Planning wireless networks by shortest path

Transmitters and receivers are the basic elements of wireless networks and are characterized by a number of radio-electrical parameters. The generic planning problem consists of establishing suitable values for these parameters so as to optimize some network performance indicator. The version here addressed, namely the Power Assignment Problem (pap), is the problem of assigning transmission powers to the transmitters of a wireless network so as to maximize the satisfied demand. This problem has relevant practical applications both in radio-broadcasting and in mobile telephony. Typical solution approaches make use of mixed integer linear programs with huge coefficients in the constraint matrix yielding numerical inaccuracy and poor bounds, and so cannot be exploited to solve large instances of practical interest. In order to overcome these inconveniences, we developed a two-phase heuristic to solve large instances of pap, namely a constructive heuristic followed by an improving local search. Both phases are based on successive shortest path computations on suitable directed graphs. Computational tests on a number of instances arising in the design of the national Italian Digital Video Broadcasting (DVB) network are presented.     

A method for solving the general parametric linear complementarity problem

This paper presents a solution method for the general (mixed integer) parametric linear complementarity problem pLCP(q(θ),M), where the matrix M has a general structure and integrality restriction can be enforced on the solution. Based on the equivalence between the linear complementarity problem and mixed integer feasibility problem, we propose a mixed integer programming formulation with an objective of finding the minimum 1-norm solution for the original linear complementarity problem. The parametric linear complementarity problem is then formulated as multiparametric mixed integer programming problem, which is solved using a multiparametric programming algorithm. The proposed method is illustrated through a number of examples.     

Solving large single allocation p-hub problems with two or three hubs

In this paper we present an efficient approach for solving single allocation p-hub problems with two or three hubs. Two different variants of the problem are considered: the uncapacitated single allocation p-hub median problem and the p-hub allocation problem. We solve these problems using new mixed integer linear programming formulations that require fewer variables than those formerly used in the literature. The problems that we solve here are the largest single allocation problems ever solved. The numerical results presented here will demonstrate the superior performance of our mixed integer linear programs over traditional approaches for large problems. Finally we present the first mixed integer linear program for solving single allocation hub location problems that requires only O(n²) variables and O(n²) constraints that is valid for any number of hubs.     

Algorithm engineering for optimal alignment of protein structure distance matrices

Protein structural alignment is an important problem in computational biology. In this paper, we present first successes on provably optimal pairwise alignment of protein inter-residue distance matrices, using the popular dali scoring function. We introduce the structural alignment problem formally, which enables us to express a variety of scoring functions used in previous work as special cases in a unified framework. Further, we propose the first mathematical model for computing optimal structural alignments based on dense inter-residue distance matrices. We therefore reformulate the problem as a special graph problem and give a tight integer linear programming model. We then present algorithm engineering techniques to handle the huge integer linear programs of real-life distance matrix alignment problems. Applying these techniques, we can compute provably optimal dali alignments for the very first time.     

Solving integer programs over monotone inequalities in three variables: A framework for half integrality and good approximations

We define a class of monotone integer programs with constraints that involve up to three variables each. A generic constraint in such integer program is of the form ax−by⩽z+c, where a and b are nonnegative and the variable z appears only in that constraint. We devise an algorithm solving such problems in time polynomial in the length of the input and the range of variables U. The solution is obtained from a minimum cut on a graph with O(nU) nodes and O(mU) arcs where n is the number of variables of the types x and y and m is the number of constraints. Our algorithm is also valid for nonlinear objective functions.Nonmonotone integer programs are optimization problems with constraints of the type ax+by⩽z+c without restriction on the signs of a and b. Such problems are in general NP-hard. We devise here an algorithm, relying on a transformation to the monotone case, that delivers half integral superoptimal solutions in polynomial time. Such solutions provide bounds on the optimum value that can only be superior to bounds provided by linear programming relaxation. When the half integral solution can be rounded to an integer feasible solution, this is a 2-approximate solution. In that the technique is a unified 2-approximation technique for a large class of problems. The results apply also for general integer programming problems with worse approximation factors that depend on a quantifier measuring how far the problem is from the class of problems we describe.The algorithm described here has a wide array of problem applications. An additional important consequence of our results is that nonmonotone problems in the framework are MAX SNP-hard and at least as hard to approximate as vertex cover.Problems that are amenable to the analysis provided here are easily recognized. The analysis itself is entirely technical and involves manipulating the constraints and transforming them to a totally unimodular system while losing no more than a factor of 2 in the integrality.     

