A quasi-variational approach to a competitive economic equilibrium problem without strong monotonicity assumption

This paper is focused on the investigation of the Walrasian economic equilibrium problem involving utility functions. The equilibrium problem is here reformulated by means of a quasi-variational inequality problem. Our goal is to give an existence result without assuming strong monotonicity conditions. To this end, we make use of a perturbation procedure. In particular, we will consider suitable perturbed utility functions whose gradient satisfies a strong monotonicity condition and whose associated equilibrium problem admits a solution. Then, we will prove that the limit solution solves the unperturbed problem. We stress out that our result allows us to consider a wide class of utility functions in which the Walrasian equilibrium problem may be solved.     

Monotonicity of control volume methods

Robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be used in a wide range of applications. Here, we discuss monotonicity for a simplified problem of single-phase flow, but where the simulation grids and media are allowed to be general, posing challenges to control-volume methods. We discuss discrete formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine-point control-volume discretizations for conforming quadrilateral grids in 2D. These criteria are less restrictive than the M-matrix property. It is shown that it is impossible to construct nine-point methods which unconditionally satisfy the monotonicity criteria when the discretization satisfies local conservation and exact reproduction of linear potential fields. Numerical examples are presented which show the validity of the criteria for monotonicity. Further, the impact of nonmonotonicity is studied. Different behavior for different discretization methods is illuminated, and simple ideas are presented for improvement in terms of monotonicity.     

Shares Allocation Methods for Generalized Game Problems with Joint Constraints

We consider an extension of a noncooperative game problem where players have joint binding constraints. We suggest a shares allocation approach, which replaces the initial problem with a sequence of Nash equilibrium problems together with an upper level set-valued variational inequality as master problem. This transformation maintains the monotonicity properties of the underlying mappings. We also show that the regularization yields a decomposable penalty method, which removes complex functions in constraints within the custom noncooperative game framework and provides the single-valued master problem with strengthened monotonicity of its cost mapping.     

Initial-boundary value problem for higher dimensional landau–lifshitz systems

This article is concerned with the partial regularity of the weak solutions satisfying energy inequality and monotonicity inequality to the initial-boundary value problem of Landau–Lifshitz system of ferromagnetic spin chain with Gilbert damping in higher dimensions. A generalized monotonicity inequality is derived. The singular set and its Hausdorff dimension are discussed.     

Aggregation of monotone reciprocal relations with application to group decision making

This paper is composed of two complementary parts. The first part is a formal investigation into the interplay of properties of reciprocal relations, how monotonicity relates to some natural and intuitive properties, including stochastic transitivity. The goal is to aggregate monotone reciprocal relations on a given set of alternatives. Monotonicity is expressed w.r.t. a linear order on the set of alternatives. The second part is a practical protocol to both determine the best fitting linear order underlying the alternatives, and construct a reciprocal relation monotone w.r.t. it. We formulate the problem as an optimization problem, where the aggregated linear order is that for which the implied stochastic monotonicity conditions are closest to being satisfied by the distribution of the input monotone reciprocal relations. We show that if stochastic monotonicity conditions are satisfied, a monotone reciprocal relation is easily found on the basis of the (possibly constructed) stochastically monotone reciprocal distributional relation.     

Iterative algorithms for generalized mixed equilibrium problems

In this paper, we consider a generalized mixed equilibrium problem in real Hilbert space. Using the auxiliary principle, we define a class of resolvent mappings. Further, using fixed point and resolvent methods, we give some iterative algorithms for solving generalized mixed equilibrium problem. Furthermore, we prove that the sequences generated by iterative algorithms converge weakly to the solution of generalized mixed equilibrium problem. These results require monotonicity (θ-pseudo monotonicity) and continuity (Lipschitz continuity) for mappings.     

Monotonicity Formula for a Problem with Hysteresis

A new monotonicity formula is obtained for a problem with discontinuous nonlinearities arising in the modeling of various processes with a memory effect.     

