Table of Content

15 March 2014, Volume 18 Issue 1

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Original Articles

An  overview of mathematical programming research in China

The Mathematical Programming Branch of Operations Research Society of China

2014, 18(1):  1-8. 

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Mathematical programming or mathematical optimization is an important branch of operations research that studies the problem of minimizing or maximizing a real function of real or integer variables,  subject to constraints on the variables.  It is one of widely used modeling tools and methodologies in operations research and management science and has numerous applications in engineering, economics and finance.  In this chapter,  we first give a brief introduction of mathematical programming problems, its history, applications and main research areas.  We then review the state-of-the-science of mathematical programming study with an overview of the development of mathematical programming in China.  Research perspectives of mathematical programming is also presented.

Vector optimization and its developments

RONG Weidong, YANG Xinmin

2014, 18(1):  9-38. 

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Vector optimization is a mathematical model which minimizes or maximizes a vector-valued function. As an important part of mathematical programming, vector optimization is a promising interdisciplinary research field with many significant applications. Since 1950, the structure of the theory of vector optimization has been very complete, as well as some important progresses have been made in the study of methods, furthermore the applications have been flourishing. In this paper, we briefly review the developments of vector optimization, introduce the main characteristics, the basic theory and methods of it, highlight some recent typical progresses achieved by Chinese researchers, and propose some possible research prospects in future.

Recent advances in integer programming

SUN Xiaoling, LI Duan

2014, 18(1):  39-68. 

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Integer programming deals with optimization problems with decision variables being all integer or partly integer. Integer programming has been one of the most active research directions in optimization due to its wide applications in operations research and management science. In this survey paper, we first briefly review the background of integer programming and summarize the fundamental results of linear and nonlinear integer programming. We then focus on some recent progress in several research topics, including semi-definite programming relaxation and randomized methods for 0-1 quadratic programs, optimization problems with cardinality and semi-continuous variables, and co-positive cone program representations and approximations of 0-1 quadratic programs. Finally, we indicate some research perspectives and open problems in integer programming.

Advances in linear and nonlinear programming

DAI Yuhong, LIU Xinwei

2014, 18(1):  69-92. 

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Linear and nonlinear programming is a classical branch in mathematical programming. We introduce some backgrounds on linear and nonlinear programming, and some new methods and new research advances in linear programming, unconstrained and constrained optimization. The alternating direction method of multipliers is very efficient in solving some constrained optimization problems with special structure and has been attracted much attentions in recent years. Global optimization is specially important for applications of optimization. These two topics are also involved.

Some topics in nonlinear positive semi-definite programming

ZHANG Liwei

2014, 18(1):  93-112. 

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Four topics in nonlinear positive semi-definite programming are discussed, which include the variational analysis about the cone of positive semi-definite matrices, optimality conditions, perturbation analysis and the augmented Lagrange method for nonlinear positive semi-definite programming.

Several developments of variational inequalities and complementarity problems, bilevel programming and MPEC

HUANG Zhenghai, LIN Guihua, XIU Naihua

2014, 18(1):  113-133. 

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This paper investigates finite-dimensional variational inequalities and complementarity problems, bilevel programming problems and mathematical programs with equilibrium constraints (MPECs).  After a brief introduction to these problems, this paper focuses on several recent rapidly developing aspects in these fields, which include theories and methods for symmetric cone complementarity problems, projection and contraction methods for variational inequality problems, models and methods for stochastic variational inequalities and stochastic complementarity problems, and new methods for bilevel programming problems and MPECs. Finally, several future research directions are proposed in this paper.

Some advances in tensor analysis and polynomial optimization

LI Zhening, LING Chen, WANG Yiju, YANG Qingzhi

2014, 18(1):  134-148. 

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 Tensor analysis (also called as numerical multilinear algebra) mainly includes tensor decomposition, tensor eigenvalue theory and relevant algorithms. Polynomial optimization mainly includes theory and algorithms for solving optimization problems with polynomial objects functions under polynomial constrains. This survey covers the most of recent advances  in these two fields. For tensor analysis, we introduce some properties and  algorithms concerning the spectral radius of nonnegative tensors' H-eigenvalue. We also discuss the optimization models and solution methods of nonnegative tensors' largest (smallest) Z-eigenvalue. For polynomial optimization problems, we mainly introduce the optimization of homogeneous polynomial function under the unit spherical constraints or binary constraints and their extended problems, and  semidefinite relaxation methods for solving them approximately. We also look into the further perspective of these research topics.

New perspectives of several fundamental problems in combinatorial optimization

CHEN Xujin, XU Dachuan, ZHANG Guochuan

2014, 18(1):  149-158. 

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Combinatorial optimization, as an interdiscipline of operations research and computer science, has been developing since the mid-20th century. It characters optimization problems with discrete structures, and explores their solution methods. Due to the structural divergence in discrete problems, there has been a wide variety of methodologies and techniques. In this paper, we briefly introduce a number of fundamental problems in combinatorial optimization, and present the recent achievements together with some open problems.