Two convex counterexamples: A discontinuous envelope function and a nondifferentiable nearest-point mapping

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology—that is, the weakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional—if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C ∞ maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C ∞ maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green’s theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction.

In this paper we extend the notion of a Lorentz cone. We call a closed convex set isotone projection set with respect to a pointed closed convex cone if the projection onto the set is isotone (i.e., monotone) with respect to the order defined by the cone. We determine the isotone projection sets with respect to an extended Lorentz cone. In particular a Cartesian product between an Euclidean space and any closed convex set in another Euclidean space is such a set. We use this property to find solutions of general mixed complementarity problems in an iterative way.

Background. The area formulae are well known for surfaces embedded into a finite-dimensional Euclidean space. However, in the case of an infinite-dimensional Banach manifold, such formulae cannot be used. Thus, a problem of finding an alternative approach to the surface measures construction appears, that, on the one hand, leads to classical results in finite-dimensional case, and on the other hand, can be used for infinite-dimensional Banach manifolds. Objective. The aim of the paper is to get a construction of surface measure induced by the Lebesgue measure and the associated form for a parametrically defined surface embedded into finite-dimensional Euclidean space. Show the consistency of surface area calculation by this construction with an area calculated by using well-known classical formulae. Methods. Basic results of mathematical analyses, measure theory and differential geometry are used. Results. An alternative construction of surface measures induced by the Lebesgue measure on surfaces in finite-dimensional space is obtained. It is shown that such approach is consistent with the classical definition of the surface area. Conclusions. The construction of surface measures suggested for infinite-dimensional spaces is a generalization of the classical approach in finite-dimensional spaces. Therefore further investigation of the described approach seems to be reasonable.

This paper studies constrained optimization problems in Banach spaces without usual differentiability and convexity assumptions on the functionals involved in the data. The aim is to give optimality conditions for the problems from which one can derive the characterization of best approximations. The objective and the inequality constraint functionals are assumed to have one-sided directional derivatives. First-order necessary conditions are given in terms of subdifferentials of the directional derivatives. The notion of “max-pseudoconvexity” weaker than pseudoconvexity is introduced for sufficiency. The optimality conditions are applied to linear and nonlinear Tchebycheff approximation problems to derive the characterization of best approximations.

We study the convergence of an iterative projection/reflection algorithm originally proposed for solving what are known as phase retrieval problems in optics. There are two features that frustrate any analysis of iterative methods for solving the phase retrieval problem: nonconvexity and infeasibility. The algorithm that we developed, called relaxed averaged alternating reflections (RAAR), was designed primarily to address infeasibility, though our strategy has advantages for nonconvex problems as well. In the present work we investigate the asymptotic behavior of the RAAR algorithm for the general problem of finding points that achieve the minimum distance between two closed convex sets in a Hilbert space with empty intersection, and for the problem of finding points that achieve a local minimum distance between one closed convex set and a closed prox-regular set, also possibly nonintersecting. The nonconvex theory includes and expands prior results limited to convex sets with nonempty intersection. To place the RAAR algorithm in context, we develop parallel statements about the standard alternating projections algorithm and gradient descent. All of the various algorithms are unified as instances of iterated averaged alternating proximal reflectors applied to a sum of regularized maximal monotone mappings.

We give a simple proof of the fact that compact, connected topological spaces have the “average distance property”. For a metric space (X, d), this asserts the existence of a unique number a = a(X) such that, given finitely many points x1, …, xn ∈ X, then there is some y ∈ X withWe examine the possible values of a(X) , for subsets of finite dimensional normed spaces. For example, if diam(X) denotes the diameter of some compact, convex set in a euclidean space, then a(X) ≤ diam(X)/√2 . On the other hand, a(X)/diam(X) can be arbitrarily close to 1 , for non-convex sets in euclidean spaces of sufficiently large dimension.

