Existence of an upper hemi-continuous and convex-valued demand sub-correspondence

In this paper we show that a strictly open, non-saturated and acyclically convex preference relation admits an extension which is ordered by inclusion (a weaker property than regularity), strictly open, locally non saturated and convex; in turn, this result permits to prove the existence of an upper hemi-continuous and convex-valued demand sub-correspondence. By directly applying standard fixed-point techniques to these sub-correspondences, it is therefore possible to demonstrate the existence of general economic equilibrium even if consumers’ preference relations are not regular.     

Isotone retraction cones in Hilbert spaces

A mapping is called isotone if it is monotone increasing with respect to the order induced by a pointed closed convex cone. Finding the pointed closed convex generating cones for which the projection mapping onto the cone is isotone is a difficult problem which was analyzed in Isac and Németh (1986, 1990, 1992) [1], [2], [3], [4] and [5]. Such cones are called isotone projection cones. In particular it was shown that any isotone projection cone is latticial (Isac (1990) [2]). This problem is extended by replacing the projection mapping with continuous retractions onto the cone. By introducing the notion of sharp mappings, it is shown that a pointed closed convex generating cone is latticial if and only if there is a continuous retraction onto the cone whose complement is sharp. Several particular cases are considered and examples are given.     

Characterization of latticial cones in Hilbert spaces by isotonicity and generalized infimum

A mapping is called isotone if it is monotone increasing with respect to the order defined by a pointed closed convex cone. Finding the pointed closed convex generating cones for which the projection mapping onto the cone is isotone is a difficult problem which was analyzed in [1, 2, 3, 4, 5]. Such cones are called isotone projection cones. In particular it was shown that any isotone projection cone is latticial [2]. This problem is extended by replacing the projection mapping with a continuous isotone retraction onto the cone. By introducing the notion of sharp mappings, it is shown that a pointed closed convex generating cone is latticial if and only if there is a continuous isotone retraction onto the cone whose complement is sharp. This result is used for characterizing a subdual latticial cone by the isotonicity of a generalization of the positive part mapping x ↦ x ⁺. This generalization is achieved by generalizing the infimum for subdual cones. The theoretical results of this paper exhibit fundamental properties of the lattice structure of the space which were not analysed before.     

Monotone and Čebyšev arcs in hyperspaces

A set in a metric space is called a ?eby?ev set if it contains a unique “nearest neighbour” to each point of the space. In this paper we introduce the concept of a monotone arc of convex sets and show that compact monotone arcs have the ?eby?ev property in the hyperspace of compact strictly convex sets. In the hyperspace of compact convex sets only certain monotone arcs are ?eby?ev ; these are characterized. Results are also obtained for affine segments and for noncompact monotone arcs.     

A Measure Of Asymmetry For Positive Semidefinite Matrices

It is known that a continuous map is the gradient of a convex function if and only if it is cyclically monotone. Also, a differentiable map F is the gradient of a function if and only if the matrices F ′(x) are symmetric for all x in the domain. Based on this connection between symmetry and monotonicity, we define a measure of asymmetry for positive semidefinite matrices.     

A Helly-Type Theorem for Semi-monotone Sets and Monotone Maps

We consider sets and maps defined over an o-minimal structure over the reals, such as real semi-algebraic or globally subanalytic sets. A monotone map is a multi-dimensional generalization of a usual univariate monotone continuous function on an open interval, while the closure of the graph of a monotone map is a generalization of a compact convex set. In a particular case of an identically constant function, such a graph is called a semi-monotone set. Graphs of monotone maps are, generally, non-convex, and their intersections, unlike intersections of convex sets, can be topologically complicated. In particular, such an intersection is not necessarily the graph of a monotone map. Nevertheless, we prove a Helly-type theorem, which says that for a finite family of subsets of $\mathbb{R }^n$ , if all intersections of subfamilies, with cardinalities at most $n+1$ , are non-empty and graphs of monotone maps, then the intersection of the whole family is non-empty and the graph of a monotone map.     

