A Robust SQP Method for Mathematical Programs with Linear Complementarity Constraints

The relationship between the mathematical program with linear complementarity constraints (MPLCC) and its inequality relaxation is studied. Based on this relationship, a new sequential quadratic programming (SQP) method is presented for solving the MPLCC. A certain SQP technique is introduced to deal with the possible infeasibility of quadratic programming subproblems. Global convergence results are derived without assuming the linear independence constraint qualification for MPEC, the nondegeneracy condition, and any feasibility condition of the quadratic programming subproblems. Preliminary numerical results are reported. Research is partially supported by Singapore-MIT Alliance and School of Business, National University of Singapore.     

鲁棒信用风险优化的线性锥优化模型

考虑了具有强健性的信用风险优化问题. 根据最差条件在值风险度量信用风险的方法,建立了信用风险优化问题的模型. 由于信用风险的损失分布存在不确定性,考虑了两类不确定性区间,即箱子型区间和椭球型区间. 把具有强健性的信用风险优化问题分别转化成线性规划问题和二阶锥规划问题. 最后,通过一个信用风险问题的例子来说明此模型的有效性.     

A Class of Quadratic Programs with Linear Complementarity Constraints

We consider a class of quadratic programs with linear complementarity constraints (QPLCC) which belong to mathematical programs with equilibrium constraints (MPEC). We investigate various stationary conditions and present new and strong necessary and sufficient conditions for global and local optimality. Furthermore, we propose a Newton-like method to find an M-stationary point in finite steps without MEPC linear independence constraint qualification. The research of this author is partially supported by NSERC, and Research Grand Council of Hong Kong.     

A Globally Convergent Sequential Quadratic Programming Algorithm for Mathematical Programs with Linear Complementarity Constraints

This paper presents a sequential quadratic programming algorithm for computing a stationary point of a mathematical program with linear complementarity constraints. The algorithm is based on a reformulation of the complementarity condition as a system of semismooth equations by means of Fischer-Burmeister functional, combined with a classical penalty function method for solving constrained optimization problems. Global convergence of the algorithm is established under appropriate assumptions. Some preliminary computational results are reported.     

Global Optimization Method for Solving Mathematical Programs with Linear Complementarity Constraints

We propose a method for finding a global optimal solution of programs with linear complementarity constraints. This problem arises for instance in bilevel programming. The main idea of the method is to generate a sequence of points either ending at a global optimal solution within a finite number of iterations or converging to a global optimal solution. The construction of such sequence is based on branch-and-bound techniques, which have been used successfully in global optimization. Results on a numerical test of the algorithm are reported.The main part of this article was written during the first authors stay as Visiting Professor at the Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Japan. The second and the third authors were supported by Grant-in-Aid for Scientific Research C(2) 13650061 of the Ministry of Education, Culture, Sports, Science, and\break Technology of Japan.The authors thank P. B. Hermanns, Department of Mathematics, University of Trier, for carrying out the numerical test reported in Section 5. The authors also thank the referees and the Associate Editor for comments and suggestions which helped improving the first version of this article.     

Distributionally robust scheduling on parallel machines under moment uncertainty

This paper investigates a distributionally robust scheduling problem on identical parallel machines, where job processing times are stochastic without any exact distributional form. Based on a distributional set specified by the support and estimated moments information, we present a min-max distributionally robust model, which minimizes the worst-case expected total flow time out of all probability distributions in this set. Our model doesn’t require exact probability distributions which are the basis for many stochastic programming models, and utilizes more information compared to the interval-based robust optimization models. Although this problem originates from the manufacturing environment, it can be applied to many other fields when the machines and jobs are endowed with different meanings. By optimizing the inner maximization subproblem, the min-max formulation is reduced to an integer second-order cone program. We propose an exact algorithm to solve this problem via exploring all the solutions that satisfy the necessary optimality conditions. Computational experiments demonstrate the high efficiency of this algorithm since problem instances with 100 jobs are optimized in a few seconds. In addition, simulation results convincingly show that the proposed distributionally robust model can hedge against the bias of estimated moments and enhance the robustness of production systems.     

