In particular, dropping the assumption that the ordering cone has a bounded base, we extend a result in Borwein and Zhuang [J.M. Borwein, D. Zhuang, Super-efficiency in vector optimization, Trans. Amer. Math. Soc. 338 (1993) 105–122] to the nonconvex setting.

Variants of the Ekeland Variational Principle for a set-valued map involving the Clarke Normal Cone

In this paper we present two set-valued variants of the Ekeland variational principle involving the Clarke normal cone and establish sufficient conditions for a set-valued map to have a weak minimizer or a properly positive minimizer when it satisfies Palais–Smale type conditions.

The Marginal Pricing Rule in Economies with Infinitely Many Commodities

Clarke's normal cone appears as the right tool to define the marginal pricing rule in finite dimensional commodity space since it allows to consider in the same framework convex, smooth as well as nonsmooth nonconvex production sets. Furthermore it has nice continuity and convexity properties. But it is not well adapted for economies with infinitely many commodities since it does satisfy minimal continuity properties. In this paper, we propose an alternative definition of the marginal pricing rule.

The Mordukhovich Normal Cone and the Foundations of Welfare Economics

The statement that Pareto optimal allocations require the equalization of marginal rates of substitution, or in an economy with public goods, require the equalization of the aggregate of the marginal rates in consumption to those in production, is formalized through the use of the Mordukhovich normal cone. Since this cone is strictly contained, in general, in the Clarke normal cone, the results generalize earlier work of Khan and Vohra, Quinzii, Yun, and Cornet. The results are an application of Mordukhovich's 1980 theorem on necessary conditions for optimality in constrained optimization problems involving functions that are not necessarily differentiable or quasi‐concave. As such, the results suggest a distinction between the mathematical programming approach to the “second welfare theorem,” as in the work of Hicks, Lange, and Samuelson, and that based on the separation of sets, as pioneered by Arrow and Debreu.

Méthodes d'approximation en théorie du point fixe et de l'équilibre

We provide a sufficient condition for the existence of (generalized) equilibria, or fixed-points for correspondences F, defined on a compact set M ⊂ ℝ n , with values in ℝ n, when the set M is neither assumed to be convex, nor smooth. We consider the class ℳ of nonempty, compact subsets of ℝ n satisfying: 0 ∉ ∂ + d M ( x ¯ ) = def lim sup x → x ¯ , x ∉ M ∂ d M ( x ) , for every xM. The correspondence F is assumed to be upper semicontinuous with nonempty, convex, compact values. If M has at least one connected component with a nonzero Euler characteristic, we prove that F admits a generalized equilibrium x * on M, i.e., x * x M and 0 x F(x *) − N M (x *), where Ñ m (x *) is the cone spanned by ∂ + d M (x *). Our approach extends previous results on the existence of generalized equilibria: ( i) by taking a (non-necessarily convex) cone Ñ M(x *) smaller than Clarke's normal cone, and ( ii) by considering a larger class of sets than the class of epi-Lipschitzian sets.

Approximation normale de sous-ensembles épi-lipschitziens de ℝ n

A sequence (M k) of closed subsets of R" converges “normally” to M ⊂ ℝ n if: (i) M = limsup m k = liminf M k in the sense of Painlevé-Kuratowski; and (ii) limsup G(N M k ) ⊂ G(N M), where G(N M) (resp. G(N M k)) denotes the graph of N M (resp. N M k), Clarke's normal cone to M (resp. m k). This Note studies the normal convergence of subsets of ℝ n , and mainly, shows two results. The first one states that every closed epi-Lipschitzian subset M of ℝ n , with a compact boundary, can be approximated by a sequence (M k) of “smooth” subsets of ℝ n , which converges normally to M, and such that the sets M k and M are lipeomorphic for every k ( i.e. the homeomorphism between M and M k, and its inverse are both Lipschitzian). The second result shows that, if a sequence (M k) of closed subsets of ℝ n converges “normally” to an epi-Lipschitzian set M, and if we additionally assume that the boundary of M k remains in a fixed compact set, then, for k large enough, the sets M k and M are lipeomorphic. In [8], direct applications of these results are given to the study (existence, stability…) of the “generalized” equation 0 ε f(x *) + n m(x *), when M is a compact epi-Lipschitzian subset of ℝ n , and f: M → ℝ n is a continuous map (or, more generally, a correspondence).