Benders's method revisited

The master problem in Benders's partitioning method is an integer program with a very large number of constraints, each of which is usually generated by solving the integer program with the constraints generated earlier. Computational experience shows that the subset B of those constraints of the master problem that are satisfied with equality at the linear programming optimum often play a crucial role in determining the integer optimum, in the sense that only a few of the remaining inequalities are needed. We characterize this subset B of inequalities. If an optimal basic solution to the linear program is nondegenerate in the continuous variables and has p integer-constrained basic variables, then the corresponding set B contains at most 2^p inequalities, none of which is implied by the others. We give an efficient procedure for generating an appropriate subset of the inequalities in B, which leads to a considerably improved version of Benders's method.     

Pruning by isomorphism in branch-and-cut

 The paper presents a branch-and-cut for solving (0, 1) integer linear programs having a large symmetry group. The group is used for pruning the enumeration tree and for generating cuts. The cuts are non-standard, cutting integer feasible solutions but leaving the optimal value of the problem unchanged. Pruning and cut generation are performed by backtracking procedures using a Schreier-Sims table for representing the group. Applications to hard set covering problems and to the generation of covering designs and error correcting codes are presented. Received: August 2001 / Accepted: October 2002 Publication online: December 9, 2002 Key Words. branch-and-cut – isomorphism pruning – symmetry     

Global solution of semi-infinite programs

Optimization problems involving a finite number of decision variables and an infinite number of constraints are referred to as semi-infinite programs (SIPs). Existing numerical methods for solving nonlinear SIPs make strong assumptions on the properties of the feasible set, e.g., convexity and/or regularity, or solve a discretized approximation which only guarantees a lower bound to the true solution value of the SIP. Here, a general, deterministic algorithm for solving semi-infinite programs to guaranteed global optimality is presented. A branch-and-bound (B&B) framework is used to generate convergent sequences of upper and lower bounds on the SIP solution value. The upper-bounding problem is generated by replacing the infinite number of real-valued constraints with a finite number of constrained inclusion bounds; the lower-bounding problem is formulated as a convex relaxation of a discretized approximation to the SIP. The SIP B&B algorithm is shown to converge finitely to –optimality when the subdivision and discretization procedures used to formulate the node subproblems are known to retain certain convergence characteristics. Other than the properties assumed by globally-convergent B&B methods (for finitely-constrained NLPs), this SIP algorithm makes only one additional assumption: For every minimizer x* of the SIP there exists a sequence of Slater points x_n for which (cf. Section 5.4). Numerical results for test problems in the SIP literature are presented. The exclusion test and a modified upper-bounding problem are also investigated as heuristic approaches for alleviating the computational cost of solving a nonlinear SIP to guaranteed global optimality.     

Minimizing the natural fuel requirement for nuclear reactor power systems: A nonstandard optimal control problem

A class of model problems in nuclear reactor economics is defined and shown to be equivalent to a linear optimal control problem to which present versions of the maximum principle apparently cannot be applied. It is shown that the search for an optimal control can be restricted tocontrols of maximum fuel utilization (Comfu), and that theComfu's are in a one-to-one correspondence with the functions which satisfy certain inequalities and are solutions of a nonlinear Volterra integral equation containing the value of the cost functional as a parameter. In the general case, one can establish an iterative procedure, involving solution of the integral equation at each iteration, for approximating the optimalComfu. For some important special cases, a point on the solution corresponding to the optimalComfu is knowna priori, and thus the optimalComfu can be obtained by solving the integral equation only once. Some possible generalizations of the original economic model are also discussed.This research was sponsored by the US Atomic Energy Commission under contract with the Union Carbide Corporation.     