Robust Stability and Performance Analysis of Uncertain Systems Using Linear Matrix Inequalities

A wide variety of problems in system and control theory can be formulated or reformulated as convex optimization problems involving linear matrix inequalities (LMIs), that is, constraints requiring an affine combination of symmetric matrices to be positive semidefinite. For a few very special cases, there are analytical solutions to these problems, but in general LMI problems can be solved numerically in a very efficient way. Thus, the reduction of a control problem to an optimization problem based on LMIs constitutes, in a sense, a solution to the original problem. The objective of this article is to provide a tutorial on the application of optimization based on LMIs to robust control problems. In the first part of the article, we provide a brief introduction to optimization based on LMIs. In the second part, we describe a specific example, that of the robust stability and performance analysis of uncertain systems, using LMI optimization.     

一类分布鲁棒线性决策随机优化研究

随机优化广泛应用于经济、管理、工程和国防等领域,分布鲁棒优化作为解决分布信息模糊下的随机优化问题近年来成为学术界的研究热点.本文基于φ-散度不确定集和线性决策方式研究一类分布鲁棒随机优化的建模与计算,构建了易于计算实现的分布鲁棒随机优化的上界和下界问题.数值算例验证了模型分析的有效性.     

A Deterministic Approach to Linear Programs with Several Additional Multiplicative Constraints

We consider a global optimization problem of minimizing a linear function subject to p linear multiplicative constraints as well as ordinary linear constraints. We show that this problem can reduce to a 2p-dimensional reverse convex program, and present an algorithm for solving the resulting problem. Our algorithm converges to a globally optimal solution and yields an -approximate solution in finite time for any > 0. We also report some results of computational experiment.     

不确定信息多目标线性优化的鲁棒方法

研究不确定信息的多目标线性优化问题,其数据不能精确给出但是属于一个给定的集合.首先,采用鲁棒方法把该问题转化为一个确定的多目标优化问题.然后,给出此问题解存在的充分条件.最后,通过实例验证了用鲁棒方法解决不确定信息的多目标线性优化问题的有效性.     

证券投资组合优化问题的强健性的锥优化方法(英文)

本文研究了具有强健性的证券投资组合优化问题.模型以最差条件在值风险为风险度量方法,并且考虑了交易费用对收益的影响.当投资组合的收益率概率分布不能准确确定但是在有界的区间内,尤其是在箱型区间结构和椭球区域结构内时,我们可以把具有强健性的证券投资组合优化问题的模型分别转化成线性规划和二阶锥规划形式.最后,我们用一个真实市场数据的算例来验证此方法.     

Linear Huber M-Estimator Under Ellipsoidal Data Uncertainty

The purpose of this note is to present a robust counterpart of the Huber estimation problem in the sense of Ben-Tal and Nemirovski when the data elements are subject to ellipsoidal uncertainty. The robust counterparts are polynomially solvable second-order cone programs with the strong duality property. We illustrate the effectiveness of the robust counterpart approach on a numerical example.     

A Geometric Approach to Global Optimization

In this paper we consider the problem of optimizing a piecewise-linear objective function over a non-convex domain. In particular we do not allow the solution to lie in the interior of a prespecified region R. We discuss the geometrical properties of this problems and present algorithms based on combinatorial arguments. In addition we show how we can construct quite complicated shaped sets R while maintaining the combinatorial properties.     

Lower Semicontinuity of the Minimum in Parametric Convex Programs

Recently, Best and Ding (Ref. 1) established a result on the lower semicontinuity of the infimum value function of a parametric convex quadratic program. In this paper, we extend this result to general convex programs. The case of semi-infinite convex optimization is included.     

On the Minimum Norm Solution of Linear Programs

This paper describes a new technique to find the minimum norm solution of a linear program. The main idea is to reformulate this problem as an unconstrained minimization problem with a convex and smooth objective function. The minimization of this objective function can be carried out by a Newton-type method which is shown to be globally convergent. Furthermore, under certain assumptions, this Newton-type method converges in a finite number of iterations to the minimum norm solution of the underlying linear program.     

