Alternative formulations of mixed integer programs

We study the formation of mixed-integer programming (MIP) constraints through the development of constructions which syntactically parallel the set operations of union, intersection, Cartesian product, and linear affine transformation. In this manner, we are able to both modularize the work of constructing representations (as the representations for base sets of a composite construction need not be disjunctively derived) and to make connections to certain of the logic-based approaches to artificial intelligence which utilize the intersection (and) and union (or) operations. We provide results which allow one to calculate the linear relaxation of a composite construction, in terms of set operations on the relaxation of the base sets. We are also able to compare the size of the relaxations for different formulations of the same MIP set, when these different formulations arise from one another through distributive laws. Utilizing these results, we generalize the Davis-Putnam algorithm of propositional logic to an MIP form, and answer a question regarding the relative efficiency of two versions of this algorithm. In this context, the subroutines of the logic algorithm correspond to list processing subroutines for MIP to be used prior to running linear programs. They are similar in nature to preprocessing routines, wherein entire MIP constraint sets are manipulated as formal symbols of logic.This work has been partially supported by National Science Foundation Grant MCS-8304075.     

A finitely converging cutting plane technique

We consider a new finitely convergent cutting plane algorithm for mixed integer linear programs in which the optimal objective value is assumed to be integral. The primary 'theoretical' contribution is the simplicity of the proof of convergence.     

Generalized Nonlinear Lagrangian Formulation for Bounded Integer Programming

Several nonlinear Lagrangian formulations have been recently proposed for bounded integer programming problems. While possessing an asymptotic strong duality property, these formulations offer a success guarantee for the identification of an optimal primal solution via a dual search. Investigating common features of nonlinear Lagrangian formulations in constructing a nonlinear support for nonconvex piecewise constant perturbation function, this paper proposes a generalized nonlinear Lagrangian formulation of which many existing nonlinear Lagrangian formulations become special cases.     

Primal cutting plane algorithms revisited

Dual fractional cutting plane algorithms, in which cutting planes are used to iteratively tighten a linear relaxation of an integer program, are well-known and form the basis of the highly successful branch-and-cut method. It is rather less well-known that various primal cutting plane algorithms were developed in the 1960s, for example by Young. In a primal algorithm, the main role of the cutting planes is to enable a feasible solution to the original problem to be improved. Research on these algorithms has been almost non-existent.  In this paper we argue for a re-examination of these primal methods. We describe a new primal algorithm for pure 0-1 problems based on strong valid inequalities and give some encouraging computational results. Possible extensions to the case of general mixed-integer programs are also discussed.     

Generalized fractional programming and cutting plane algorithms

In this paper, we introduce a variant of a cutting plane algorithm and show that this algorithm reduces to the well-known Dinkelbach-type procedure of Crouzeix, Ferland, and Schaible if the optimization problem is a generalized fractional program. By this observation, an easy geometrical interpretation of one of the most important algorithms in generalized fractional programming is obtained. Moreover, it is shown that the convergence of the Dinkelbach-type procedure is a direct consequence of the properties of this cutting plane method. Finally, a class of generalized fractional programs is considered where the standard positivity assumption on the denominators of the ratios of the objective function has to be imposed explicitly. It is also shown that, when using a Dinkelbach-type approach for this class of programs, the constraints ensuring the positivity on the denominators can be dropped.The authors like to thank the anonymous referees and Frank Plastria for their constructive remarks on an earlier version of this paper.This research was carried out at Erasmus University, Rotterdam, The Netherlands and was supported by JNICT, Lisboa, Portugal, under Contract BD/707/90-RM.     

Generalized Convex Disjunctive Programming: Nonlinear Convex Hull Relaxation

Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative to mixed-integer programming for representing discrete/continuous optimization problems. The basic idea of GDP consists of representing these problems in terms of sets of disjunctions in the continuous space, and logic propositions in terms of Boolean variables. In this paper we consider GDP problems involving convex nonlinear inequalities in the disjunctions. Based on the work by Stubbs and Mehrotra [21] and Ceria and Soares [6], we propose a convex nonlinear relaxation of the nonlinear convex GDP problem that relies on the convex hull of each of the disjunctions that is obtained by variable disaggregation and reformulation of the inequalities. The proposed nonlinear relaxation is used to formulate the GDP problem as a Mixed-Integer Nonlinear Programming (MINLP) problem that is shown to be tighter than the conventional big-M formulation. A disjunctive branch and bound method is also presented, and numerical results are given for a set of test problems.     

New upper bounds for the two-dimensional orthogonal non-guillotine cutting stock problem

The two-dimensional orthogonal non-guillotine cutting stockproblem (NGCP) appears in many industries (e.g. the wood andsteel industries) and consists of cutting a rectangular mastersurface into a number of rectangular pieces, each with a givensize and value. The pieces must be cut with their edges alwaysparallel to the edges of the master surface (orthogonal cuts).The objective is to maximize the total value of the pieces cut. New upper bounds on the optimal solution to the NGCP are described.The new bounding procedures are obtained by different relaxationsof a new mathematical formulation of the NGCP. Various proceduresfor strengthening the resulting upper bounds and reducing thesize of the original problem are discussed. The proposed newupper bounds have been experimentally evaluated on test problemsderived from the literature. Comparisons with previous boundingprocedures from the literature are given. The computationalresults indicate that these bounds are significantly betterthan the bounds proposed in the literature.     