A reformulation framework for global optimization

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called αBB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.     

Moment information and entropy evaluation for probability densities

How much information does the sequence of integer moments carry about the corresponding unknown absolutely continuous distribution? We prove that a reliable evaluation of the corresponding Shannon entropy can be done by exploiting some known theoretical results on the entropy convergence, uniquely involving exact moments without solving the underlying moment problem. All the procedure essentially rests on the solution of linear systems, with nearly singular matrices, and hence it requires both calculations in high precision and a pre-conditioning technique. Numerical examples are provided to support the theoretical results.     

A o(n logn) algorithm for LP knapsacks with GUB constraints

A specialization of the dual simplex method is developed for solving the linear programming (LP) knapsack problem subject to generalized upper bound (GUB) constraints. The LP/GUB knapsack problem is of interest both for solving more general LP problems by the dual simplex method, and for applying surrogate constraint strategies to the solution of 0–1 Multiple Choice integer programming problems. We provide computational bounds for our method of o(n logn), wheren is the total number of problem variables. These bounds reduce the previous best estimate of the order of complexity of the LP/GUB knapsack problem (due to Witzgall) and provide connections to computational bounds for the ordinary knapsack problem.We further provide a variant of our method which has only slightly inferior worst case bounds, yet which is ideally suited to solving integer multiple choice problems due to its ability to post-optimize while retaining variables otherwise weeded out by convex dominance criteria.     

The efficient solution of large-scale linear programming problems—some algorithmic techniques and computational results

First, this paper presents the results of experiments with algorithmic techniques for efficiently solving medium and large scale linear and mixed integer programming problems. The techniques presented here are either original or recent.The solution of a great number of problems has shown that efficient problem solving requires automatic adaptation of algorithmic techniques upon problem characteristics. We show when a given technique should be used for a particular problem.The last part of this paper describes an attempt to provide a powerful mathematical programming language, allowing an easy programming of specific studies on medium-size models such as the recursive use of LP or the build-up of algorithms based on the simplex method.All these features have been implemented in the IBM Mathematical Programming System, MPSX/370, and its feature MIP/370. Extensive numerical results and comparisons on real-life problems are provided and commented upon.Presented at the IXth International Symposium on Mathematical Programming in Budapest (1976).     

A column generation based decomposition algorithm for a parallel machine just-in-time scheduling problem

We propose a column generation based exact decomposition algorithm for the problem of scheduling n jobs with an unrestrictively large common due date on m identical parallel machines to minimize total weighted earliness and tardiness. We first formulate the problem as an integer program, then reformulate it, using Dantzig–Wolfe decomposition, as a set partitioning problem with side constraints. Based on this set partitioning formulation, a branch and bound exact solution algorithm is developed for the problem. In the branch and bound tree, each node is the linear relaxation problem of a set partitioning problem with side constraints. This linear relaxation problem is solved by column generation approach where columns represent partial schedules on single machines and are generated by solving two single machine subproblems. Our computational results show that this decomposition algorithm is capable of solving problems with up to 60 jobs in reasonable cpu time.     

Algebraic generation of minimum size orthogonal fractional factorial designs: an approach based on integer linear programming

Generation of orthogonal fractional factorial designs (OFFDs) is an important and extensively studied subject in applied statistics. In this paper we show how searching for an OFFD that satisfies a set of constraints, expressed in terms of orthogonality between simple and interaction effects, is, in many applications, equivalent to solving an integer linear programming problem. We use a recent methodology, based on polynomial counting functions and strata, that represents OFFDs as the positive integer solutions of a system of linear equations. We use this system to set up an optimization problem where the cost function to be minimized is the size of the OFFD and the constraints are represented by the system itself. Finally we search for a solution using standard integer programming techniques. Some applications are also presented in the computational results section. It is worth noting that the methodology does not put any restriction either on the number of levels of each factor or on the orthogonality constraints and so it can be applied to a very wide range of designs, including mixed orthogonal arrays.     

