凹整数规划的分枝定界解法

凹整数规划是一类重要的非线性整数规划问题，也是在经济和管理中有着广泛应用的最优化问题．本文主要研究用分枝定界方法求解凹整数规划问题，这一方法的基本思想是对目标函数进行线性下逼近，然后用乘子搜索法求解连续松弛问题．数值结果表明，用这种分枝定界方法求解凹整数规划是有效的．     

A Constraint Programming Approach to Extract the Maximum Number of Non-Overlapping Test Forms

This paper introduces a novel approach for extracting the maximum number of non-overlapping test forms from a large collection of overlapping test sections assembled from a given item bank. The approach involves solving maximum set packing problems (MSPs). A branch-and-bound MSP algorithm is developed along with techniques adapted from constraint programming to estimate lower and upper bounds on the optimal MSP solution. The algorithm is general and can be applied in other applications including combinatorial auctions. The results of computer simulations and experiments with an operational item bank are presented. An erratum to this article is available at .     

Speedup estimates for some variants of the parallel implementations of the branch-and-bound method

The efficiency of parallel implementations of the branch-and-bound method in discrete optimization problems is considered. A theoretical analysis and comparison of two parallel implementations of this method is performed. A mathematical model of the computation process is constructed and used to obtain estimates of the maximum possible speedup. Examples of problems in which none of these two parallel implementations can speed up the computations are considered.     

Heuristic Rejection in Interval Global Optimization

Based on the investigation carried out in Ref. 1, this paper incorporates new studies about the properties of inclusion functions on subintervals while a branch-and-bound algorithm is solving global optimization problems. It is found that the relative place of the global minimum value within the inclusion function value of the objective function at the current interval indicates mostly whether the given interval is close to a minimizer point. This information is used in a heuristic interval rejection rule that can save a considerable amount of computation. Illustrative examples are discussed and an extended numerical study shows the advantages of the new approach.     

A Finite Algorithm for Global Minimization of Separable Concave Programs

Researchers first examined the problem of separable concave programming more than thirty years ago, making it one of the earliest branches of nonlinear programming to be explored. This paper proposes a new algorithm that finds the exact global minimum of this problem in a finite number of iterations. In addition to proving that our algorithm terminates finitely, the paper extends a guarantee of finiteness to all branch-and-bound algorithms for concave programming that (1) partition exhaustively using rectangular subdivisions and (2) branch on the incumbent solution when possible. The algorithm uses domain reduction techniques to accelerate convergence; it solves problems with as many as 100 nonlinear variables, 400 linear variables and 50 constraints in about five minutes on an IBM RS/6000 Power PC. An industrial application with 152 nonlinear variables, 593 linear variables, and 417 constraints is also solved in about ten minutes.     

On consistency of bounding operations in deterministic global optimization

This technical comment refers to the discussion of strong consistency of several bounding procedures in Lemma 2.1 and Proposition 2.1 of Ref. 1. A necessary clarification is given of the notion of convergence _q in Lemma 2.1, and a derivation of Proposition 2.1 is presented that includes a new and simple consistency proof of the classical bounding by convex envelopes used in many branch-and-bound procedures.     

An exact algorithm for the capacitated shortest spanning arborescence

We are given a complete and loop-free digraphG=(V, A), whereV={1,...,n} is the vertex set,A={(i, j) :i, j V} the arc set, andr V is a distinguishedroot vertex. For each arc (i, j) A, letc _ij be the associatedcost, and for each vertexi, letq _i 0 be the associateddemand (withq _r =0). Moreover, a nonnegativebranch capacity, Q, is defined.A Capacitated Shortest Spanning Arborescence rooted at r (CSSA _r ) is a minimum cost partial digraph such that: (i) each vertexj r has exactly one entering arc; (ii) for each vertexj r, a path fromr toj exists; (iii) for each branch leaving vertexr, the total demand of the vertices does not exceed the branch capacity,Q. A variant of theCSSA _r problem (calledD-CSSA _r ) arises when the out-degree of the root vertex is constrained to be equal to a given valueD. These problems are strongly NP-hard, and find practical applications in routing and network design. We describe a new Lagrangian lower bound forCSSA _r andD-CSSA _r problems, strengthened in a cutting plane fashion by iteratively adding violated constraints to the Lagrangian problem. We also present a new lower bound based on projection leading to the solution of min-cost flow problems. The two lower bounds are then combined so as to obtain an overall additive lower bounding procedure. The additive procedure is then imbedded in a branch-and-bound algorithm whose performance is enhanced by means of reduction procedures, dominance criteria, feasibility checks and upper bounding. Computational tests on asymmetric and symmetric instances from the literature, involving up to 200 vertices, are given, showing the effectiveness of the proposed approach.     

An exact algorithm for the maximum stable set problem

We describe a new branch-and-bound algorithm for the exact solution of the maximum cardinality stable set problem. The bounding phase is based on a variation of the standard greedy algorithm for finding a colouring of a graph. Two different node-fixing heuristics are also described. Computational tests on random and structured graphs and very large graphs corresponding to real-life problems show that the algorithm is competitive with the fastest algorithms known so far.This work has been supported by Agenzia Spaziale Italiana.     

Global Optimization Algorithm for the Nonlinear Sum of Ratios Problem

This article presents a branch-and-bound algorithm for globally solving the nonlinear sum of ratios problem (P). The algorithm economizes the required computations by conducting the branch-and-bound search in ^p, rather than in ⁿ, where p is the number of ratios in the objective function of problem (P) and n is the number of decision variables in problem (P). To implement the algorithm, the main computations involve solving a sequence of convex programming problems for which standard algorithms are available.     

Local branching

The availability of effective exact or heuristic solution methods for general Mixed-Integer Programs (MIPs) is of paramount importance for practical applications. In the present paper we investigate the use of a generic MIP solver as a black-box ``tactical' tool to explore effectively suitable solution subspaces defined and controlled at a ``strategic' level by a simple external branching framework. The procedure is in the spirit of well-known local search metaheuristics, but the neighborhoods are obtained through the introduction in the MIP model of completely general linear inequalities called local branching cuts. The new solution strategy is exact in nature, though it is designed to improve the heuristic behavior of the MIP solver at hand. It alternates high-level strategic branchings to define the solution neighborhoods, and low-level tactical branchings to explore them. The result is a completely general scheme aimed at favoring early updatings of the incumbent solution, hence producing high-quality solutions at early stages of the computation. The method is analyzed computationally on a large class of very difficult MIP problems by using the state-of-the-art commercial software ILOG-Cplex 7.0 as the black-box tactical MIP solver. For these instances, most of which cannot be solved to proven optimality in a reasonable time, the new method exhibits consistently an improved heuristic performance: in 23 out of 29 cases, the MIP solver produced significantly better incumbent solutions when driven by the local branching paradigm. Mathematics Subject Classification (2000):90C06, 90C10, 90C11, 90C27, 90C59