A new linearization technique for multi-quadratic 0-1 programming problems

We consider the reduction of multi-quadratic 0-1 programming problems to linear mixed 0-1 programming problems. In this reduction, the number of additional continuous variables is O(kn) (n is the number of initial 0-1 variables and k is the number of quadratic constraints). The number of 0-1 variables remains the same.     

On complexity of unconstrained hyperbolic 0-1 programming problems

Single- and multiple-ratio unconstrained hyperbolic 0-1 programming problems are considered. We prove that checking whether these problems have a unique solution is NP-hard. Furthermore, we show that finding the global maximizer of problems with unique solution remains NP-hard. We also discuss complexity of local search and approximability for multiple-ratio problems.     

Reducibility of minimax to minisum 0–1 programming problems

We consider 0–1 programming problems with a minimax objective function and any set of constraints. Upon appropriate transformations of its cost coefficients, such a minimax problem can be reduced to a linear minisum problem with the same set of feasible solutions such that an optimal solution to the latter will also solve the original minimax problem.Although this reducibility applies for any 0–1 programming problem, it is of particular interest for certain locational decision models. Among the obvious implications are that an algorithm for solving a p-median (minisum) problem in a network will also solve a corresponding p-center (minimax) problem.It should be emphasized that the results presented will in general only hold for 0–1 problems due to intrinsic properties of the minimax criterion.     

A polynomially solvable class of quadratic semi-assignment problems

The Quadratic Semi-Assignment Problem (QSAP) models a large variety of practical applications. In the present note we will consider a particular class of QSAP that can be solved by determining the maximum cost flow on a network. This class of problems arises in schedule synchronization and in transportation.     

Testing optimality for quadratic 0-1 problems

Received June 4, 1996 / Revised version received November 19, 1997 Published online November 24, 1998     

Cutting plane algorithms for 0-1 programming based on cardinality cuts

We present new valid inequalities for 0-1 programming problems that work in similar ways to well known cover inequalities. Discussion and analysis of these cuts is followed by their revision and use in integer programming as a new generation of cuts that excludes not only portions of polyhedra containing noninteger points, also parts with some integer points that have been explored in search of an optimal solution. Our computational experimentations demonstrate that this new approach has significant potential for solving large scale integer programming problems.     

A class of polynomially solvable linear complementarity problems

Although the general linear complementarity problem (LCP) is NP-complete, there are special classes that can be solved in polynomial time. One example is the type where the defining matrix is nondegenerate and for which the n-step property holds. In this paper we consider an extension of the property to the degenerate case by introducing the concept of an extended n-step vector and matrix. It is shown that the LCP defined by such a matrix is polynomially solvable as well.     

Lifted inequalities for 0-1 mixed integer programming: Basic theory and algorithms

We study the mixed 0-1 knapsack polytope, which is defined by a single knapsack constraint that contains 0-1 and bounded continuous variables. We develop a lifting theory for the continuous variables. In particular, we present a pseudo-polynomial algorithm for the sequential lifting of the continuous variables and we discuss its practical use.This research was supported by NSF grants DMI-0100020 and DMI-0121495Mathematics Subject Classification (2000): 90C11, 90C27     

Lifted inequalities for 0-1 mixed integer programming: Superlinear lifting

We study the mixed 0-1 knapsack polytope, which is defined by a single knapsack constraint that contains 0-1 and bounded continuous variables, through the lifting of continuous variables fixed at their upper bounds. We introduce the concept of a superlinear inequality and show that, in this case, lifting is significantly simpler than for general inequalities. We use the superlinearity theory, together with the traditional lifting of 0-1 variables, to describe families of facets of the mixed 0-1 knapsack polytope. Finally, we show that superlinearity results can be extended to nonsuperlinear inequalities when the coefficients of the variables fixed at their upper bounds are large.This research was supported by NSF grants DMI-0100020 and DMI-0121495Mathematics Subject Classification (1991): 90C11, 90C27     

线性约束0-1 二次规划的一个定界技术

本文给出确定线性约束0－1二次规划问题最优值下界的方法,该方法结合McBride和Yormark的思想和总体优化中定下界的方法,证明了所定的界较McBride和Yormark的要好．求解线性约束0－1二次规划问题的分支定界算法可以利用本文的定界技术．     

求解0-1线性整数规划问题的有界单纯形法

提出了一种求解0-1线性整数规划问题的有界单纯形法, 不仅通过数学论证, 讨论了该方法的合理性, 奠定了其数学理论基础, 而且通过求解无容量设施选址问题, 验证了该方法的可行性. 在此基础上, 就该有界单纯形法的不足和存在的问题, 给出了进一步改进的途径和手段.     

