On Extracting Maximum Stable Sets in Perfect Graphs Using Lovász's Theta Function

We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions of different formulations of Lovász's theta function. We propose reductions from feasible solutions corresponding to a graph to those corresponding to its induced subgraphs. We develop an efficient, polynomial-time algorithm to extract a maximum stable set in a perfect graph using the theta function. Our algorithm iteratively transforms an approximate solution of the semidefinite formulation of the theta function into an approximate solution of another formulation, which is then used to identify a vertex that belongs to a maximum stable set. The subgraph induced by that vertex and its neighbors is removed and the same procedure is repeated on successively smaller graphs. We establish that solving the theta problem up to an adaptively chosen, fairly rough accuracy suffices in order for the algorithm to work properly. Furthermore, our algorithm successfully employs a warm-start strategy to recompute the theta function on smaller subgraphs. Computational results demonstrate that our algorithm can efficiently extract maximum stable sets in comparable time it takes to solve the theta problem on the original graph to optimality. This work was supported in part by NSF through CAREER Grant DMI-0237415. Part of this work was performed while the first author was at the Department of Applied Mathematics and Statisticsat Stony Brook University, Stony Brook, NY, USA.     

Generalized fractional programming duality: A parametric approach

Using a parametric approach, duality is presented for a minimax fractional programming problem that involves several ratios in the objective function.The first author is thankful to Natural Science and Engineering Research Council of Canada for financial support through Grant A-5319, and the authors are thankful to the anonymous referees for useful suggestions.     

A finite cutting plane method for facial disjunctive programs

This paper shows that any linear disjunctive program with a finite number of constraints can be transformed into an equivalent facial program. Based upon linear programming technique, a new, finite cutting plane method is presented for the facial programs.

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Zusammenfassung Die Arbeit zeigt, daß jedes lineare disjunktive Optimierungsproblem mit endlich vielen Restriktionen in ein äquivalentes Fazetten-Problem transformiert werden kann. Auf der Grundlage von linearer Optimierungstechnik wird für das Fazetten-Problem ein neues, endliches Schnittebenenverfahren vorgestellt.

     

A new trust region technique for the maximum weight clique problem

A new simple generalization of the Motzkin-Straus theorem for the maximum weight clique problem is formulated and directly proved. Within this framework a trust region heuristic is developed. In contrast to usual trust region methods, it regards not only the global optimum of a quadratic objective over a sphere, but also a set of other stationary points of the program. We formulate and prove a condition when a Motzkin-Straus optimum coincides with such a point. The developed method has complexity O(n³), where n is the number of vertices of the graph. It was implemented in a publicly available software package QUALEX-MS.Computational experiments indicate that the algorithm is exact on small graphs and very efficient on the DIMACS benchmark graphs and various random maximum weight clique problem instances.     

变量有广义界线性规划的直接对偶单纯形法

本文讨论变量有广义界线性规划问题借助标准形线性规划同单纯形法技术，建立问题的一个直接对偶单纯形法。分析了方法的性质，给出了初始对偶可行基的计算方法，并用实例说明方法的具体操作。     

Redefining Event Variables for Efficient Modeling of Continuous-Time Batch Processing

We define events so as to reduce the number of events and decision variables needed for modeling batch-scheduling problems such as described in [15]. We propose a new MILP formulation based on this concept, defining non-uniform time periods as needed and decision variables that are not time-indexed. It can handle complicated multi-product/multi-stage machine processes, with production lines merging and diverging, and with minimum and maximum batch sizes. We compare it with earlier models and show that it can solve problems with small to medium demands relative to batch sizes in reasonable computer times.     

Towards Strong Duality in Integer Programming

We consider in this paper the Lagrangian dual method for solving general integer programming. New properties of Lagrangian duality are derived by a means of perturbation analysis. In particular, a necessary and sufficient condition for a primal optimal solution to be generated by the Lagrangian relaxation is obtained. The solution properties of Lagrangian relaxation problem are studied systematically. To overcome the difficulties caused by duality gap between the primal problem and the dual problem, we introduce an equivalent reformulation for the primal problem via applying a pth power to the constraints. We prove that this reformulation possesses an asymptotic strong duality property. Primal feasibility and primal optimality of the Lagrangian relaxation problems can be achieved in this reformulation when the parameter p is larger than a threshold value, thus ensuring the existence of an optimal primal-dual pair. We further show that duality gap for this partial pth power reformulation is a strictly decreasing function of p in the case of a single constraint. Dedicated to Professor Alex Rubinov on the occasion of his 65th birthday. Research supported by the Research Grants Council of Hong Kong under Grant CUHK 4214/01E, and the National Natural Science Foundation of China under Grants 79970107 and 10571116.     

解可分离约束双凹规划问题的一种外逼近方法[英文]

在本文中，我们提出了双凹规划问题和更一般的广义凹规划问题。我们给出了双凹规划问题的整体最优性条件，并构造了一个有限终止外逼近算法。     

Numerical treatment of an asset price model with non-stochastic uncertainty

In contrast to stochastic differential equation models used for the calculation of the term structure of interest rates, we develop an approach based on linear dynamical systems under non-stochastic uncertainty with perturbations. The uncertainty is described in terms of known feasible sets of varying parameters. Observations are used in order to estimate these parameters by minimizing the maximum of the absolute value of measurement errors, which leads to a linear or nonlinear semi-infinite programming problem. A regularized logarithmic barrier method for solving (ill-posed) convex semi-infinite programming problems is suggested. In this method a multi-step proximal regularization is coupled with an adaptive discretization strategy in the framework of an interior point approach. A special deleting rule permits one to use only a part of the constraints of the discretized problems. Convergence of the method and its stability with respect to data perturbations in the cone of convexC ¹-functions are studied. On the basis of the solutions of the semi-infinite programming problems a technical trading system for future contracts of the German DAX is suggested and developed. Supported by the Stiftung Rheinland/Pfalz für Innovation, No. 8312-386261/307.     

Comparison of Two Reformulation-Linearization Technique Based Linear Programming Relaxations for Polynomial Programming Problems

In this paper, we compare two strategies for constructing linear programmingrelaxations for polynomial programming problems using aReformulation-Linearization Technique (RLT). RLT involves an automaticreformulation of the problem via the addition of certain nonlinear impliedconstraints that are generated by using the products of the simple boundingrestrictions (among other products), and a subsequent linearization based onvariable redefinitions. We prove that applying RLT directly to the originalpolynomial program produces a bound that dominates in the sense of being atleast as tight as the value obtained when RLT is applied to the jointcollection of all equivalent quadratic problems that could be constructed byrecursively defining additional variables as suggested by Shor.