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     

A branch-and-bound algorithm for 0–1 parametric mixed integer programming

A branch-and-bound algorithm to solve 0–1 parametric mixed integer linear programming problems has been developed. The present algorithm is an extension of the branch-and-bound algorithm for parametric analysis on pure integer programming. The characteristic of the present method is that optimal solutions for all values of the parameter can be obtained.     

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.     

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.     

Lifted flow cover inequalities for mixed 0-1 integer programs

We investigate strong inequalities for mixed 0-1 integer programs derived from flow cover inequalities. Flow cover inequalities are usually not facet defining and need to be lifted to obtain stronger inequalities. However, because of the sequential nature of the standard lifting techniques and the complexity of the optimization problems that have to be solved to obtain lifting coefficients, lifting of flow cover inequalities is computationally very demanding. We present a computationally efficient way to lift flow cover inequalities based on sequence independent lifting techniques and give computational results that show the effectiveness of our lifting procedures. Received May 15, 1996 / Revised version received August 7, 1998 Published online June 28, 1999     

Marginal values in mixed integer linear programming

For a given optimization problem, P, considered as a function of the data, its marginal values are defined as the directional partial derivatives of the value of P with respect to perturbations in that data. For linear programs, formulas for the marginal values were given by Mills, [10], and further developed by the current author [16]. In this paper, the marginal value formulas are extended to the case of mixed integer linear programming (MIP). As in ordinary linear programming, discontinuities in the value can occur, and the analysis here identifies them. This latter aspect extends previous work on continuity by the current author, [18], Geoffrion and Nauss, [5], Nauss, [11], and Radke, [12], and work on the value function of Blair and Jeroslow, [2]. Application is made to model formulation and to post-optimal analysis.Supported in part by the Air Force Office of Scientific Research, Grant # AFSOR-0271 to Rutgers University.     

On a pair of nonlinear mixed integer programming problems

We consider maximin and minimax nonlinear mixed integer programming problems which are nonsymmetric in duality sense. Under weaker (pseudo-convex/pseudo-concave) assumptions, we show that the supremum infimum of the maximin problem is greater than or equal to the infimum supremum of the minimax problem. As a particular case, this result reduces to the weak duality theorem for minimax and symmetric dual nonlinear mixed integer programming problems. Further, this is used to generalize available results on minimax and symmetric duality in nonlinear mixed integer programming.     

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.     

Cutting plane algorithms for the inverse mixed integer linear programming problem

We present cutting plane algorithms for the inverse mixed integer linear programming problem (InvMILP), which is to minimally perturb the objective function of a mixed integer linear program in order to make a given feasible solution optimal.     

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.     

On the facets of the mixed–integer knapsack polyhedron

We study the mixed–integer knapsack polyhedron, that is, the convex hull of the mixed–integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet–defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable. These inequalities strengthen and/or generalize known inequalities for several special cases. We report computational results on using the inequalities as cutting planes for mixed–integer programming.Supported, in part, by NSF grants DMII–0070127 and DMII–0218265.Mathematics Subject Classification (2000): 90C10, 90C11, 90C57     

A branch-and-cut method for 0-1 mixed convex programming

We generalize the disjunctive approach of Balas, Ceria, and Cornuéjols [2] and devevlop a branch-and-cut method for solving 0-1 convex programming problems. We show that cuts can be generated by solving a single convex program. We show how to construct regions similar to those of Sherali and Adams [20] and Lovász and Schrijver [12] for the convex case. Finally, we give some preliminary computational results for our method. Received January 16, 1996 / Revised version received April 23, 1999?Published online June 28, 1999     

