Parametric Duality Models for Semi-infinite Discrete Minmax Fractional Programming Problems Involving Generalized （η,ρ）-Invex Functions

A semi-infinite programming problem is a mathematical programming problem with a finite number of variables and infinitely many constraints. Duality theories and generalized convexity concepts are important research topics in mathematical programming. In this paper, we discuss a fairly large number of paramet- ric duality results under various generalized （η,ρ）-invexity assumptions for a semi-infinite minmax fractional programming problem.     

MONOTONIZATION IN GLOBAL OPTIMIZATION

A general monotonization method is proposed for converting a constrained programming problem with non-monotone objective function and monotone constraint functions into a monotone programming problem. An equivalent monotone programming problem with only inequality constraints is obtained via this monotonization method. Then the existing convexification and concavefication methods can be used to convert the monotone programming problem into an equivalent better-structured optimization problem.     

具有V-不变凸的多目标规划的另一种方法

In this paper, optimality conditions for multiobjective programming problems having V-invex objective and constraint functions are considered. An equivalent multiobjective programming problem is constructed by a modification of the objective function.Furthermore, a (α, η)-Lagrange function is introduced for a constructed multiobjective programming problem, and a new type of saddle point is introduced. Some results for the new type of saddle point are given.     

广义几何规划的压缩共轭梯度路径非单调信赖域算法(英文)

In this paper,on the basis of making full use of the characteristics of unconstrained generalized geometric programming（GGP）,we establish a nonmonotonic trust region algorithm via the conjugate path for solving unconstrained GGP problem.A new type of condensation problem is presented,then a particular conjugate path is constructed for the problem,along which we get the approximate solution of the problem by nonmonotonic trust region algorithm,and further prove that the algorithm has global convergence and quadratic convergence properties.     

SOLVING A CLASS OF INVERSE QP PROBLEMS BY A SMOOTHING NEWTON METHOD

We consider an inverse quadratic programming （IQP） problem in which the parameters in the objective function of a given quadratic programming （QP） problem are adjusted as little as possible so that a known feasible solution becomes the optimal one. This problem can be formulated as a minimization problem with a positive semidefinite cone constraint and its dual （denoted IQD（A, b）） is a semismoothly differentiable （SC^1） convex programming problem with fewer variables than the original one. In this paper a smoothing Newton method is used for getting a Karush-Kuhn-Tucker point of IQD（A, b）. The proposed method needs to solve only one linear system per iteration and achieves quadratic convergence. Numerical experiments are reported to show that the smoothing Newton method is effective for solving this class of inverse quadratic programming problems.     

具有随机区间收益的效用优化分析(英文)

This article discusses the problem of utility maximization in a market with random-interval payoffs without short-selling prohibition. A novel expected utility model is given to measure an investor’s subjective view toward random interval wealth. Some techniques are proposed to transfer a complex programming involving interval numbers into a simple non-linear programming. Under the existence of the optimal strategy, relations between the optimal strategy and assets’ prices are discussed. Some properties of the maximal utility function with respect to the endowment are given.     

A Dynamic Programming Algorithm for the κ-Haplotyping Problem

The Minimum Fragments Removal （MFR） problem is one of the haplotyping problems： given a set of fragments, remove the minimum number of fragments so that the resulting fragments can be partitioned into k classes of non-conflicting subsets. In this paper, we formulate the κ-MFR problem as an integer linear programming problem, and develop a dynamic programming approach to solve the κ-MFR problem for both the gapless and gap eases.     

本刊英文版2014年57卷第12期摘要(英文)

<正>An SDP randomized approximation algorithm for max hypergraph cut with limited unbalance XU Bao Gang,YU Xing Xing,ZHANG Xiao YanZHANG Zan-Bo Abstract We consider the design of semidefinite programming(SDP)based approximation algorithm for the problem Max Hypergraph Cut with Limited Unbalance(MHC-LU):Find a partition of the vertices of a weighted hypergraph H=(V,E)into two subsets V1,V2with||V2|-|V1||u for some given u and maximizing the total weight of the edges meeting both V1and V2.The problem MHC-LU generalizes several other combinatorial     

A NEW DUAL PROBLEM FOR NONDIFFERENTIABLE CONVEX PROGRAMMING

This paper gives a new dual problem for nondifferentiable convex programming and provesthe properties of weak duality and strong duality and offers a necessary and sufficient condition ofstrong duality.     

A trust region algorithm for solving bilevel programming problems

In this paper, we present a new trust region algorithm for a nonlinear bilevel programming problem by solving a series of its linear or quadratic approximation subproblems. For the nonlinear bilevel programming problem in which the lower level programming problem is a strongly convex programming problem with linear constraints, we show that each accumulation point of the iterative sequence produced by this algorithm is a stationary point of the bilevel programming problem.     

A DUAL NON-LINEAR PROGRAM

In the paper a dual of a nonlinear fractional functional programming problem has beenformulated.This problem can be used to obtain a solution of the mixed 0—1 integer linearprogramming problem.Some properties related to the primal and dual have been given.     