Nonparametric Estimation of Hazard Rate Under the Constraint of Monotonicity

This article shows how to smoothly “monotonize” standard kernel estimators of hazard rate, using bootstrap weights. Our method takes a variety of forms, depending on choice of kernel estimator and on the distance function used to define a certain constrained optimization problem. We confine attention to a particularly simple kernel approach and explore a range of distance functions. It is straightforward to reduce “quadratic” inequality constraints to “linear” equality constraints, and so our method may be implemented using little more than conventional Newton–Raphson iteration. Thus, the necessary computational techniques are very familiar to statisticians. We show both numerically and theoretically that monotonicity, in either direction, can generally be imposed on a kernel hazard rate estimator regardless of the monotonicity or otherwise of the true hazard rate. The case of censored data is easily accommodated. Our methods have straightforward extension to the problem of testing for monotonicity of hazard rate, where the distance function plays the role of a test statistic.     

Noncooperative Games with Vector Payoffs Under Relative Pseudomonotonicity

We consider the Nash equilibrium problem with vector payoffs in a topological vector space. By employing the recent concept of relative (pseudo) monotonicity, we establish several existence results for vector Nash equilibria and vector equilibria. The results strengthen in a major way existence results for vector equilibrium problems which were based on the usual (generalized) monotonicity concepts.     

Convexity with respect to a differential equation

The concepts of convexity of a set, convexity of a function and monotonicity of an operator with respect to a second-order ordinary differential equation are introduced in this paper. Several well-known properties of usual convexity are derived in this context, in particular, a characterization of convexity of function and monotonicity of an operator. A sufficient optimality condition for a optimization problem is obtained as an application. A number of examples of convex sets, convex functions and monotone operators with respect to a differential equation are presented.     

广义均值保持变换的随机单调性及其在库存系统中的应用

本文提出一类非线性且均值可能不等的广义均值保持变换,研究实现其变换前后随机变量比较的充分条件或充分必要条件,并用此变换来定量刻画需求不确定性对库存系统决策和利润的影响。首先给出变换前后或不同参数下分布函数的关系及其满足一阶随机占优和割准则序的充分条件,特征刻画此变换与广义TTT变换之间的关系。进一步,用三类特殊的广义均值保持变换进行验证。最后,将此变换应用到报童模型中,得出该变换对包含最小化成本及最大化利润的一致化报童问题的随机单调性。     

Progressive Decoupling of Linkages in Optimization and Variational Inequalities with Elicitable Convexity or Monotonicity

Algorithms for problem decomposition and splitting in optimization and the solving of variational inequalities have largely depended on assumptions of convexity or monotonicity. Here, a way of “eliciting” convexity or monotonicity is developed which can get around that limitation. It supports a procedure called the progressive decoupling algorithm, which is derived from the proximal point algorithm through passing to a partial inverse, localizing and rescaling. In the optimization setting, elicitability of convexity corresponds to a new and very general kind of second-order sufficient condition for a local minimum. Applications are thereby opened up to problem decomposition and splitting even in nonconvex optimization, moreover with augmented Lagrangians for subproblems assisting in the implementation.

     

旋转摆的周期单调性

考虑了一类旋转摆系统.该系统拥有至多5个周期轨族,通过计算椭圆积分得到了参数在不同取值范围内该系统各个周期轨族的周期单调性.该文的分析过程也为讨论这类周期单调性问题提供了一个比Abel积分更基本的方法.最后给出的数值仿真结果映证了文中的结论.     

Characterizations and applications of generalized invexity and monotonicity in Asplund spaces

In this paper, the concepts of invexity, monotonicity, and their generalizations in Asplund spaces are studied. Some characterizations for several kinds of generalized invexity and monotonicity concepts are given, using the properties of Mordukhovich limiting subdifferentials in Asplund spaces; and some applications in mathematical programming are provided. Also, some necessary and sufficient weak Pareto-optimality conditions for a multiple-objective optimization problem are proved.     