An efficient linearization technique for mixed 0-1 polynomial problem

This paper addresses a new and efficient linearization technique to solve mixed 0-1 polynomial problems to achieve a global optimal solution. Given a mixed 0-1 polynomial term z=c_tx₁x₂…x_ny, where x₁,x₂,…,x_n are binary (0-1) variables and y is a continuous variable. Also, c_t can be either a positive or a negative parameter. We transform z into a set of auxiliary constraints which are linear and can be solved by exact methods such as branch and bound algorithms. For this purpose, we will introduce a method in which the number of additional constraints is decreased significantly rather than the previous methods proposed in the literature. As is known in any operations research problem decreasing the number of constraints leads to decreasing the mathematical computations, extensively. Thus, research on the reducing number of constraints in mathematical problems in complicated situations have high priority for decision makers. In this method, each n-auxiliary constraints proposed in the last method in the literature for the linearization problem will be replaced by only 3 novel constraints. In other words, previous methods were dependent on the number of 0-1 variables and therefore, one auxiliary constraint was considered per 0-1 variable, but this method is completely independent of the number of 0-1 variables and this illustrates the high performance of this method in computation considerations. The analysis of this method illustrates the efficiency of the proposed algorithm.     

Checking solvability of systems of interval linear equations and inequalities via mixed integer programming

This paper deals with the problems of checking strong solvability and feasibility of linear interval equations, checking weak solvability of linear interval equations and inequalities, and finding control solutions of linear interval equations. These problems are known to be NP$\mathit{NP}$-hard. We use some recently developed characterizations in combination with classical arguments to show that these problems can be equivalently stated as optimization tasks and provide the corresponding linear mixed 0–1 programming formulations.     

A recursive procedure to generate all cuts for 0–1 mixed integer programs

We study several ways of obtaining valid inequalities for mixed integer programs. We show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and we show how the superadditive inequalities relate to Gomory's mixed integer cuts. We also show how all valid inequalities for mixed 0–1 programs can be generated recursively from a simple subclass of the disjunctive inequalities.The research of this author was supported by NSF Contract No. ECS-8540898.     

Generalized mixed integer rounding inequalities: facets for infinite group polyhedra

We present a generalization of the mixed integer rounding (MIR) approach for generating valid inequalities for (mixed) integer programming (MIP) problems. For any positive integer n, we develop n facets for a certain (n + 1)-dimensional single-constraint polyhedron in a sequential manner. We then show that for any n, the last of these facets (which we call the n-step MIR facet) can be used to generate a family of valid inequalities for the feasible set of a general (mixed) IP constraint, which we refer to as the n-step MIR inequalities. The Gomory Mixed Integer Cut and the 2-step MIR inequality of Dash and günlük  (Math Program 105(1):29–53, 2006) are the first two families corresponding to n = 1,2, respectively. The n-step MIR inequalities are easily produced using periodic functions which we refer to as the n-step MIR functions. None of these functions dominates the other on its whole period. Finally, we prove that the n-step MIR inequalities generate two-slope facets for the infinite group polyhedra, and hence are potentially strong.      

Strong formulations for mixed integer programming: A survey

We attempt to motivate and survey recent research on the use of strong valid inequalities and reformulation to solve mixed integer programming problems.     

天然气稳态运行优化的混合整数模型及其算法

天然气稳态运行优化问题的难点在于网络结构复杂、规模大、目标函数及约束高度非线性.针对其混合整数非线性规划模型,基于网络约简和线性化技术,建立了线性近似模型,并提出一种新的求解算法.将新算法用于优化我国西部天然气管网系统,结果表明所提算法是有效的.     

Pivot,Cut, and Dive: a heuristic for 0-1 mixed integer programming

This paper describes a heuristic for 0-1 mixed-integer linear programming problems, focusing on “stand-alone” implementation. Our approach is built around concave “merit functions” measuring solution integrality, and consists of four layers: gradient-based pivoting, probing pivoting, convexity/intersection cutting, and diving on blocks of variables. The concavity of the merit function plays an important role in the first and third layers, as well as in connecting the four layers. We present both the mathematical and software details of a test implementation, along with computational results for several variants.