Introduction: New approaches to linear programming

This issue ofAlgorithmica present papers on various aspects of nonlinear methods for solving linear programming problems, inspired by the work of Karmarkar. This introduction describes some of these aspects and briefly mentions other recent developments in the field. A bibliography of recent articles is included.     

New integer linear programming approaches for course timetabling

The most complete form of academic timetabling problem is the population and course timetabling problem. In this problem, there may be multiple classes of each subject, and the decision on which students are to constitute each class is made in concert with the decision on the timetable for each class. In order to solve this problem, it is normally simplified or decomposed in some fashion. One simplification commonly used in practice is known as blocking: it is assumed that the classes can be partitioned into sets of classes (or blocks) that will be timetabled in parallel. This restricts clashing to occur only between classes in the same block, and essentially removes the timetabling aspect of the problem, which can be carried out once the blocks are constituted and the classes populated. The problem of constituting the blocks and populating the classes, known as the course blocking and population problem, is nevertheless a challenging problem, and provides the focus of this paper. We demonstrate, using data provided by a local high school, that integer linear programming approaches can solve the problem in a matter of seconds. Key features include remodelling to remove symmetry caused by students with identical subject selection, and the observation that in practice, only integrality of the block composition variables needs to be enforced; the class population aspects of the model have strong integrality properties.     

An introduction to nonlinear programming—II: The linear programming problem

Linear programming problems represent the most thoroughly analyzed and widely solved class of parameter optimization problems. In Part II, we shall restrict our attention to this general class of problems. The characteristics of the admissible region are investigated and established. The Kuhn-Tucker conditions developed in Part I are applied to establish necessary and sufficient conditions that must be satisfied at a minimum. Included in the discussion is a consideration of dual linear programming problems. Then, we direct our attention to the question of determining the solution of specific problems. A general algorithm known as the Simplex Method is described and applied to several examples.     

An experimental comparison of the new goal programming and the linear programming approaches in the two-group discriminant problems

The aim of this article is to consider a new linear programming and two goal programming models for two-group classification problems. When these approaches are applied to the data of real life or of simulation, our proposed new models perform well both in separating the groups and the group–membership predictions of new objects. In discriminant analysis some linear programming models determine the attribute weights and the cut-off value in two steps, but our models determine simultaneously all of these values in one step. Moreover, the results of simulation experiments show that our proposed models outperform significantly than existing linear programming and statistical approaches in attaining higher average hit-ratios.     

Bicriterion linear programming

Basically leaning on the concept of “best” compromise, the technique seeks the optimal solution by fair relaxations of the objectives commensurate with their degrees of importance until the optimum feasible compromise is reached. The concept is made operational by deriving a linear constraint (referred to as the compromise constraint) to be added into the original set. An offspring of both objective functions, the compromise constraint cuts the original feasible region and forces both objectives to settle on a common point along this added restriction. The resulting singular equivalent of the bicriterion problem optimizes any one of the two objective functions and their equivalent sum (referred to as the supergoal) subject to the new set of constraints. Post optimality analysis is employed to minimize the computational effort usually done by a computer. A very attractive feature of this new practical technique is its ability to search for the optimum in any point in the feasible region, even other than the vertices of the convex set.     

A linear programming approach to max-sum problem: a review

The max-sum labeling problem, defined as maximizing a sum of binary (i.e., pairwise) functions of discrete variables, is a general NP-hard optimization problem with many applications, such as computing the MAP configuration of a Markov random field. We review a not widely known approach to the problem, developed by Ukrainian researchers Schlesinger et al. in 1976, and show how it contributes to recent results, most importantly, those on the convex combination of trees and tree-reweighted max-product. In particular, we review Schlesinger et al.'s upper bound on the max-sum criterion, its minimization by equivalent transformations, its relation to the constraint satisfaction problem, the fact that this minimization is dual to a linear programming relaxation of the original problem, and the three kinds of consistency necessary for optimality of the upper bound. We revisit problems with Boolean variables and supermodular problems. We describe two algorithms for decreasing the upper bound. We present an example application for structural image analysis.     

Rule-based intelligence to support linear programming analysis

Increased size and complexity of linear programs make it difficult to understand results and manage the model. This difficulty is overcome, at least partially, with new advances in computing environments and techniques from artificial intelligence and operations research. A software system, called ANALYZE, is presented within the larger context of developing an artificially intelligent environment for mathematical programming modeling and analysis. With heuristics governed by rule-based reasoning and with syntax-driven translations into English, the ANALYZE system enables a form of intelligence to support analysis. Particular problems that illustrate this capability are explanations of dual prices, diagnoses of infeasibilities, and reasoning about redundancy.     

Multi-media instructional approach to teaching linear programming

This paper describes different strategies employed in converting a lecture-oriented mathematical programming course to a Personalized Self-Paced Instructional (PSI) format. This is an elective course for students in science, engineering and management. A multi media instructional approach is used in the PSI system which combines traditional lectures, self-paced and individualized learning assisted by interactive computer programs and video taped instructional materials. This unique PSI system for mathematical programming provides maximum learning opportunity and flexibility to students. The author's experiences with the PSI system and the students' evaluation of the self-paced system are also discussed.     