Robust Solution to Fuzzy Identification Problem with Uncertain Data by Regularization

This study considers the robust identification of the parameters describing a Sugeno type fuzzy inference system with uncertain data. The objective is to minimize the worst-case residual error using a numerically efficient algorithm. The Sugeno type fuzzy systems are linear in consequent parameters but nonlinear in antecedent parameters. The robust consequent parameters identification problem can be formulated as second-order cone programming problem. The optimal solution of this second-order cone problem can be interpreted as solution of a Tikhonov regularization problem with a special choice of regularization parameter which is optimal for robustness (Ghaoui and Lebret (1997). SAIM Journal of Matrix Analysis and Applications 18, 1035–1064). The final regularized nonlinear optimization problem allowing simultaneous identification of antecedent and consequent parameters is solved iteratively using a generalized Gauss–Newton like method. To illustrate the approach, several simulation studies on numerical examples including the modelling of a spectral data function (one-dimensional benchmark example) is provided. The proposed robust fuzzy identification scheme has been applied to approximate the physical fitness of patients with a fuzzy expert system. The identified fuzzy expert system is shown to be capable of capturing the decisions (experiences) of a medical expert.     

Continuity of the Solution Map in Quadratic Programs under Linear Perturbations

It is well known that the solution map of a quadratic program where only the linear part of the data is subject to perturbation is an upper Lipschitz multifunction. This paper characterizes the continuity and lower semicontinuity of that solution map.This work was supported by the Brain Korea 21 Project in 2003, the APEC Postdoctoral Fellowships Program, and the KOSEF Brain Pool Program. The authors thank Professor F. Giannessi and two anonymous referees for helpful comments.Communicated by F. Giannessi     

A Cutting Plane Algorithm for Linear Reverse Convex Programs

In this paper, global optimization of linear programs with an additional reverse convex constraint is considered. This type of problem arises in many applications such as engineering design, communications networks, and many management decision support systems with budget constraints and economies-of-scale. The main difficulty with this type of problem is the presence of the complicated reverse convex constraint, which destroys the convexity and possibly the connectivity of the feasible region, putting the problem in a class of difficult and mathematically intractable problems. We present a cutting plane method within the scope of a branch-and-bound scheme that efficiently partitions the polytope associated with the linear constraints and systematically fathoms these portions through the use of the bounds. An upper bound and a lower bound for the optimal value is found and improved at each iteration. The algorithm terminates when all the generated subdivisions have been fathomed.     

Solving Stochastic Linear Programs with Restricted Recourse Using Interior Point Methods

In this paper we present a specialized matrix factorization procedure for computing the dual step in a primal-dual path-following interior point algorithm for solving two-stage stochastic linear programs with restricted recourse. The algorithm, based on the Birge-Qi factorization technique, takes advantage of both the dual block-angular structure of the constraint matrix and of the special structure of the second-stage matrices involved in the model. Extensive computational experiments on a set of test problems have been conducted in order to evaluate the performance of the developed code. The results are very promising, showing that the code is competitive with state-of-the-art optimizers.     

New Relaxation Method for Mathematical Programs with Complementarity Constraints

In this paper, we present a new relaxation method for mathematical programs with complementarity constraints. Based on the fact that a variational inequality problem defined on a simplex can be represented by a finite number of inequalities, we use an expansive simplex instead of the nonnegative orthant involved in the complementarity constraints. We then remove some inequalities and obtain a standard nonlinear program. We show that the linear independence constraint qualification or the Mangasarian–Fromovitz constraint qualification holds for the relaxed problem under some mild conditions. We consider also a limiting behavior of the relaxed problem. We prove that any accumulation point of stationary points of the relaxed problems is a weakly stationary point of the original problem and that, if the function involved in the complementarity constraints does not vanish at this point, it is C-stationary. We obtain also some sufficient conditions of B-stationarity for a feasible point of the original problem. In particular, some conditions described by the eigenvalues of the Hessian matrices of the Lagrangian functions of the relaxed problems are new and can be verified easily. Our limited numerical experience indicates that the proposed approach is promising.