Table of Content

15 March 2011, Volume 15 Issue 1

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Original Articles

An Integral Optimality Condition for Global Optimization

ZHANG Li-Li, LI Jian-Yu, LI Xing-Si

2011, 15(1):  1-10. 

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An integral optimality condition for global optimization problem is investigated by using a level set auxiliary function. The auxiliary function has one variant that represents an estimated optimal value of the objective function in primal optimization problem and one controlling parameter for accuracy. Necessary and sufficient condition for global optimality in terms of the behavior of the auxiliary function is derived. The integral global optimality condition is obtained via a limiting process of this auxiliary function. Furthermore, if the measure is the Lebesgue measure and the integral region takes a finite subset of the Natural Number set, then this integral global optimality condition divergences to the approximation scheme that used aggregate function to approximate the max-function in the finite minimax problem. So the integral global optimality condition is an extension of this approximation scheme in continuous maximum problem.

Optimization of Truss Vibration with Reduction  of Symmetric Semidefinite Programming

ZHOU Yi-Kai, BAI Yan-Qin, SUN Yan

2011, 15(1):  11-24. 

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A truss vibration optimization problem is to minimize the total weight of truss subject to a given fundamental vibration frequency. This paper focuses on the recent results of matrix algebraic approach to solve the truss vibration optimization problem expressed in terms of symmetric semidefinite programming problem. We derive two sufficient conditions on constructing the symmetric group representation to reduce the problems size. An example of eight-bar truss design problem is given to illustrate how to construct a group representation and to demonstrate its effectiveness.

A New Class of Penalty Functions and Penalty Algorithm

ZHANG Yu-Huan, WANG Chang-Yu

2011, 15(1):  25-34. 

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In this paper, we propose a new class of penalty functions for solving nonlinear programming problems with inequality constraints, a subclass of which smoothly approximates the $l_1$ penalty function. Based on the new class of penalty functions, we consider a penalty algorithm, the characteristic of which is at each iteration, an exact global optimal solution or an inexact global optimal solution is obtained. Under very weak conditions, the algorithm is always applicable. We present the global convergence without any constraint qualification. Finally, numerical experiments are given.

A Generalization for Directed Scale-Free Graphs

YAN Yun-Zhi, WANG Han-Xing

2011, 15(1):  35-45. 

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We study a dynamically evolving directed random graph which randomly adds vertices and directed edges using preferential attachment and prove that its vertice degree obey power law and has elaborate power law exponents.

A New Filter Method

PU Ding-Guo, SHAO Wen-Qiong, LIU Mei-Ling, LIU Ci-Wen

2011, 15(1):  46-58. 

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In this paper, we define a new filter  and propose a filter QP-free infeasible method with some piecewise linear relational NCP function   for constrained nonlinear optimization  problems. This iterative method is  based on the solution of nonsmooth equations which  are obtained by the multipliers and the NCP function for the KKT first-order optimality conditions.  Locally, each iteration of this method can be viewed as a perturbation of a mixed Newton-quasi Newton iteration on both the primal and dual variables for the solution of the KKT optimality conditions. We also use the filter on line searches.  This method is  implementable and globally convergent. We also prove that the method has superlinear convergence rate under some mild conditions.

Block-Iterative Subgradient Projection Algorithms  for the Convex Feasibility problem

DANG Ya-Zheng, GAO Yan, ZHI Li-Ping

2011, 15(1):  59-70. 

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In this paper,   sequential block-iterative subgradient projection algorithm and parallel block-iterative subgradient projection algorithm for solving the convex feasibility problem expressed by  the system of inequalities are presented. Each step in these methods consists of finding the approximation projection of the current point on the subsystem which is constructed  through  parting  the system of inequalities into several blocks.  The convergence for both of sequential block-iterative subgradient projection algorithm and parallel block-iterative subgradient projection algorithm are obtained under some weak conditions.

New SDP Relaxations  for Unconstrained  0-1 Polynomial Problems 

JI Shu-Hui

2011, 15(1):  71-84. 

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In this paper, we present new semidefinite programming (SDP) relaxation schemes for 0-1 unconstrained polynomial programming (0-1UPP) problems. We first construct an SDP relaxation based on matrix cone decomposition and (piecewise) linear approximation for (0-1UPP).  It is shown that this SDP bound is tighter than the standard linear form (SLF). We then use Lagrangian dual and sum of squares (SOS) relaxation to obtain SDP relaxations which are equivalent to Lasserre's SDP relaxations for (0-1UPP). This provides a new way to derive Lasserre's hierarchy of SDP relaxations for (0-1UPP).

Global Convergence of HZ's Conjugate Gradient  Method with Armijo-type Line Search

WEI Jing-Guang, ZHANG Jian-Jun

2011, 15(1):  85-94. 

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HZ's  conjugate gradient method (proposed by William W. Hager and Hongchao  Zhang)  has been proved to be an efficient method. In this paper, we prove the global convergence of  HZ's method with Armijo-type line search.  Our numerical experiments show that the new algorithm are more efficient and competitive with HZ's method with Wolfe line search in most cases.

Second Order Sufficient Conditions for Mathematical Programs Governed by Second-Order   Cone Constrained  Generalized Equations 

WU Jia, ZHANG Li-Wei

2011, 15(1):  95-103. 

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 In this paper, we consider a class of mathematical programs governed by second-order cone constrained generalized equations. We define the critical cone  in terms of the directional derivative of a nonsmooth  mapping and derive its equivalent characterization at a feasible point.  With the help of the critical cone,  we propose a set of second order sufficient conditions for the mathematical program governed by second-order cone constrained generalized equations. We demonstrate, under some conditions, that the set of second order sufficient conditions is sufficient for the second order growth of an M-stationary point.

The Global Optimum Conditions of a New Class   of Level-value Estimation Methods

LI Feng, LOU Ye

2011, 15(1):  104-112. 

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 In this paper, we propose global optimum conditions of a new class of level-value estimate methods for global optimization. Through researching into the equivalence between the root of variance or the $\nu$-variance function and the optimal value of the original problem, we get the corresponding optimum conditions.

Global  Convergence Results of BFGS Methods with New Nonmonotone Step Size Rule (Chinese)

GUO Yuan-Bao, HUANG Bing-Jia

2011, 15(1):  113-121. 

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We propose a new nonmonotone step size rule and analyze the global convergence of new BFGS quasi-Newton method. The new step size rule is similar to Zhang H.C. nonmonotone step size rule and contains it as a special case. Numerical experiments have been conducted which show that the proposed algorithm is encouraging.  

 Constructing Integral Spectra Graphs by the  Super Generalized Line Graph Methods

ZHANG Hong-Rui, WANG Li-Gong

2011, 15(1):  122-128. 

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 Line graph plays an important role in spectral graph theory. In this article, we gain a new method by reaching on the sufficient conditions under which the super generlized line graph is the integral graph. And using this method we can construct infinite new integral graphs.