The multiple-sets split feasibility problem (MSFP) is to find a point belongs to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation belongs to the intersection of another family of closed convex sets in the image space. Many iterative methods can be employed to solve the MSFP. Jinling Zhao et al. proposed a modification for the CQ algorithm and a relaxation scheme for this modification to solve the MSFP. The strong convergence of these algorithms are guaranteed in finite-dimensional Hilbert spaces. Recently López et al. proposed a relaxed CQ algorithm for solving split feasibility problem, this algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, this algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed self-adaptive CQ algorithm for solving the MSFP where closed convex sets are level sets of some convex functions such that the strong convergence is guaranteed in the framework of infinite-dimensional Hilbert spaces. Our result extends and improves the corresponding results.

Let K K be a noncompact convex subset of a normed space X X . It is shown that if K K is not totally-bounded then there exists a Lipschitz self map f f of K K with inf ⁡ { ‖ x − f ( x ) ‖ : x ∈ K } > 0 \operatorname {inf}\left \{ {\left \| {x - f\left ( x \right )} \right \|:x \in K} \right \} > 0 , while if K K is totally-bounded then such a map does not exist, but still K K lacks the fixed point property for Lipschitz mappings. It follows that a closed convex set in a normed space has the fixed point property for Lipschitz maps if and only if it is compact.

We develop a theory of best simultaneous approximation for closed convex sets in a conditionally complete lattice Banach space X with a strong unit. We study best simultaneous approximation in X by elements of closed convex sets, and give necessary and sufficient conditions for the uniqueness of best simultaneous approximation. We give a characterization of simultaneous pseudo-Chebyshev and quasi-Chebyshev closed convex sets in X. Also, we present various characterizations of best simultaneous approximation of elements by closed convex sets in terms of the extremal points of the closed unit ball B X* of X*.

We develop a theory of simultaneous metric projection in a normed linear space $X$ and present various characterizations of simultaneous metric projection onto closed convex sets in terms of the elements of $X^{\ast}$. Also, we characterize the elements of simultaneous metric projection onto closed convex sets in terms of extreme points of the closed unit ball $B_{X^{\ast}}.$ Finally, as an application, we give various characterizations of simultaneous metric projection onto subspaces of the Banach space $C(Q,Y).$

A note on non-support points, negligible sets, Gâteaux differentiability and Lipschitz embeddings

Let F F be a continuous mapping from an open subset D D of a separable Banach space X X into a Banach space Y Y . We show that if the one sided directional derivative D x + F ( a ) D_x^ + F(a) of F F at a a in the direction x x exists for each ( a , x ) (a,x) from a dense G δ {G_\delta } subset S S of an open set D × U ⊂ X × X D \times U \subset X \times X , then F F is Gâteaux differentiable on a dense G δ {G_\delta } subset of D D . Similar results are obtained for Fréchet differentiability when X X is finite-dimensional and for w ∗ {w^ * } -Gâteaux differentiability.

Let S be a 3-convex subset of the plane.If (cl S ~ £>) S int (cl S) or if (cl S ~ S) g bdry (cl S), then S is expressible as a union of four or fewer convex sets.Otherwise, S is a union of six or fewer convex sets.In each case, the bound is best possible.1* Introduction.Let S be a subset of R d .Then S is said to be 3-convex iff for every three distinct points in S, at least one of the segments determined by these points lies in S. Valentine [2] has proved that for S a closed, 3-convex subset of the plane, S is expressible as a union of three or fewer closed convex sets.We are interested in obtaining a similar decomposition without requiring the set S to be closed.The following definitions and results obtained by Valentine will be useful.

A method for the approximate solution of quasi-static problems for hardening bodies

We show that if E is a closed convex set in {mathbb {C}}^n(n>1) contained in a closed halfspace H such that Ecap bH is nonempty and bounded, then the concave domain Omega = {mathbb {C}}^n{setminus } E contains images of proper holomorphic maps f:Xrightarrow {mathbb {C}}^n from any Stein manifold X of dimension <n, with approximation of a given map on closed compact subsets of X. If in addition 2dim X+1le n then f can be chosen an embedding, and if 2dim X=n, then it can be chosen an immersion. Under a stronger condition on E, we also obtain the interpolation property for such maps on closed complex subvarieties.