AN EFFECTIVE CONTINUOUS ALGORITHM FOR APPROXIMATE SOLUTIONS OF LARGE SCALE MAX-CUT PROBLEMS

An effective continuous algorithm is proposed to find approximate solutions of NP-hardmax-cut problems.The algorithm relaxes the max-cut problem into a continuous nonlinearprogramming problem by replacing n discrete constraints in the original problem with onesingle continuous constraint.A feasible direction method is designed to solve the resultingnonlinear programming problem.The method employs only the gradient evaluations ofthe objective function,and no any matrix calculations and no line searches are required.This greatly reduces the calculation cost of the method,and is suitable for the solutionof large size max-cut problems.The convergence properties of the proposed method toKKT points of the nonlinear programming are analyzed.If the solution obtained by theproposed method is a global solution of the nonlinear programming problem,the solutionwill provide an upper bound on the max-cut value.Then an approximate solution to themax-cut problem is generated from the solution of the nonlinear programming and providesa lower bound on the max-cut value.Numerical experiments and comparisons on somemax-cut test problems(small and large size)show that the proposed algorithm is efficientto get the exact solutions for all small test problems and well satisfied solutions for mostof the large size test problems with less calculation costs.     

ON SOME PROPERTIES OF SOLUTIONS TO SEMIDEFINITE PROGRAMMING

It is well known that for symmetric linear programming there exists a strictly complementary solution if the primal and the dual problems are both feasible. However, this is not necessary true for symmetric or general semide finite programming even if both the primal problem and its dual problem are strictly feasible. Some other properties are also concerned.     

AN ITERATIVE METHOD FOR THE MINIMAX PROBLEM

In this paper a class of iterative methods for the minimax problem i; proposed.We present a sequence of the extented linear-quadratic programming (ELQP) problems as subproblems of the original minimal problem and solve the ELQP problem iteratively.The locally linear and su-perlinear convergence results of the algorithm are established.     

Global optimization method for linear multiplicative programming

In this paper, a new global algorithm is presented to globally solve the linear multiplicative programming(LMP). The problem(LMP) is firstly converted into an equivalent programming problem(LMP(H))by introducing p auxiliary variables. Then by exploiting structure of(LMP(H)), a linear relaxation programming(LP(H)) of(LMP(H)) is obtained with a problem(LMP) reduced to a sequence of linear programming problems. The algorithm is used to compute the lower bounds called the branch and bound search by solving linear relaxation programming problems(LP(H)). The proposed algorithm is proven that it is convergent to the global minimum through the solutions of a series of linear programming problems. Some examples are given to illustrate the feasibility of the proposed algorithm.     

向量优化问题的对偶性(英文)

Duality framework on vector optimization problems in a locally convex topological vector space are established by using scalarization with a cone-strongly increasing function.The dualities for the scalar convex composed optimization problems and for general vector optimization problems are studied.A general approach for studying duality in vector optimization problems is presented.     

二阶($F, \alpha,\rho, d,p$) 一致凸性与极小极大分式规划的对偶

In this paper,we introduce a class of generalized second order(F,α,ρ,d,p)-univex functions.Two types of second order dual models are considered for a minimax fractional programming problem and the duality results are established by using the assumptions on the functions involved.     

带有极值函数的多目标双层规划问题的对偶

For a multiobjective bilevel programming problem(P) with an extremal-value function,its dual problem is constructed by using the Fenchel-Moreau conjugate of the functions involved.Under some convexity and monotonicity assumptions,the weak and strong duality assertions are obtained.     

Higher-Order (F,α,β,ρ,d,E)-Convexity in Fractional Programming

In this paper we define higher order (F,α,β,ρ,d,E)-convex function with respect to E-differentiable function K and obtain optimality conditions for nonlinear programming problem (NP) from the concept of higher order (F,α,β,ρ,d)-convexity. Here, we establish Mond-Weir and Wolfe duality for (NP) and utilize these duality in nonlinear fractional programming problem.     

Single machine stochastic JIT scheduling problem subject to machine breakdowns

In this paper we research the single machine stochastic JIT scheduling problem subject to the machine breakdowns for preemptive-resume and preemptive-repeat.The objective function of the problem is the sum of squared deviations of the job-expected completion times from the due date.For preemptive-resume,we show that the optimal sequence of the SSDE problem is V-shaped with respect to expected processing times.And a dynamic programming algorithm with the pseudopolynomial time complexity is given.We discuss the difference between the SSDE problem and the ESSD problem and show that the optimal solution of the SSDE problem is a good approximate optimal solution of the ESSD problem,and the optimal solution of the SSDE problem is an optimal solution of the ESSD problem under some conditions.For preemptive-repeat,the stochastic JIT scheduling problem has not been solved since the variances of the completion times cannot be computed.We replace the ESSD problem by the SSDE problem.We show that the optimal sequence of the SSDE problem is V-shaped with respect to the expected occupying times.And a dynamic programming algorithm with the pseudopolynomial time complexity is given.A new thought is advanced for the research of the preemptive-repeat stochastic JIT scheduling problem.