Parametric Integer Programming Analysis: A Contraction Approach

An algorithm is presented for solving families of integer linear programming problems in which the problems are "related" by having identical objective coefficients and constraint matrix coefficients. The righthand-side constants have the form b + θd where b and d are conformable vectors and θ varies from zero to one.The approach consists primarily of solving the most relaxed problem (θ = 1) using cutting planes and then contracting the region of feasible integer solutions in such a manner that the current optimal integer solution is eliminated.The algorithm was applied to 1800 integer linear programming problems with reasonable success. Integer programming problems which have proved to be unsolvable using cutting planes have been solved by expanding the region of feasible integer solutions (θ = 1) and then contracting to the original region.     

Decomposition strategy for the stochastic pooling problem

The stochastic pooling problem is a type of stochastic mixed-integer bilinear program arising in the integrated design and operation of various important industrial networks, such as gasoline blending, natural gas production and transportation, water treatment, etc. This paper presents a rigorous decomposition method for the stochastic pooling problem, which guarantees finding an ${\epsilon}$ -optimal solution with a finite number of iterations. By convexification of the bilinear terms, the stochastic pooling problem is relaxed into a lower bounding problem that is a potentially large-scale mixed-integer linear program (MILP). Solution of this lower bounding problem is then decomposed into a sequence of relaxed master problems, which are MILPs with much smaller sizes, and primal bounding problems, which are linear programs. The solutions of the relaxed master problems yield a sequence of nondecreasing lower bounds on the optimal objective value, and they also generate a sequence of integer realizations defining the primal problems which yield a sequence of nonincreasing upper bounds on the optimal objective value. The decomposition algorithm terminates finitely when the lower and upper bounds coincide (or are close enough), or infeasibility of the problem is indicated. Case studies involving two example problems and two industrial problems demonstrate the dramatic computational advantage of the proposed decomposition method over both a state-of-the-art branch-and-reduce global optimization method and explicit enumeration of integer realizations, particularly for large-scale problems.     

Haplotyping populations by pure parsimony based on compatible genotypes and greedy heuristics

The population haplotype inference problem based on the pure parsimony criterion (HIPP) infers an m × n genotype matrix for a population by a 2m × n haplotype matrix with the minimum number of distinct haplotypes. Previous integer programming based HIPP solution methods are time-consuming, and their practical effectiveness remains unevaluated. On the other hand, previous heuristic HIPP algorithms are efficient, but their theoretical effectiveness in terms of optimality gaps has not been evaluated, either. We propose two new heuristic HIPP algorithms (MGP and GHI) and conduct more complete computational experiments. In particular, MGP exploits the compatible relations among genotypes to solve a reduced integer linear programming problem so that a solution of good quality can be obtained very quickly; GHI exploits a weight mechanism to selects better candidate haplotypes in a greedy fashion. The computational results show that our proposed algorithms are efficient and effective, especially for solving cases with larger recombination rates.     

Solving Large Airline Crew Scheduling Problems: Random Pairing Generation and Strong Branching

The airline crew scheduling problem is the problem of assigning crew itineraries to flights. We develop a new approach for solving the problem that is based on enumerating hundreds of millions random pairings. The linear programming relaxation is solved first and then millions of columns with best reduced cost are selected for the integer program. The number of columns is further reduced by a linear programming based heuristic. Finally an integer solution is obtained with a commercial integer programming solver. The branching rule of the solver is enhanced with a combination of strong branching and a specialized branching rule. The algorithm produces solutions that are significantly better than ones found by current practice.