New trajectory-following polynomial-time algorithm for linear programming problems

A new interior point method for the solution of the linear programming problem is presented. It is shown that the method admits a polynomial time bound. The method is based on the use of the trajectory of the problem, which makes it conceptually very simple. It has the advantage above related methods that it requires no problem transformation (either affine or projective) and that the feasible region may be unbounded. More importantly, the method generates at each stage solutions of both the primal and the dual problem. This implies that, contrary to the simplex method, the quality of the present solution is known at each stage. The paper also contains a practical (i.e., deepstep) version of the algorithm.The author is indebted to J. Bisschop, P. C. J. M. Geven, J. H. Van Lint, J. Ponstein, and J. P. Vial for their remarks on an earlier version of this paper.     

与0-1二次规划问题等价的连续问题(英)

在较一般的条件下，证明了线性约束0-1二次规划问题等价于一个凹二次规划问题，改进了已有的结果．     

A Tabu Search Algorithm for Quadratic 0—1 Programming Problem

1.IntroductionThemathematicalmodelofaquaduatico-1programmingproblemisasfollows:MinimizesubjecttwhereI,AsfaraspaperL1'2Jcanseemedel(I)(fordu=O)isveryimPOrtantinthemarshallingofsinglegrouptrainbetweenmarshallingstationsinrailwaynetworkandthemarshallingoftraininnetw0rkwiththetw0types0fvehiclefl0w,butproblem(I)isNP-C.C0nsiderarelax-ationproblemasf0llows:MinimizeIngeneral,solvingrelaxati0nproblemiseasierthansolvingcombinatiorial0ptimalpr0b-lem,thesameaslinearpr0grammingproblemissolvableinPOly…     

Block-diagonal semidefinite programming hierarchies for 0/1 programming

Lovász and Schrijver, and later Lasserre, proposed hierarchies of semidefinite programming relaxations for 0/1 linear programming problems. We revisit these two constructions and propose two new, block-diagonal hierarchies, which are at least as strong as the Lovász-Schrijver hierarchy, but less costly to compute. We report experimental results for the stable set problem of Paley graphs.     

Decomposition and Linearization for 0-1 Quadratic Programming

This paper presents a general decomposition method to compute bounds for constrained 0-1 quadratic programming. The best decomposition is found by using a Lagrangian decomposition of the problem. Moreover, in its simplest version this method is proved to give at least the bound obtained by the LP-relaxation of a non-trivial linearization. To illustrate this point, some computational results are given for the 0-1 quadratic knapsack problem.     

Untersuchungen zu speziellen linearen gemischt-ganzzahligen 0-1-optimierungsaufgaben

In this paper criteria for solution of a linear mixed-integer 0-1-programming -problem with one restriction and any coefficients in the objective function respectively in the restriction are given. Under certain conditions the given problem can be reduced to a mixed-integer problem of a special structure. An algorithm to solve this problem is described concisely.     

A linearization framework for unconstrained quadratic (0-1) problems

In this paper, we are interested in linearization techniques for the exact solution of the Unconstrained Quadratic (0-1) Problem. Our purpose is to propose “economical” linear formulations. We first extend current techniques in a general linearization framework containing many other schemes and propose a new linear formulation. Numerical results comparing classical, Glover’s and the new linearization are reported.     

编组站调度计划0-1规划法

针对货车编组问题,采用半分离式两阶段0-1线性规划模型对各阶段联合求解,对局部最优解采用调度时序图可视化表述.首先,对无、有调车辆分离,无调车采用启发式安排.有调车推峰顺序可以转化为零件加工问题,以驼峰总工作量最大、等待时间最小为目标建立模型I.列车解体时间与解体方向数成正比增长,但在未确定具体解体方案时无法确定(即模型I的独立),通过在模型II中对解体时间模糊化来处理两步独立的缺陷,从而达到两步规划的连续特性.车辆新编,决策变量属于多维结构,通过将多维稀疏变量转化为一维序列,有效解除其稀疏特性,形成二维决策变量建立规划模型II直接求解.其次,通过仿真创建模拟数据,运用主模型求解,得到了驼峰是编组站主要瓶颈的结论.最后,我们还对铁路资源的紧缺性、编组效率建模给出了较详细改进措施.     

Unconstrained 0–1 nonlinear programming: A nondifferentiable approach

The purpose of this paper is to give new formulations for the unconstrained 0–1 nonlinear problem. The unconstrained 0–1 nonlinear problem is reduced to nonlinear continuous problems where the objective functions are piecewise linear. In the first formulation, the objective function is a difference of two convex functions while the other formulations lead to concave problems. It is shown that the concave problems we obtain have fewer integer local minima than has the classical concave formulation of the 0–1 unconstrained 0–1 nonlinear problem.