Camassa-Holm方程凹凸尖峰及光滑孤立子解

研究一类完全可积的新型浅水波方程Camassa-Holm方程的行波孤立子解及双孤立子解.引入凹凸尖峰孤立子及光滑孤立子的概念,研究得到该方程的行波解中具有尖峰性质的凹凸尖峰孤立子解及光滑孤立子解.同时利用Backlund变换给出该类方程的新的双孤立子解.     

Discretize-then-relax approach for convex/concave relaxations of the solutions of parametric ODEs

This paper presents a discretize-then-relax methodology to compute convex/concave bounds for the solutions of a wide class of parametric nonlinear ODEs. The procedure builds upon interval methods for ODEs and uses the McCormick relaxation technique to propagate convex/concave bounds. At each integration step, a two-phase procedure is applied: a priori convex/concave bounds that are valid over the entire step are calculated in the first phase; then, pointwise-in-time convex/concave bounds at the end of the step are obtained in the second phase. An approach that refines the interval state bounds by considering subgradients and affine relaxations at a number of reference parameter values is also presented. The discretize-then-relax method is implemented in an object-oriented manner and is demonstrated using several numerical examples.     

Convex solutions of the polynomial-like iterative equation with variable coefficients

In this paper, by applying the Schauder''s fixed point theorem we prove the existence of increasing and decreasing solutions of the polynomial-like iterative equation with variable coefficients and further completely investigate increasing convex (or concave) solutions and decreasing convex (or concave) solutions of this equation. The uniqueness and continuous dependence of those solutions are also discussed     

Remark on invariant curves for planar mappings

Bounded solutions for a functional equation were investigated by Ng and Zhang [C. T. Ng, W. Zhang, Invariant curves for planar mapping, J. Difference Eqn. Appl. 3 (1997) 147-168]. In this paper, we consider its solutions which may be unbounded. We give not only existence and uniqueness of its solutions but also the convex and concave solutions. Moreover, some examples are given.     

一类凹与凸算子的推广

运用锥的性质以及单调迭代技巧给出了一类广义凹算子与凸算子正不动点的存在唯一性,并讨论了算子特征值方程的性质.作为应用,利用所获结果研究了一类积分方程正解的存在唯一性.     

Some Farkas-type results for fractional programming problems with DC functions

For a kind of fractional programming problem that the objective functions are the ratio of two DC (difference of convex) functions with finitely many convex constraints, in this paper, its dual problems are constructed, weak and strong duality assertions are given, and some sufficient and necessary optimality conditions which characterize their optimal solutions are obtained. Some recently obtained Farkas-type results for fractional programming problems that the objective functions are the ratio of a convex function to a concave function with finitely many convex constraints are the special cases of the general results of this paper.     

一类非线性方程的解的存在性及其应用

设Ａ是Ａｍａｎｎ意义下的凹（凸）算子,本文提出序Ｌｉｐｓｃｈｉｔｚ条件,无需考虑任何紧性或连续性条件,由Ｍａｎｎ迭代技巧证明了方程Ａｘ＝ｘ的解的存在性,将所得结果应用于无辊域ａｍｍｅｒｓｔｅｉｎ发方程,得到了新结果。     

一类非线性算子的不动点定理及其应用

本文利用半序方法讨论了u0凹、-u0凸算子的不动点存在唯一性定理及迭代序列的收敛性问题,还讨论了序凹、序凸算子及u0凸算子的有关问题及它们在Ham- merstein型积分方程中的应用.所得结论推广并改进了已有的相关结论.     

Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients

We deal with a Hamilton-Jacobi equation with a Hamiltonian that is discontinuous in the space variable. This is closely related to a conservation law with discontinuous flux. Recently, an entropy framework for single conservation laws with discontinuous flux has been developed which is based on the existence of infinitely many stable semigroups of entropy solutions based on an interface connection. In this paper, we characterize these infinite classes of solutions in terms of explicit Hopf-Lax type formulas which are obtained from the viscosity solutions of the corresponding Hamilton-Jacobi equation with discontinuous Hamiltonian. This also allows us to extend the framework of infinitely many classes of solutions to the Hamilton-Jacobi equation and obtain an alternative representation of the entropy solutions for the conservation law. We have considered the case where both the Hamiltonians are convex (concave). Furthermore, we also deal with the less explored case of sign changing coefficients in which one of the Hamiltonians is convex and the other concave. In fact in convex-concave case we cannot expect always an existence of a solution satisfying Rankine-Hugoniot condition across the interface. Therefore the concept of generalised Rankine-Hugoniot condition is introduced and prove existence and uniqueness.     

Two inequalities for differentiable mappings and applications to weighted trapezoidal formula,weighted midpoint formula and random variable

In this paper, we establish two inequalities for differentiable convex mappings and differentiable concave mappings which are connected with Fejér’s inequality holding for convex mappings and concave mappings. Some error estimates for the weighted trapezoidal formula and the weighted midpoint formula are given.     

Fenchel duality theorem in multiobjective programming problems with set functions

In this paper, we characterize a vector-valued convex set function by its epigraph. The concepts of a vector-valued set function and a vector-valued concave set function are given respectively. The definitions of the conjugate functions for a vector-valued convex set function and a vector-valued concave set function are introduced. Then a Fenchel duality theorem in multiobjective programming problem with set functions is derived.     