Sensitivity, controllability, and necessary conditions of optimal control problems governed by integral equations

Infinite-dimensional perturbations in all constraints of an optimal control problem governed by a Volterra integral equation with the presence of a state constraint are considered. These perturbations give rise to a value function, whose analysis through the proximal normal technique provides sensitivity, controllability, and even necessary conditions for the basic problem. Actually all information about the value function is contained in Clarke's normal cone of its epigraph, which can be characterized by the proximal normal formula.

Cost Minimization of Nonconvex Firms under Prices in Normal Cones

Clarke's normal cone has been frequently used to formulate the marginal cost pricing rule for nonconvex firms. The author provides examples where a firm with convex iso-output sets is not minimizing its cost at a price vector in the normal cone. For a firm to be cost minimizing under any price in the normal cone, lower hemicontinuity of the iso-output correspondence is sufficient. The author also provides an extension of the second welfare theorem with a price vector in the normal cone of each firm and cost minimization of firms. Copyright 1994 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Approximation and regularisation of arbitrary sets in finite dimension

We define the regularisation of a set by using two kernels. This procedure generalizes that defined by Laory-Lions for the functions. We privilege the geometrical point of view and we obtain in particular some new results about the asymptotic behaviour of Clarke's normal cone at a point of a set belonging to the regularized family

Existence of Marginal Cost Pricing Equilibria: The Nonsmooth Case

In this article we report a general existence theorem of a marginal (cost) pricing equilibrium for an economy which may exhibit increasing returns to scale or more general types of in the production sector. Our model considers an arbitrary number of firms and no smoothness assumption is made on their production sets or on the aggregate production set. In this article we report a general existence theorem of marginal (cost) pricing equilibria for an economy, which may exhibit increasing returns to scale or more general types of nonconvexities in the production sector. In our model, which considers a positive finite numbers 1 of commodities, m, of consumers, and, n, of firms, (a) consumers maximize their preferences subject to their budget constraints, (b) convex producers maximize profits, while nonconvex producers are instructed to follow the marginal pricing rule, i.e., they fulfill the first-order necessary condition for profit maximization in a precise mathematical sense, formalized with Clarke's normal cone. Our treatment of convex and nonconvex producers will in fact be symmetric. The technological possibilities of the jth producer (j = 1, ..., n)

On the non-existence of a marginal cost pricing equilibrium and the Ioffe normal cone

In his recent paper, Khan (1987) proved a version of the second fundamental theorem of welfare economics with the Ioffe normal cone as a formalization of marginal rates of substitution. His introduction of the Ioffe normal cone was motivated by the fact that, in the absence of free disposal, the Clarke normal cone may be too "big" and the result based on it, by Khan and Vohra (1987) and Cornet (1986), is not sharp enough to be satisfactory. An interesting property of the Ioffe normal cone is that it is not necessarily convex. Khan showed that the lack of convexity is of no consequence for the second welfare theorem, but left open the question as to whether the convexity property is essential for the existence of a marginal cost pricing equilibrium as originally formulated by Guesnerie (1975); see also Beato and Mas-Colell (1985) and Brown et al. (1986). In this note we answer this question in the negative.

A remark on Clarke's normal cone and the marginal cost pricing rule

This paper constructs a closed set Y in R l such that for all y in the boundary of Y, Clarke's normal cone to Y at y is equal to R l +. If Y is the production set of a firm, then the marginal cost pricing rule imposes no restriction. The existence of Y is shown to be equivalent to the existence of Lipschitzian function ƒ from R l-1 to R such that the generalized gradient of ƒ is everywhere equal to the convex hull of 0 and the simplex of R l-1.

A remark on Clarke's normal cone and the marginal cost pricing rule

This paper constructs a closed set Y in R l such that for all y in the boundary of Y, Clarke's normal cone to Y at y is equal to R l +. If Y is the production set of a firm, then the marginal cost pricing rule imposes no restriction. The existence of Y is shown to be equivalent to the existence of a Lipschitzian function f from R l−1 to R such that the generalized gradient of f is everywhere equal to the convex hull of 0 and the simplex of R l−1 .