Optimal Pricing for Optimal Transport

Suppose that c(x, y) is the cost of transporting a unit of mass from x ∈ X to y ∈ Y and suppose that a mass distribution μ on X is transported optimally (so that the total cost of transportation is minimal) to the mass distribution ν on Y. Then, roughly speaking, the Kantorovich duality theorem asserts that there is a price f(x) for a unit of mass sold (say by the producer to the distributor) at x and a price g(y) for a unit of mass sold (say by the distributor to the end consumer) at y such that for any x ∈ X and y ∈ Y, the price difference g(y) ? f(x) is not greater than the cost of transportation c(x, y) and such that there is equality g(y) ? f(x) = c(x, y) if indeed a nonzero mass was transported (via the optimal transportation plan) from x to y. We consider the following optimal pricing problem: suppose that a new pricing policy is to be determined while keeping a part of the optimal transportation plan fixed and, in addition, some prices at the sources of this part are also kept fixed. From the producers’ side, what would then be the highest compatible pricing policy possible? From the consumers’ side, what would then be the lowest compatible pricing policy possible? We have recently introduced and studied settings in c-convexity theory which gave rise to families of c-convex c-antiderivatives, and, in particular, we established the existence of optimal c-convex c-antiderivatives and explicit constructions of these optimizers were presented. In applications, it has turned out that this is a unifying language for phenomena in analysis which used to be considered quite apart. In the present paper we employ optimal c-convex c-antiderivatives and conclude that these are natural solutions to the optimal pricing problems mentioned above. This type of problems drew attention in the past and existence results were previously established in the case where X = Y = ? ⁿ under various specifications. We solve the above problem for general spaces X, Y and real-valued, lower semicontinuous cost functions c. Furthermore, an explicit construction of solutions to the general problem is presented.     

Towards Lim

The paper contains an elegant extension of the Nadler fixed point theorem for multivalued contractions (see Theorem 21). It is based on a new idea of the α-step mappings (see Definition 17) being more efficient than α-contractions. In the present paper this theorem is a tool in proving some fixed point theorems for “nonexpansive” mappings in the bead spaces (metric spaces that, roughly speaking, are modelled after convex sets in uniformly convex spaces). More precisely the mappings are nonexpansive on a set with respect to only one point - the centre of this set (see condition (4)). The results are pretty general. At first we assume that the value of the mapping under consideration at this central point looks “sharp” (see Definition 6). This idea leads to a group of theorems (based on Theorem 7). Their proofs are compact and the theorems, in particular, are natural extensions of the classical results for (usual) nonexpansive mappings. In the second part we apply the idea of Lim to investigate the regular sequences and here the proofs are based on our extension of Nadler's Theorem. In consequence we obtain some fixed point theorems that generalise the classical Lim Theorem for multivalued nonexpansive mappings (see e.g. Theorem 26).     

The Lipschitzianity of convex vector and set-valued functions

It is well known that every scalar convex function is locally Lipschitz on the interior of its domain in finite dimensional spaces. The aim of this paper is to extend this result for both vector functions and set-valued mappings acting between infinite dimensional spaces with an order generated by a proper convex cone C. Under the additional assumption that the ordering cone C is normal, we prove that a locally C-bounded C-convex vector function is Lipschitz on the interior of its domain by two different ways. Moreover, we derive necessary conditions for Pareto minimal points of vector-valued optimization problems where the objective function is C-convex and C-bounded. Corresponding results are derived for set-valued optimization problems.     

2-Coherent and 2-convex conditional lower previsions

In this paper we explore relaxations of (Williams) coherent and convex conditional previsions that form the families of n-coherent and n-convex conditional previsions, at the varying of n. We investigate which such previsions are the most general one may reasonably consider, suggesting (centered) 2-convex or, if positive homogeneity and conjugacy is needed, 2-coherent lower previsions. Basic properties of these previsions are studied. In particular, we prove that they satisfy the Generalised Bayes Rule and always have a 2-convex or, respectively, 2-coherent natural extension. The role of these extensions is analogous to that of the natural extension for coherent lower previsions. On the contrary, n-convex and n-coherent previsions with $n\ge 3$ either are convex or coherent themselves or have no extension of the same type on large enough sets. Among the uncertainty concepts that can be modelled by 2-convexity, we discuss generalisations of capacities and niveloids to a conditional framework and show that the well-known risk measure Value-at-Risk only guarantees to be centered 2-convex. In the final part, we determine the rationality requirements of 2-convexity and 2-coherence from a desirability perspective, emphasising how they weaken those of (Williams) coherence.     

