Harvey J. Greenberg

New approaches for heuristic search: A bilateral linkage with artificial intelligence

Abstract This survey considers emerging approaches of heuristic search for solutions to combinatorially complex problems. Such problems arise in business applications, of traditional interest to operations research, such as in manufacturing operations, financial investment, capital budgeting and resource management. Artificial intelligence is a revived approach to problem-solving that requires heuristic search intrinsically in knowledge-base operations, especially for logical and analogical reasoning mechanisms. Thus, one bilateral linkage between operations research and artificial intelligence is their common interest in solving hard problems with heuristic search. That is the focus here. But longstanding methods of directed tree search with classical problem heuristics, such as for the Traveling Salesman Problem—a paradigm for combinatorially difficult problems—are not wholly satisfactory. Thus, new approaches are needed, and it is at least stimulating that some of these are inspired by natural phenomena.

Surrogate Mathematical Programming

This paper presents an approach, similar to penalty functions, for solving arbitrary mathematical programs. The surrogate mathematical program is a lesser constrained problem that, in some cases, may be solved with dynamic programming. The paper deals with the theoretical development of this surrogate approach.

Opportunities for Combinatorial Optimization in Computational Biology

This is a survey designed for mathematical programming people who do not know molecular biology and want to learn the kinds of combinatorial optimization problems that arise. After a brief introduction to the biology, we present optimization models pertaining to sequencing, evolutionary explanations, structure prediction, and recognition. Additional biology is given in the context of the problems, including some motivation for disease diagnosis and drug discovery. Open problems are cited with an extensive bibliography, and we offer a guide to getting started in this exciting frontier.

A Multiple-Objective Analysis of Sensor Placement Optimization in Water Networks

Terrorism concerns have recently led to increased interest in the potential use of sensors to detect malicious attacks on municipal water systems. A key deployment issue is identifying where the sensors should be placed in order to maximize the level of protection. Researchers have proposed several algorithms for constructing such sensor placements, each optimizing with respect to a different design objective. The use of disparate objectives raises several questions, in particular (1) What is the relationship between optimal placements obtained under different design objectives? and (2) Is there any risk in focusing on speci?c design objectives? To answer these questions, we develop mixed-integer linear programming models for the sensor placement problem over a range of design objectives. Using two real-world water systems, we show that optimal solutions with respect to one design objective are typically highly sub-optimal with respect to other design objectives. The implication is that robust algorithms for the sensor placement problem must carefully and simultaneously consider multiple, disparate design objectives.

A Review of Quasi-Convex Functions

Many theorems involving convex functions have appeared in the literature since the pioneering work of Jensen. Recently some results have been obtained for a larger class of functions: quasi-convex. This review summarizes in condensed form results known to date, providing some refinements to gain further generality. An additional objective of this review is to clarify the structure underlying quasi-convex functions by presenting analogues to properties of convex functions, and by illustrating where analogues do not exist.

Design and implementation of optimization software

History of mathematical programming systems.- Scope of mathematical programming software.- Anatomy of a mathematical programming system.- Elements of numerical linear algebra.- A tutorial on matricial packing.- Pivot selection tactics.- An interactive query system for MPS solution information.- Modeling and solving network problems.- Integer programming codes.- Some considerations in using branch-and bound codes.- Quadratic programming.- Nonlinear programming using a general mathematical programming system.- The design and implementation of software for unconstrained optimization.- The GRG method for nonlinear programming.- Generalized reduced gradient software for linearly and nonlinearly constrained problems.- The ALGOL 60 procedure minifun for solving non-linear optimization problems.- An accelerated conjugate gradient algorithm.- Global optima without convexity.- Computational aspects of geometric programming.- A proposal for the classification and documentation of test problems in the field of nonlinear programming.- Guidelines for reporting computational experiments in mathematical programming.- COAL session summary.- List of participants.

An Annotated Bibliography for Post-solution Analysis in Mixed Integer Programming and Combinatorial Optimization

This annotated bibliography focuses on what has been published since the 1977 Geoffrion-Nauss survey, and it is in BibTEX format, so it can be searched on the World Wide Web. In addition to postoptimal sensitivity analysis, this survey includes debugging a run, such as when the integer program is unbounded, anomalous or infeasible.

Robust optimization of contaminant sensor placement for community water systems

We present a series of related robust optimization models for placing sensors in municipal water networks to detect contaminants that are maliciously or accidentally injected. We formulate sensor placement problems as mixed-integer programs, for which the objective coefficients are not known with certainty. We consider a restricted absolute robustness criteria that is motivated by natural restrictions on the uncertain data, and we define three robust optimization models that differ in how the coefficients in the objective vary. Under one set of assumptions there exists a sensor placement that is optimal for all admissible realizations of the coefficients. Under other assumptions, we can apply sorting to solve each worst-case realization efficiently, or we can apply duality to integrate the worst-case outcome and have one integer program. The most difficult case is where the objective parameters are bilinear, and we prove its complexity is NP-hard even under simplifying assumptions. We consider a relaxation that provides an approximation, giving an overall guarantee of near-optimality when used with branch-and-bound search. We present preliminary computational experiments that illustrate the computational complexity of solving these robust formulations on sensor placement applications.

Consistency, Redundancy, and Implied Equalities in Linear Systems

Systems of linear inequalities have been studied for more than a century, but many of the results were developed during the early years of linear programming (1950s). New developments in linear programming plus interests in artificial intelligence have recently produced new results. One question is that of consistency: Does there exist a solution to satisfy all linear relations simultaneously? If so, are some of the relations redundant — that is, implied by the others? Are there implied equalities — that is, does some (weak) inequality have to hold with equality in every feasible solution? This paper brings together the main theorems that address these questions, unifies the framework, and presents some new results.

The use of the optimal partition in a linear programming solution for postoptimal analysis

Over the years we have learned to use an optimal basic solution to perform sensitivity analysis. Recently, the importance of an optimal partition, induced by a strictly complementary solution, has surfaced in connection with the interior point method. This paper gives examples where the partition is what is needed or desired to perform the analysis.