Penalty schemes with inertial effects for monotone inclusion problems

We introduce a penalty term-based splitting algorithm with inertial effects designed for solving monotone inclusion problems involving the sum of maximally monotone operators and the convex normal cone to the (nonempty) set of zeros of a monotone and Lipschitz continuous operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the monotone inclusion problem, provided a condition expressed via the Fitzpatrick function of the operator describing the underlying set of the normal cone is verified. Under strong monotonicity assumptions we can even show strong nonergodic convergence of the iterates. This approach constitutes the starting point for investigating from a similar perspective monotone inclusion problems involving linear compositions of parallel-sum operators and, further, for the minimization of a complexly structured convex objective function subject to the set of minima of another convex and differentiable function.     

Monotone and cash-invariant convex functions and hulls

This paper provides some useful results for convex risk measures. In fact, we consider convex functions on a locally convex vector space E which are monotone with respect to the preference relation implied by some convex cone and invariant with respect to some numeraire (‘cash’). As a main result, for any function f, we find the greatest closed convex monotone and cash-invariant function majorized by f. We then apply our results to some well-known risk measures and problems arising in connection with insurance regulation.     

Monotonicity and Market Equilibrium

Economic theory relates prices to quantities via ” market curves.” Typically, such curves are monotone, hence they admit functional representations. The latter invoke linear pricing of quantities so as to obtain market values. Specifically, if higher prices call forward greater supply, a convex function, bounded below by market values, represents the resulting supply curve. Likewise, if demand decreases at higher prices, a concave function, bounded above by market values, represents the attending demand curve. In short, grantedmonotonicity, market curves are described by bivariate functions, either convex or concave, appropriately bounded by linear valuations of quantities. The bounding supply (demand) function generates ask (resp. bid)valuations. Exchange and trade, as modelled here, are driven by valuation differentials, called bid-ask spreads. These disappear, and market equilibrium prevails, if all ”inverse market curves” intersect in a common price. A main issue is whether and how market agents, by themselves, may reach such equilibrium. The paper provides positive and constructive answers. As vehicle it contends with bilateral transactions.     

Characterization of a coherent upper conditional prevision as the Choquet integral with respect to its associated Hausdorff outer measure

A model of coherent upper conditional prevision for bounded random variables is proposed in a metric space. It is defined by the Choquet integral with respect to Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension. Otherwise, when the conditioning event has Hausdorff outer measure equal to zero or infinity in its Hausdorff dimension, it is defined by a 0–1 valued finitely, but not countably, additive probability. If the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension it is proven that a coherent upper conditional prevision is uniquely represented by the Choquet integral with respect to the upper conditional probability defined by Hausdorff outer measure if and only if it is monotone, comonotonically additive, submodular and continuous from below.     

Some characterizations for SOC-monotone and SOC-convex functions

We provide some characterizations for SOC-monotone and SOC-convex functions by using differential analysis. From these characterizations, we particularly obtain that a continuously differentiable function defined in an open interval is SOC-monotone (SOC-convex) of order n ≥ 3 if and only if it is 2-matrix monotone (matrix convex), and furthermore, such a function is also SOC-monotone (SOC-convex) of order n ≤ 2 if it is 2-matrix monotone (matrix convex). In addition, we also prove that Conjecture 4.2 proposed in Chen (Optimization 55:363–385, 2006) does not hold in general. Some examples are included to illustrate that these characterizations open convenient ways to verify the SOC-monotonicity and the SOC-convexity of a continuously differentiable function defined on an open interval, which are often involved in the solution methods of the convex second-order cone optimization.     

Fixed-Point Theorems for Generalized Nonexpansive Mappings

Using the concept of asymptotic center we obtain the existence of fixed points having preassigned location for a wider class of asymptotic nonexpansive mappings in a uniformly convex Banach space. This generalization leads us to get a recent result of Alfuraidan and Khamsi for continuous monotone asymptotic nonexpansive mappings as well as the classical fixed-point result of Geobel and Kirk for asymptotic nonexpansive mappings in a uniformly convex Banach space. Also we prove a fixed-point theorem for order preserving continuous maps on a quasiordered closed convex subset of a uniformly convex Banach sapce having monotone norm.     