Stability with respect to the initial data and monotonicity of an implicit difference scheme for a homogeneous porous medium equation with a quadratic nonlinearity

We study the stability and monotonicity of a conservative difference scheme approximating an initial-boundary value problem for a porous medium equation with a quadratic nonlinearity under certain conditions imposed only on the input data of the problem. We prove a grid analog of the Bihari lemma, which is used to obtain a priori estimates for higher derivatives; these estimates are needed both in the proof of the continuous dependence of the solution on small perturbations in the input data and for the analysis of monotonicity in the nonlinear case. We show that, regardless of the smoothness of the initial condition, the higher derivatives can become infinite in finite critical time. We give an example in which there arises a runningwave solution, which justifies the theoretical conclusions.     

Monotonicity preserving transformations of MOT and SEP

Recently, Beiglböck and Juillet (2016) and Beiglböck et al. (2015) established that optimizers to the martingale optimal transport problem (MOT) are concentrated on $c$-monotone sets. In this article we characterize monotonicity preserving transformations revealing certain symmetries between optimizers of MOT for different cost functions. Due to the intimate connection of MOT and the Skorokhod embedding problem (SEP) these transformations are also monotonicity preserving and disclose symmetries for certain solutions to the optimal SEP. Furthermore, the SEP picture allows to easily understand the geometry of these transformations once we have established the SEP counterparts to the known solutions of MOT based on the monotonicity principle for SEP which in turn allows to directly read off the structure of the MOT optimizers.     

Conditions for Error Bounds and Bounded Level Sets of Some Merit Functions for the Second-Order Cone Complementarity Problem

Recently this author studied several merit functions systematically for the second-order cone complementarity problem. These merit functions were shown to enjoy some favorable properties, to provide error bounds under the condition of strong monotonicity, and to have bounded level sets under the conditions of monotonicity as well as strict feasibility. In this paper, we weaken the condition of strong monotonicity to the so-called uniform P ^*-property, which is a new concept recently developed for linear and nonlinear transformations on Euclidean Jordan algebra. Moreover, we replace the monotonicity and strict feasibility by the so-called R ₀₁ or R ₀₂-functions to keep the property of bounded level sets. This work is partially supported by National Science Council of Taiwan.     

Global s-type error bound for the extended linear complementarity problem and applications

For the extended linear complementarity problem over an affine subspace, we first study some characterizations of (strong) column/row monotonicity and (strong) R ₀-property. We then establish global s-type error bound for this problem with the column monotonicity or R ₀-property, especially for the one with the nondegeneracy and column monotonicity, and give several equivalent formulations of such error bound without the square root term for monotone affine variational inequality. Finally, we use this error bound to derive some properties of the iterative sequence produced by smoothing methods for solving such a problem under suitable assumptions. Received: May 2, 1999 / Accepted: February 21, 2000?Published online July 20, 2000     

Log-supermodularity of weight functions, ordering weighted losses, and the loading monotonicity of weighted premiums

The paper is motivated by a problem concerning the monotonicity of insurance premiums with respect to their loading parameter: the larger the parameter, the larger the insurance premium is expected to be. This property, usually called the loading monotonicity, is satisfied by premiums that appear in the literature. The increased interest in constructing new insurance premiums has raised a question as to what weight functions would produce loading-monotonic premiums. In this paper, we demonstrate a decisive role of log-supermodularity or, equivalently, of total positivity of order 2 (TP₂) in answering this question. As a consequence, we establish-at a stroke-the loading monotonicity of a number of well-known insurance premiums, and offer a host of further weight functions, and consequently of premiums, thus illustrating the power of the herein suggested methodology for constructing loading-monotonic insurance premiums.     

Combined partial regularization and descent method for a generalized primal–dual system

A variational inequality system, which is a generalization of the saddle point problem, is considered. The system does not possess monotonicity properties and the feasible set is unbounded in general. To solve the problem we propose a completely implementable iterative scheme, whose convergence is proved under certain coercivity type conditions.