John J. Dinkel

Entropy maximization and geometric programming

This paper shows the equivalence of entropy-maximization models to geometric programs. As a result we derive a dual geometric program which consists of the minimization of an unconstrained convex function. We develop the necessary duality equivalencies between the two dual programs and show the computational attractiveness of our approach. We also develop some characterizations of the optimal solution of the entropy model which have important implications with regard to postoptimal or sensitivity analysis.

An interval arithmetic approach to sensitivity analysis in geometric programming

Geometric programming has been an important optimization approach in several engineering design areas. In this paper, we present an approach to sensitivity analysis in geometric programming using interval arithmetic. The study of the effect of changes in the problem parameters is important for theoretical as well as for practical reasons. To date, most of the numerical approaches to sensitivity analysis have characterized the solution in terms of differential changes in the parameters. We use interval arithmetic to generate an interval of solution values associated with an interval of parameter values. These results indicate a new approach to characterizing solutions to geometric programs in terms of changes in the problem parameters.

An approach to the numerical solutions of geometric programs

An algorithm for solving ordinary geometric programs is presented. The algorithm is based on the reduced system associated with geometric programs and is highly flexible in that it allows the use of several nonlinear optimization techniques.

Computational aspects of cutting-plane algorithms for geometric programming problems

This paper presents the results of computational studies of the properties of cutting plane algorithms as applied to posynomial geometric programs. The four cutting planes studied represent the gradient method of Kelley and an extension to develop tangential cuts; the geometric inequality of Duffin and an extension to generate several cuts at each iteration. As a result of over 200 problem solutions, we will draw conclusions regarding the effectiveness of acceleration procedures, feasible and infeasible starting point, and the effect of the initial bounds on the variables. As a result of these experiments, certain cutting plane methods are seen to be attractive means of solving large scale geometric programs.

Absolute Deviations Curve Fitting: An Alternative to Least Squares

The determination of minimum absolute deviation estimators of regression coefficients by linear programming is reviewed. The linear program is shown to be equivalent to a geometric program. As an illustration of the small sample properties of the estimators, a Monte Carlo simulation experiment is conducted using Normal, Cauchy, and Laplace error distributions.

Technical Note-On Sensitivity Analysis in Geometric Programming

This note develops efficient sensitivity procedures for posynomial geometric programs. These procedures provide ranging information for the primal coefficients, means for dealing with problems with loose primal constraints, and an incremental procedure for improving the estimated solutions. These sensitivity procedures are independent of the method of solution of the geometric program.

On the solution of regional planning models via geometric programming

The purpose of this paper is to demonstrate the applicability of geometric programming to regional land-use planning models. The regional models are of the type where the criterion is maximum accessibility, and the model is constrained by population limits on each district. After a brief discussion of geometric programming, the relationship of total accessibility models to geometric programs is developed and a method of obtaining numerical solutions is presented. Several models are analyzed using the geometric programming approach, including a game theoretic model which is used to generate decentralized plans.

Technical Note-An Implementation of Surrogate Constraint Duality

This paper presents an implementation of surrogate constraint duality in mathematical programming. Motivated by the use of linear programming duality for surrogate constraints in integer linear programs, this implementation is based on geometric programming duality. As a result of this formulation we are able to present an algorithm for surrogate constraint duality and discuss several important properties of the algorithm.

Environmental inspection routes and the constrained travelling system salesman problem

Abstract The constrained version of the classical travelling salesman problem (TSP) is seen to be a generic model for a wide variety of problems. Our concern here is limited to those problems which impinge directly on the environmental issues. Some potential applications are in the areas of resource management, energy conservation and transportation. The version of the problem we are faced with can be stated as: Given 1 or more persons (or vehicles) that must visit a set of n sites (plants, bus stops, pickup point, and so on) how can one develop a route which is of minimum mileage and meets certain restrictions (8 hour work day, bus seating capacity, vehicle capacity, length of trip and so on). This paper addresses the following issues in light of this problem: 1. 1. Data problems: in particular, efficient means of gathering and maintaining data. 2. 2. Numerical results: the study of efficient algorithms so that the model can be used efficiently on large problems and also on a day-to-day basis. 3. 3. Areas of potential application: for example, locational problems (office location), regionalization studies (development of efficient regional boundaries, efficient inspection and delivery routes, long range and short term resource management). 4. 4. Decision making applications in terms of analysis of operations, planning and analysis of resources via sensitivity analysis.

A computational study of methods for solving polynomial geometric programs

A computational comparison of several methods for dealing with polynomial geometric programs is presented. Specifically, we compare the complementary programs of Avriel and Williams (Ref. 1) with the reversed programs and the harmonic programs of Duffin and Peterson (Refs. 2, 3). These methods are used to generate a sequence of posynomial geometric programs which are solved using a dual algorithm.