均衡约束数学规划的约束规格和最优性条件综述

约束规格在约束优化问题的最优性条件中起着重要的作用,介绍了近几年国际上关于均衡约束数学规划(简记为MPEC)的约束规格以及最优性条件的研究成果, 包括以下主要内容: (1) MPEC常用的约束规格(如线性无关约束规格 (MPEC-LICQ)、Mangasarian-Fromovitz约束规格 (MPEC-MFCQ)等)和新的约束规格(如恒秩约束规格、常数正线性相关约束规格等), 以及它们之间的关系; (2) MPEC常用的稳定点; (3) MPEC的最优性条件. 最后还对MPEC的约束规格和最优性条件的研究前景进行了探讨.     

MATLAB在动力学实验数据处理中的应用

将MATLAB 数学软件编写程序用于物理化学实验“丙酮碘化反应速率方程”的数据处理与绘图,根据孤立法设计实验步骤以此推得反应级数,所得结果准确度高,避免了传统数据处理方法所带来的人为误差,具有方便、快速与直观等优点,可以补充和改进物理化学实验的教学。     

Minimum principle sufficiency

We characterize the property of obtaining a solution to a convex program by minimizing over the feasible region a linearization of the objective function at any of its solution points (Minimum Principle Sufficiency). For the case of a monotone linear complementarity problem this MPS property is completely equivalent to the existence of a nondegenerate solution to the problem. For the case of a convex quadratic program, the MPS property is equivalent to the span of the Hessian of the objective function being contained in the normal cone to the feasible region at any solution point, plus the cone generated by the gradient of the objective function at any solution point. This in turn is equivalent to the quadratic program having a weak sharp minimum. An important application of the MPS property is that minimizing on the feasible region a linearization of the objective function at a point in a neighborhood of a solution point gives an exact solution of the convex program. This leads to finite termination of convergent algorithms that periodically minimize such a linearization.This material is based on research supported by National Science Foundation Grants DCR-8521228 and CCR-8723091, and Air Force Office of Scientific Research Grants AFOSR 86-0172 and AFOSR and AFOSR 89-0410.     

Computing the Matrix Cosine

An algorithm is developed for computing the matrix cosine, building on a proposal of Serbin and Blalock. The algorithm scales the matrix by a power of 2 to make the -norm less than or equal to 1, evaluates a Padé approximant, and then uses the double angle formula cos(2A)=2cos(A)²–I to recover the cosine of the original matrix. In addition, argument reduction and balancing is used initially to decrease the norm. We give truncation and rounding error analyses to show that an [8,8] Padé approximant produces the cosine of the scaled matrix correct to machine accuracy in IEEE double precision arithmetic, and we show that this Padé approximant can be more efficiently evaluated than a corresponding Taylor series approximation. We also provide error analysis to bound the propagation of errors in the double angle recurrence. Numerical experiments show that our algorithm is competitive in accuracy with the Schur–Parlett method of Davies and Higham, which is designed for general matrix functions, and it is substantially less expensive than that method for matrices of -norm of order 1. The dominant computational kernels in the algorithm are matrix multiplication and solution of a linear system with multiple right-hand sides, so the algorithm is well suited to modern computer architectures.     

用MATLAB解决高等数学中的图形问题

本文通过实例将 MATLAB与 Mathematica的图形功能做了比较 ,指出了 MATLAB在解决高等数学图形问题时的优势 .     

Global Optimization of Nonconvex Polynomial Programming Problems Having Rational Exponents

This paper considers the solution of nonconvex polynomial programming problems that arise in various engineering design, network distribution, and location-allocation contexts. These problems generally have nonconvex polynomial objective functions and constraints, involving terms of mixed-sign coefficients (as in signomial geometric programs) that have rational exponents on variables. For such problems, we develop an extension of the Reformulation-Linearization Technique (RLT) to generate linear programming relaxations that are embedded within a branch-and-bound algorithm. Suitable branching or partitioning strategies are designed for which convergence to a global optimal solution is established. The procedure is illustrated using a numerical example, and several possible extensions and algorithmic enhancements are discussed.     

A strong duality theorem for the minimum of a family of convex programs

We produce a duality theorem for the minimum of an arbitrary family of convex programs. This duality theorem provides a single concave dual maximization and generalizes recent work in linear disjunctive programming. Homogeneous and symmetric formulations are studied in some detail, and a number of convex and nonconvex applications are given.This work was partially funded by National Research Council of Canada, Grant No. A4493. Thanks are due to Mr. B. Toulany for many conversations and to Dr. L. MacLean who suggested the chance-constrained model.     

An extension lemma and homogeneous programming

An extension lemma, which is equivalent to the generalized Gordan's theorem of the alternative, due to Fan, Glicksberg, and Hoffman, is applied to present a duality theory for a general class of homogeneous programs, with and without a constraint qualification of Slater type. In addition, an existence theorem for optimal solutions of homogeneous programs is given.The author thanks an anonymous referee for valuable suggestions about an earlier draft of this paper.     

Recovering optimal dual solutions in Karmarkar's polynomial algorithm for linear programming

This paper presents a “standard form” variant of Karmarkar's algorithm for linear programming. The tecniques of using duality and cutting objective are combined in this variant to maintain polynomial-time complexity and to bypass the difficulties found in Karmarkar's original algorithm. The variant works with problems in standard form and simultaneously generates sequences of primal and dual feasible solutions whose objective function values converge to the unknown optimal value. Some computational results are also reported.     

Stability analysis for stochastic programs

For stochastic programs with recourse and with (several joint) probabilistic constraints, respectively, we derive quantitative continuity properties of the relevant expectation functionals and constraint set mappings. This leads to qualitative and quantitative stability results for optimal values and optimal solutions with respect to perturbations of the underlying probability distributions. Earlier stability results for stochastic programs with recourse and for those with probabilistic constraints are refined and extended, respectively. Emphasis is placed on equipping sets of probability measures with metrics that one can handle in specific situations. To illustrate the general stability results we present possible consequences when estimating the original probability measure via empirical ones.