Theorems of the Alternative for Multivalued Mappings and Applications to Mixed Convex\Concave Systems of Inequalities

We provide a purely algebraic theorem of the alternative (and its topological variant) involving multivalued mappings under relaxed convexity assumptions. Various more or less classical applications are given, specially for nonconvex quadratic systems. In the second part of the paper we introduce an alternative formulation for a mixed convex\concave statement. The theory is applied to systems of mixed convex\concave inequalities.     

Optimal Financial Portfolios

The classes of reward‐risk optimization problems that arise from different choices of reward and risk measures are considered. In certain examples the generic problem reduces to linear or quadratic programming problems. An algorithm based on a sequence of convex feasibility problems is given for the general quasi‐concave ratio problem. Reward‐risk ratios that are appropriate in particular for non‐normal assets return distributions and are not quasi‐concave are also considered.     

Classifying convex extremum problems over linear topologies having separation properties

It is shown that any convex or concave extremum problem possesses a subsidiary extremum problem which has certain homogeneous properties. Analogous to the given problem, the “homogenized” extremum problem seeks the minimum of a convex function or the maximum of a concave function over a convex domain. By using homogenized extremum problems, new relationships are developed between any given convex extremum problem (P) and a concave extremum problem (P¹) (also having a convex domain), called the “dual” problem of (P). This is achieved by combining all possibilities in tabular form of (1) the values of the extremum functions and (2) the nature of the convex domains including perturbations of all problems (P), (P¹), and each of their respective homogenized extremum problems.This detailed and refined classification is contrasted to the relationships obtainable by combining only the possible values of the extremum functions of the problems (P) and (P¹) and the possible limiting values of these functions stemming from perturbations of the convex constraint domains of (P) and (P¹), respectively.The extremum problems in this paper and classification results are set forth in real topologically paired vector spaces having the Hahn-Banach separation property.     

Minimarg and maximarg operators

Two operators on the set ofn-person cooperative games are introduced, the minimarg operator and the maximarg operator. These operators can be seen as dual to each other. Some nice properties of these operators are given, and classes of games for which these operators yield convex (respectively, concave) games are considered. It is shown that, if these operators are applied iteratively on a game, in the limit one will yield a convex game and the other a concave game, and these two games will be dual to each other. Furthermore, it is proved that the convex games are precisely the fixed points of the minimarg operator and that the concave games are precisely the fixed points of the maximarg operator.     

Three kinds of convexity and concavity properties of optimal value function in parametric nonlinear programming

This work is concerned with exploring the new convexity and concavity properties of the optimal value function in parametric programming. Some convex (concave) functions are discussed and sufficient conditions for new convexity and concavity of the optimal value function in parametric programming are given. Many results in this paper can be considered as deepen the convexity and concavity studies of convex (concave) functions and the optimal value functions.     

On ranking of feasible solutions of a bottleneck linear programming problem

Summary An algorithm for the ranking of the feasible solutions of a bottleneck linear programming problem in ascending order of values of a concave bottleneck objective function is developed in this paper. The “best” feasible solution for a given value of the bottleneck objective is obtained at each stage. It is established that only the extreme points of a convex polytope need to be examined for the proposed ranking. Another algorithm, involving partitioning of the nodes, to rank the feasible solutions of the bottleneck linear programming problem is proposed, and numerical illustrations for both are provided.     

Minimizing the maximum network flow: models and algorithms with resource synergy considerations

In this paper, we model and solve the network interdiction problem of minimizing the maximum flow through a network from a given source node to a terminus node, while incorporating different forms of superadditive synergy effects of the resources applied to the arcs in the network. Within this context, we examine linear, concave, and convex–concave synergy relationships, illustrate their relative effect on optimal solution characteristics, and accordingly develop and test effective solution procedures for the underlying problems. For a concave synergy relationship, which yields a convex programme, we propose an inner-linearization procedure that significantly outperforms the competitive commercial solver SBB by improving the quality of solutions found by the latter by 6.2% (within a time limit of 1800 CPU?s), while saving 84.5% of the required computational effort. For general non-concave synergy relationships, we develop an outer-approximation-based heuristic that achieves solutions of objective value 0.20% better than the commercial global optimization software BARON, with a 99.3% reduction in computational effort for the subset of test problems for which BARON could identify a feasible solution within the set time limit.     

Multi-objective optimization over convex disjunctive feasible sets using reference points

In this paper we consider the Multiple Objective Optimization Problem (MOOP), where concave functions are to be maximized over a feasible set represented as a union of compact convex sets. To solve this problem we consider two auxiliary scalar optimization problems which use reference points. The first one contains only continuous variables, it has higher dimensionality, however it is convex. The second scalar problem is a mixed integer programming problem. The solutions of both scalar problems determine nondominated points. Some other properties of these problems are also discussed.     

A Farkas lemma for difference sublinear systems and quasidifferentiable programming

A new generalized Farkas theorem of the alternative is presented for systems involving functions which can be expressed as the difference of sublinear functions. Various other forms of theorems of the alternative are also given using quasidifferential calculus. Comprehensive optimality conditions are then developed for broad classes of infinite dimensional quasidifferentiable programming problems. Applications to difference convex programming and infinitely constrained concave minimization problems are also discussed.