Bala Shetty

The nonlinear knapsack problem - algorithms and applications

Abstract We present a survey of algorithms and applications for the nonlinear knapsack problem (or, the nonlinear resource allocation problem). In its most general form, the nonlinear knapsack problem will be defined as a nonlinear optimization problem with just one constraint, bounds on the variables, and, in some cases, a set of specially structured constraints such as generalized upper bounds (GUBs). This problem is encountered either directly, or as a subproblem, in a variety of applications, including production planning, financial modeling, stratified sampling, and capacity planning in manufacturing, health care, and computer networks. By taking advantage of the special structure of the problem, efficient solution methods can be developed. Problem classes addressed here include continuous and integer problems, convex and nonconvex problems, separable and nonseparable problems, and problems with additional specially structured constraints.

The Nonlinear Resource Allocation Problem

In this paper we study the nonlinear resource allocation problem, defined as the minimization of a convex function over one convex constraint and bounded integer variables. This problem is encountered in a variety of applications, including capacity planning in manufacturing and computer networks, production planning, capital budgeting, and stratified sampling. Despite its importance to these and other applications, the nonlinear resource allocation problem has received little attention in the literature. Therefore, we develop a branch-and-bound algorithm to solve this class of problems. First we present a general framework for solving the continuous-variable problem. Then we use this framework as the basis for our branch-and-bound method. We also develop reoptimization procedures and a heuristic that significantly improve the performance of the branch-and-bound algorithm. In addition, we show how the algorithm can be modified to solve nonconvex problems so that a concave objective function can be handled. The general algorithm is specialized for the applications mentioned above and computational results are reported for problems with up to 200 integer variables. A computational comparison with a 0, 1 linearization approach is also provided.

Strategic financial risk management and operations research

Abstract Risk management has become a vital topic for financial institutions in the 1990s. Strategically, asset/liability management systems are important tools for controlling a firms financial risks. They manage these risks by dynamically balancing the firms asset and liabilities to achieve the firms objectives. We discuss such leading international firms as Towers Perrin, Frank Russell, and Falcon Asset Management, which apply asset/liability management for efficiently managing risk over extended time periods. Three components of asset/liability management are described: 1) a multi-stage stochastic program for coordinating the asset/liability decisions; 2) a scenario generation procedure for modeling the stochastic parameters; and 3) solution algorithms for solving the resulting large-scale optimization problem.

Financial planning via multi-stage stochastic optimization

This paper describes a framework for modeling significant financial planning problems based on multi-stage optimization under uncertainty. Applications include risk management for institutions, banks, government entities, pension plans, and insurance companies. The approach also applies to individual investors who are interested in integrating investment choices with savings and borrowing strategies. A dynamic discrete-time structure addresses realistic financial issues. The resulting stochastic program is enormous by current computer standards, but it possesses a special structure that lends itself to parallel and distributed optimization algorithms. Interior-point methods are particularly attractive. Solving these stochastic programs presents a major challenge for the computational operations research and computer science community.

A pegging algorithm for the nonlinear resource allocation problem

In this paper we present a new algorithm for solving the nonlinear resource allocation problem. The nonlinear resource allocation problem is defined as the minimization of a convex function over a single convex constraint and bounded integer variables. We first present a pegging algorithm for solving the continuous variable problem, and then incorporate the pegging method in a branch and bound algorithm for solving the integer variable problem. We compare the computational performance of the pegging branch and bound algorithm with three other methods: a multiplier search branch and bound algorithm, dynamic programming, and a 0,1 linearization method. The computational results demonstrate that the pegging branch and bound algorithm advances the state of the art in methods for solving the nonlinear resource allocation problem.

A generalization of Polyak's convergence result for subgradient optimization

This paper generalizes a practical convergence result first presented by Polyak. This new result presents a theoretical justification for the step size which has been successfully used in several specialized algorithms which incorporate the subgradient optimization approach.

A Branch and Bound Algorithm for Integer Quadratic Knapsack Problems

We present a branch and bound algorithm for solving separable convex quadratic knapsack problems with lower and upper bounds on the integer variables. The algorithm solves a series of continuous quadratic knapsack problems via their Kuhn-Tucker conditions. We also discuss reoptimization procedures for efficiently solving the continuous subproblems. Computational results for problems with up to 300 integer variables are reported. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

The equal flow problem

Abstract This paper presents a new algorithm for the solution of a network problem with equal flow side constraints. The solution technique is motivated by the desire to exploit the special structure of the side constraints and to maintain as much of the characteristics of pure network problems as possible. The proposed algorithm makes use of Lagrangean relaxation to obtain a lower bound and decomposition by right-hand-side allocation to obtain upper bounds. The lagrangean dual serves not only to provide a lower bound used to assist in termination criteria for the upper bound, but also allows an initial allocation of equal flows for the upper bound. The algorithm has been tested on problems with up to 1500 nodes and 6000 arcs. Computational experience indicates that solutions whose objective function value is well within 1% of the optimum can be obtained in 1%–65% of the MPSX time depending on the amount of imbalance inherent in the problem. Incumbent integer solutions which are within 99.99% feasible and well within 1% of the proven lower bound are obtained in a straightforward manner requiring, on the average, 30% of the MPSX time required to obtain a linear optimum.

Model management systems: a survey

This paper provides a survey of model management literature within the mathematical modeling domain. The first part of the survey is a review and a summary of the literature. After giving some basic definitions of modeling, modeling life cycle, and model management, two representative algebraic modeling languages followed by three approaches to modeling are introduced. These approaches are database, graph-based, and knowledge-based. The discussion is followed by a review of two specialized model management systems. The second part of the survey is a categorization of various modeling systems based on the modeling functions they provide and some of their features. These functions include life cycle support and model base administration. The degree of model independence provided by model management systems and the implemented environment systems is also summarized. The last part of the paper provides directions for future research.

Multicommodity Network Problems: Applications and Computations

Abstract It is well documented that pure network problems can be solved from 10 to 100 times faster using specialized primal simplex software as compared to general linear programming systems. For multi-commodity network flow problems, the computational savings are a function of the number of tight-side constraints. In this study, we present three real-world multicommodity models and data concerning the number of tight-side constraints. We also present the results of a computational study on a set of 25 randomly generated test problems which have a wide range of number of tight-side constraints. We conclude that a specialized multicommodity network code is three times as fast as a general code, while a specialized network with general side constraints code has twice the speed of a general LP code.