Propositional distances and compact preference representation

Distances between possible worlds play an important role in logic-based knowledge representation (especially in belief change, reasoning about action, belief merging and similarity-based reasoning). We show here how they can be used for representing in a compact and intuitive way the preference profile of an agent, following the principle that given a goal G, then the closer a world w to a model of G, the better w. We give an integrated logical framework for preference representation which handles weighted goals and distances to goals in a uniform way. Then we argue that the widely used Hamming distance (which merely counts the number of propositional symbols assigned a different value by two worlds) is generally too rudimentary and too syntax-sensitive to be suitable in real applications; therefore, we propose a new family of distances, based on Choquet integrals, in which the Hamming distance has a position very similar to that of the arithmetic mean in the class of Choquet integrals.     

Induction and Skolemization in saturation theorem proving

We consider a typical integration of induction in saturation-based theorem provers and investigate the effects of Skolem symbols occurring in the induction formulas. In a practically relevant setting we establish a Skolem-free characterization of refutation in saturation-based proof systems with induction. Finally, we use this characterization to obtain unprovability results for a concrete saturation-based induction prover.     

Boole代数上的度量结构及其在命题逻辑中的应用

设B是一个Boole代数, Ω是从B到Boole代数{0,1}的全体同态之集,μ是Ω上的概率测度.本文基于μ在B中引入了元素的尺寸概念以及元素对之间的相似度概念,并由此在B上建立了度量结构.作为应用,本文改进了新近提出的命题逻辑中的近似推理理论.     

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.      

Soft Constraint Logic Programming and Generalized Shortest Path Problems

In this paper we study the relationship between Constraint Programming (CP) and Shortest Path (SP) problems. In particular, we show that classical, multicriteria, partially ordered, and modality-based SP problems can be naturally modeled and solved within the Soft Constraint Logic Programming (SCLP) framework, where logic programming is coupled with soft constraints. In this way we provide this large class of SP problems with a high-level and declarative linguistic support whose semantics takes care of both finding the cost of the shortest path(s) and also of actually finding the path(s). On the other hand, some efficient algorithms for certain classes of SP problems can be exploited to provide some classes of SCLP programs with an efficient way to compute their semantics.     

Cutting planes for integer programs with general integer variables

We investigate the use of cutting planes for integer programs with general integer variables. We show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0–1 variables. We also explore the use of Gomory's mixed-integer cuts. We address both theoretical and computational issues and show how to embed these cutting planes in a branch-and-bound framework. We compare results obtained by using our cut generation routines in two existing systems with a commercially available branch-and-bound code on a range of test problems arising from practical applications. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.This research was partly performed when the author was affiliated with CORE, Université Catholique de Louvain.     

A cutting plane procedure for the travelling salesman problem on road networks

A cutting plane algorithm for the exact solution of the symmetric travelling salesman problem (TSP) is proposed. The real tours on a usually incomplete road network, which are in general non-Hamiltonian, are characterized directly by an integer linear programming model. The algorithm generates special cutting planes for this model. Computational results for real road networks with up to 292 visiting places are reported, as well as for classical problems of the literature with up to 120 cities. Some of the latter problems have been solved for the first time with a pure cutting plane approach.     

Horn clause computability

It is proved that a Turing computable functionf is computable in binary Horn clauses, which are a subset of first order logic. Moreover, it is proved that the binary Horn clauses do not need more than one function symbol. The proofs comprise computable relations that can be run efficiently as logic programs on a computer.     

基于可信性理论的模糊广义指派问题研究

鉴于广义指派问题的参数确定上通常包含不确定性,因此,将模型的主要参数,即单位费用、资源消耗量,用梯形模糊变量来刻画,从而建立模糊广义指派模型.在模型求解过程中,结合到决策者的实际要求,利用可信性理论将目标函数和约束条件进行清晰化处理,进而通过参数分解法求解.最后,通过数值例子说明模糊广义指派问题的应用,并检验所提方法的有效性.     

Generalized metarationalities in the graph model for conflict resolution

A metarational tree is defined within the graph model for conflict resolution paradigm, providing a general framework within which rational behavior in models with two decision makers (DMs) can be described more comprehensively. A new definition of stability for a DM that depends on the total number, h, of moves and counter-moves allowed is proposed. Moreover, the metarational tree can be refined so that all moves must be unilateral improvements, resulting in a new set of stability definitions for each level of the tree. Relationships among stabilities at various levels of the basic and refined trees are explored, and connections are established to existing stability definitions including Nash stability, general metarationality, symmetric metarationality, sequential and limited-move stability, and policy equilibria.     

A view of automated proof checking and proving

Different techniques of automated formal reasoning are described and their performance and requirements on the human user are evaluated. The main trade-off is between autonomy and flexibility in conducting proofs. Examples of the use of techniques and existing systems are given, but not attempt of an exhaustive overview is made. The goal is to provide the reader with an idea of what to look for when selecting an approach for his/her application.     

整数规划的广义填充函数算法

文[9,10]设计了直接求整数规划问题近似解的填充函数算法,但其所利用的文[2,3]的填充函数均带有参数,需要在算法过程中逐步调节。本文建立整数规划的广义填充函数的定义,说明了文[9,10]所利用的填充函数是整数规划问题的广义填充函数,并构造了一类不带参数的广义填充函数。进而本文设计了整数规划的一类不带参数的广义填充函数算法,数值试验表明算法是有效的。     

A continuous approach to inductive inference

In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean function:{0, 1} ⁿ {0, 1} using outputs obtained by applying a limited number of random inputs to the hidden function. Given this input—output sample, we give a method to synthesize a Boolean function that describes the sample. We pose the Boolean Function Synthesis Problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. A similar integer programming implementation has been used in a previous study to solve randomly generated instances of the Satisfiability Problem. In this paper we introduce a new variant of this algorithm, where the Riemannian metric used for defining the search region is dynamically modified. Computational results on 8-, 16- and 32-input, 1-output functions are presented. Our implementation successfully identified the majority of hidden functions in the experiment.