Existence of equilibria of set-valued maps on bounded epi-Lipschitz domains in Hilbert spaces without invariance conditions

In this paper, we provide a new result of the existence of equilibria for set-valued maps on bounded closed subsets K of Hilbert spaces. We do not impose either convexity or compactness assumptions on K but we assume that K has epi-Lipschitz sections, i.e. its intersection with suitable finite dimensional spaces is locally the epigraph of Lipschitz functions. In finite dimensional spaces, the famous Brouwer theorem asserts the existence of a fixed point for a continuous function from a compact convex set K to itself. Our result could be viewed as a kind of generalization of this classical result in the context of Hilbert spaces and when the function (or the set-valued map) does not necessarily map K into itself (K is not invariant under the map). Our approach is based firstly on degree theory for compact and for condensing set-valued maps and secondly on flows generated by trajectories of differential inclusions.     

Book review of Israel Gohberg,Seymour Goldberg,Marinus A. Kaashoek,Basic Classes of Linear Operators,Birkhäuser Verlag,Basel–Boston–Berlin 2003, 423 p. ISBN 3-7643-6930-2

In projective space three notions of convexity (weak convexity, strong convexity, p-convexity) are regarded systematically. Since these notions are defined only by incidence relations, there can be introduced dual notions. We consider relations.between the introduced notions and the most essential properties of convex sets. To all assertions. can be formulated dual assertions, too. The most important theorems given by Fenchel can be generalised. The property of a point set (a set of hyperplanes) to be strongly convex or p-convex, respectively, is invariant with respect to correlations.     

BMO and uniform estimates for multi-well problems

We establish optimal local regularity results for vector-valued extremals and minimizers of variational integrals whose integrand is the squared distance function to a compact set K in matrix space ${{{\mathbb M}^{N \times n}}}$ . The optimality is illustrated by explicit examples showing that, in the nonconvex case, minimizers need not be locally Lipschitz. This is in contrast to the case when the set K is suitably convex, where we show that extremals are locally Lipschitz continuous. The results rely on the special structure of the integrand and elementary Cordes–Nirenberg type estimates for elliptic systems.     

Convergence of Ishikawa’s iteration method for pseudocontractive mappings

Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let T_i:C→C,i=1,2,…,N, be a finite family of Lipschitz pseudocontractive mappings. It is our purpose, in this paper, to prove strong convergence of Ishikawa’s method to a common fixed point of a finite family of Lipschitz pseudocontractive mappings provided that the interior of the common fixed points is nonempty. No compactness assumption is imposed either on T or on C. Moreover, computation of the closed convex set C_n for each n≥1 is not required. The results obtained in this paper improve on most of the results that have been proved for this class of nonlinear mappings.     

Improved gradient method for monotone and lipschitz continuous mappings in banach spaces

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {A_n}_(n∈N) be a family of monotone and Lipschitz continuos mappings of C into E~*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [10] for solving the variational inequality problem for{A_n} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.     

Products of resolvents and multivalued hybrid mappings in CAT(0) spaces

In this article, we introduce and investigate the concept of multivalued hybrid mappings in CAT(0) spaces by using the concept of quasilinearization. Also, we present a new iterative algorithm involving products of Moreau-Yosida resolvents for finding a common element of the set of minimizers of a finite family of convex functions and a common fixed point of two multivalued hybrid mappings in CAT(0) spaces.     

Fine tuning Nesterov’s steepest descent algorithm for differentiable convex programming

We modify the first order algorithm for convex programming described by Nesterov in his book (in Introductory lectures on convex optimization. A basic course. Kluwer, Boston, 2004). In his algorithms, Nesterov makes explicit use of a Lipschitz constant L for the function gradient, which is either assumed known (Nesterov in Introductory lectures on convex optimization. A basic course. Kluwer, Boston, 2004), or is estimated by an adaptive procedure (Nesterov 2007). We eliminate the use of L at the cost of an extra imprecise line search, and obtain an algorithm which keeps the optimal complexity properties and also inherit the global convergence properties of the steepest descent method for general continuously differentiable optimization. Besides this, we develop an adaptive procedure for estimating a strong convexity constant for the function. Numerical tests for a limited set of toy problems show an improvement in performance when compared with the original Nesterov’s algorithms.     