A sequential linear programming approach to solve mixed integer programming problems

The optimal operation of pumps in a large water supply system under time-of-use electricity rates is formulated as a mixed integer programming (MIP) problem. The problem is solved using an iterative linear programming (LP) scheme. The scheme is applied to an actual world problem, the City of Inglewood Water Supply System. Computational results are presented and termination criteria for the solution scheme are discussed.     

Investment planning for the development of a national resource —linear programming based approaches

This paper will review and generalize the work of Gearing, Swart and Var dealing with the development of a mathematical model to aid the government of Turkey in determining the “best” allocation of the capital budget for tourism among a large group of competing proposals. One of the more unique aspects of this work was the development of a measure of benefit for particular allocation plans which allowed for subjective information to be an integral part of the investment planning model.The computational procedures developed heretofore to derive investment strategies from the model have been based on dynamic programming, integer programming or combined dynamic programming-integer programming approaches. This paper will show how various methods based on linear programming yield satisfactory answers to many questions regarding development policies. The primary advantage of these linear programming methods is that they can be implemented with standard linear programming computer packages which are generally available, and hence eliminate the high cost of developing special purpose computer codes.     

Spectral approach to linear programming bounds on codes

We give new proofs of asymptotic upper bounds of coding theory obtained within the frame of Delsarte’s linear programming method. The proofs rely on the analysis of eigenvectors of some finite-dimensional operators related to orthogonal polynomials. Examples of the method considered in the paper include binary codes, binary constant-weight codes, spherical codes, and codes in projective spaces.     

A linear programming approach to reasoning about probabilities

We continue our study, initiated in [9], of the following computational problem proposed by Nilsson: Several clauses (Boolean functions of several variables) are given, and for each clause the probability that the clause is true is specified. We are asked whether these probabilities are consistent. They are if there is a probability distribution on the truth assignments such that the probability of each clause is the measure of its satisfying set of assignments. Since this is a generalization of the satisfiability problem of predicate calculus, it is immediately NP-hard. In [9] we showed certain restricted cases of the problem to be NP-complete, and used the Ellipsoid Algorithm to show that a certain special case is in P. In this paper we use the Simplex method, column generation techniques, and variable-depth local search to derive an effective heuristic for the general problem. Experiments show that our heuristic performs successfully on instances with many dozens of variables and clauses. We also prove several interesting complexity results that answer open questions in [9] and motivate our approach.     

A linear programming approach to constrained robust predictivecontrol

A receding horizon predictive control algorithm for systems with model uncertainty and input constraints is developed. The proposed algorithm adopts the receding horizon dual-mode (i.e., free control moves and invariant set) paradigm. The approach is novel in that it provides a convenient way of combining predictions of control moves, which are optimal in the sense of worst case performance, with large target invariant sets. Thus, the proposed algorithm has a large stabilizable set of states corresponding to a cautious state feedback law while enjoying the good performance of a tightly tuned but robust control law. Unlike earlier approaches which are based on QP or semidefinite programming, here computational complexity is reduced through the use of LP     

The progressive party problem: Integer linear programming and constraint programming compared

Many discrete optimization problems can be formulated as either integer linear programming problems or constraint satisfaction problems. Although ILP methods appear to be more powerful, sometimes constraint programming can solve these problems more quickly. This paper describes a problem in which the difference in performance between the two approaches was particularly marked, since a solution could not be found using ILP.The problem arose in the context of organizing a progressive party at a yachting rally. Some yachts were to be designated hosts; the crews of the remaining yachts would then visit the hosts for six successive half-hour periods. A guest crew could not revisit the same host, and two guest crews could not meet more than once. Additional constraints were imposed by the capacities of the host yachts and the crew sizes of the guests.Integer linear programming formulations which included all the constraints resulted in very large models, and despite trying several different strategies, all attempts to find a solution failed. Constraint programming was tried instead and solved the problem very quickly, with a little manual assistance. Reasons for the success of constraint programming in this problem are identified and discussed.     

Introduction to coordinating programming concepts in university courses

This article details an experiment which began in the academic year 1977–1978 and is still being performed, concerning the use of computers in teaching the first two year courses towards a university degree in mathematics. The experiment took place in algebra, geometry and calculus I courses for first year students, and. in 1979–1980. in the calculus II course for second year students as well.In performing this experiment we wanted to modify, as compared to the experiments we conducted in the years up to and including 1974, the teaching methods used, both qualitatively (by coordinating the activity in more courses and directly involving the professors) and quantitatively (by going from courses for only a few students to classes of approx. 150 students working toward a mathematics degree).In this experiment, we used video-graphics terminals and teletype machines connected to a time-sharing system and programmable pocket calculators.