Monotonicity and complex convexity in Banach lattices

The goal of this article is to study the relations among monotonicity properties of real Banach lattices and the corresponding convexity properties in the complex Banach lattices. We introduce the moduli of monotonicity of Banach lattices. We show that a Banach lattice E is uniformly monotone if and only if its complexification EC is uniformly complex convex. We also prove that a uniformly monotone Banach lattice has finite cotype. In particular, we show that a Banach lattice is of cotype q for some 2?q<∞ if and only if there is an equivalent lattice norm under which it is uniformly monotone and its complexification is q-uniformly PL-convex. We also show that a real Köthe function space E is strictly (respectively uniformly) monotone and a complex Banach space X is strictly (respectively uniformly) complex convex if and only if Köthe-Bochner function space E(X) is strictly (respectively uniformly) complex convex.     

单调全局最优化问题的凸化外逼近算法

单调优化是指目标函数与约束函数均为单调函数的全局优化问题．本文提出一种新的凸化变换方法把单调函数化为凸函数，进而把单调优化问题化为等价的凸极大或凹极小问题，然后采用Hoffman的外逼近方法来求得问题的全局最优解．我们把这种凸化方法同Tuy的Polyblock外逼近方法作了比较，通过数值比较可以看出本文提出的凸化的方法在收敛速度上明显优于Polyblock方法．     

To the theory of operator monotone and operator convex functions

We prove that a real function is operator monotone (operator convex) if the corresponding monotonicity (convexity) inequalities are valid for some normal state on the algebra of all bounded operators in an infinite-dimensional Hilbert space. We describe the class of convex operator functions with respect to a given von Neumann algebra in dependence of types of direct summands in this algebra. We prove that if a function from ℝ⁺ into ℝ⁺ is monotone with respect to a von Neumann algebra, then it is also operator monotone in the sense of the natural order on the set of positive self-adjoint operators affiliated with this algebra.     

A sequential parametric convex approximation method with applications to nonconvex truss topology design problems

We describe a general scheme for solving nonconvex optimization problems, where in each iteration the nonconvex feasible set is approximated by an inner convex approximation. The latter is defined using an upper bound on the nonconvex constraint functions. Under appropriate conditions, a monotone convergence to a KKT point is established. The scheme is applied to truss topology design (TTD) problems, where the nonconvex constraints are associated with bounds on displacements and stresses. It is shown that the approximate convex problem solved at each inner iteration can be cast as a conic quadratic programming problem, hence large scale TTD problems can be efficiently solved by the proposed method.     

单调Minkowski泛函与Henig真有效性的标量化

没有凸锥的闭性和点性假设,该文考虑由一般凸锥生成的单调Minkowski泛函并研究其性质.由此,在偏序局部凸空间的框架下,通过利用单调连续Minkowski泛函和单调连续半范,该文分别获得了一般集合及锥有界集合的弱有效点的标量化.利用此弱有效性的标量化,该文分别推导出一般集合及锥有界集合的Henig真有效点的标量化.进而,当序锥具备有界基时,该文获得局部凸空间中超有效性的一些标量化结果.最后,该文给出Henig真有效性和超有效性的稠密性结果.这些结果推广并改进了有关的已知结果.     

Application of the Proximal Point Method to Nonmonotone Equilibrium Problems

We consider a general equilibrium problem defined on a convex set, whose cost bifunction may not be monotone. We show that this problem can be solved by the inexact proximal point method if there exists a solution to the dual problem. An application of this approach to nonlinearly constrained problems is also suggested.     

SINGLE PROJECTION ALGORITHM FOR VARIATIONAL INEQUALITIES IN BANACH SPACES WITH APPLICATION TO CONTACT PROBLEM

We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space. The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous. A weak convergence result is obtained under reasonable assumptions on the variable step-sizes. We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous. For this strong convergence case, the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters, rather, the variable step-sizes are diminishing and non-summable. The asymptotic estimate of the convergence rate for the strong convergence case is also given. For completeness, we give another strong convergence result using the idea of Halpern's iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function.Finally, we give an example of a contact problem where our proposed method can be applied.     

Spectral gradient projection method for solving nonlinear monotone equations

An algorithm for solving nonlinear monotone equations is proposed, which combines a modified spectral gradient method and projection method. This method is shown to be globally convergent to a solution of the system if the nonlinear equations to be solved is monotone and Lipschitz continuous. An attractive property of the proposed method is that it can be applied to solving nonsmooth equations. We also give some preliminary numerical results to show the efficiency of the proposed method.