Strongly convex functions of higher order

The notion of strongly n-convex functions with modulus c>0 is introduced and investigated. Relationships between such functions and n-convex functions in the sense of Popoviciu as well as generalized convex functions in the sense of Beckenbach are given. Characterizations by derivatives are presented. Some results on strongly Jensen n-convex functions are also given.     

Six Kinds of Roughly Convex Functions

This paper considers six kinds of roughly convex functions, namely: δ-convex, midpoint δ-convex, ρ-convex, γ-convex, lightly γ-convex, and midpoint γ-convex functions. The relations between these concepts are presented. It is pointed out that these roughly convex functions have two optimization properties: each r-local minimizer is a global minimizer, and if they assume their maximum on a bounded convex domain D (in a Hilbert space), then they do so at least at one r-extreme point of D, where r denotes the roughness degree of these functions. Furthermore, analytical properties are investigated, such as boundedness, continuity, and conservation properties.     

Critical types of krasnoselskii fixed point theorems in weak topologies

Abstract

In this note, by means of the technique of measures of weak noncom- pactness, we establish a generalized form of fixed point theorem for the sum of T+S in weak topology setups of a metrizable locally convex space, where S is not weakly compact, I?T allows to be noninvertible, and T is not necessarily continuous. The obtained results unify and significantly extend a lot of previously known extensions of Krasnoselskii fixed-point theorems. The analysis presented here reveals the essential characteristics of the Krasnoselskii type fixed-point theorem in weak topology settings.     

A Geometric Lower Bound Theorem

We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their toric varieties. The proof uses higher notions of chordality. Further, for C ²-convex bodies, asymptotically tight lower bounds on the g-numbers of the approximating polytopes are given, in terms of their Hausdorff distance from the convex body.     

The Busemann theorem for complex p-convex bodies

The Busemann theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper, we prove a version of the Busemann theorem for complex p-convex bodies. Namely that the complex intersection body of an origin-symmetric complex p-convex body is γ-convex for certain γ. The result is the complex analogue of the work of Kim, Yaskin, and Zvavitch on (real) p-convex bodies. Furthermore, we show that the generalized radial qth mean body of a p-convex body is γ-convex for certain γ.     

The common minimal-norm fixed point of a finite family of nonexpansive mappings

Consider a finite family of nonexpansive mappings which are defined on a closed convex subset of a Hilbert space H. Suppose the set of common fixed points of this family is nonempty. We then address the problem of finding the minimum-norm element of this common fixed point set. We introduce both cyclic and parallel iteration methods to find this minimal-norm element.     

Uniformly normal structure and uniformly Lipschitzian semigroups

Assume that X is a real Banach space with uniformly normal structure and C is a nonempty closed convex subset of X. We show that a κ-uniformly Lipschitzian semigroup of nonlinear self-mappings of C admits a common fixed point if the semigroup has a bounded orbit and if κ is appropriately greater than one. This result applies, in particular, to the framework of uniformly convex Banach spaces.     

Minimal antiderivatives and monotonicity

We consider settings in convex analysis which give rise to families of convex functions that contain their lower envelope. Given certain partial data regarding a subdifferential, we consider the family of all convex antiderivatives that comply with the given data. We prove that this family is not empty and, in particular, contains a minimal antiderivative under a fairly general assumption on the given data. It turns out that the representation of monotone operators by convex functions fits naturally in these settings. Duality properties of representing functions are also captured by these settings, and the gap between the Fitzpatrick function and the Fitzpatrick family is filled by this broader sense of minimality of the Fitzpatrick function.     

Equilibrium existence theorems in KKM spaces

An abstract convex space satisfying the KKM principle is called a KKM space. This class of spaces contains G$G$-convex spaces properly. In this work, we show that a large number of results in KKM theory on G$G$-convex spaces also hold on KKM spaces. Examples of such results are theorems of Sperner and Alexandroff–Pasynkoff, Fan–Browder type fixed point theorems, Horvath type fixed point theorems, Ky Fan type minimax inequalities, variational inequalities, von Neumann type minimax theorems, Nash type equilibrium theorems, and Himmelberg type fixed point theorems.