Modified geometric programming problem and its applications

In this paper, we propose unconstrained and constrained posynomial Geometric Programming (GP) problem with negative or positive integral degree of difficulty. Conventional GP approach has been modified to solve some special typer of GP problems. In specific case, when the degree of difficulty is negative, the normality and the orthogonality conditions of the dual program give a system of linear equations. No general solution vector exists for this system of linear equations. But an approximate solution can be determined by the least square and also max-min method. Here, modified form of geometric programming method has been demonstrated and for that purpose necessary theorems have been derived. Finally, these are illustrated by numerical examples and applications.     

A new algorithm for geometric programming based on the linear structure of its dual problem

In this article, we present an algorithm for the resolution of a nonlinear optimization problem, concretely the posynomial geometric programming model. The solution procedure that we develop extends the condensation techniques for geometric programming, allowing us to find the optimal solutions to the dual geometric problems that we get from the interior of the corresponding feasible regions, in the line that interior point methods for linear programming work, which leads us to obtain considerable computational advantages with respect of the classical solution procedures.     

一般的正项几何规划的一种分解方法

关于几何规划的分解方法，［１３］均给出特殊类型的几何规划的分解方法，本文则对一般的正项几何规划给出一种直接分解方法。     

An infeasible interior-point algorithm for solving primal and dual geometric programs

In this paper an algorithm is presented for solving the classical posynomial geometric programming dual pair of problems simultaneously. The approach is by means of a primal-dual infeasible algorithm developed simultaneously for (i) the dual geometric program after logarithmic transformation of its objective function, and (ii) its Lagrangian dual program. Under rather general assumptions, the mechanism defines a primal-dual infeasible path from a specially constructed, perturbed Karush-Kuhn-Tucker system.Subfeasible solutions, as described by Duffin in 1956, are generated for each program whose primal and dual objective function values converge to the respective primal and dual program values. The basic technique is one of a predictor-corrector type involving Newton’s method applied to the perturbed KKT system, coupled with effective techniques for choosing iterate directions and step lengths. We also discuss implementation issues and some sparse matrix factorizations that take advantage of the very special structure of the Hessian matrix of the logarithmically transformed dual objective function. Our computational results on 19 of the most challenging GP problems found in the literature are encouraging. The performance indicates that the algorithm is effective regardless of thedegree of difficulty, which is a generally accepted measure in geometric programming. Research supported in part by the University of Iowa Obermann Fellowship and by NSF Grant DDM-9207347.     

Posynomial geometric programming as a special case of semi-infinite linear programming

This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual problem is equivalent to a generalized linear program. Furthermore, the duality results that are available for the traditionally defined primal-dual pair are readily obtained from the duality theory for semi-infinite linear programs. It is also shown that two efficient algorithms (one primal based and the other dual based) for geometric programming actually operate on the semi-infinite linear program and its dual.     

Rough posynomial geometric programming

A rough posynomial geometric programming is put forward by the author. This model is advantageous for us to consider questions not only from the quantity of aspect, but from the quality because it contains more information than a traditional geometric programming one. Here, a rough convex function concept is advanced in rough value sets on foundation of rough sets and rough convex sets. Besides, a knowledge expression model in rough posynomial geometric programming is established and so is a mathematical one. Thirdly, solution properties are studied in mathematical model of rough posynomial geometric programming, and antinomy of the more-for-less paradox is solved with an arithmetic in rough posynomial geometric programming given, which can be changed into a rough linear programming after monomial rough posynomial geometric programming is solved. Finally, validity in model and algorithm is verified by examples.     

New method to posynomial geometric programming of trapezoidal fuzzy numbers

This paper presents a method for solving posynomial geometric programming with fuzzy coefficients. By utilizing comparison of fuzzy numbers with a method, the programming with fuzzy coefficients is reduced to the programming with constant coefficients. Then the programming with fuzzy coefficients can be solved by using a method for posynomial geometric programming. Finally, one comparative example is used to illustrate advantage of the new method.     

A comparison between a primal and a dual cutting plane algorithm for posynomial geometric programming problems

In this paper, primal and dual cutting plane algorithms for the solution of posynomial geometric programming problems are presented. It is shown that these cuts are deepest, in the sense that they cut off as much of the infeasible set as possible. Problems of nondifferentiability in the dual cutting plane are circumvented by the use of a subgradient. Although the resulting dual problem seems easier to solve, the computational experience seems to show that the primal cutting plane outperforms the dual.     

A perturbation approach to the main duality theorem of quadratic geometric programming

Quadratic geometric programming as introduced by Hough and Goforth is an extension of posynomial geometric programming. By using the theory of generalized geometric programming, Jefferson and Scott defined its exact geometric dual. The fundamental relationship between the geometric dual and the original problem is that they assume a common value at their respective optima. This result formally stated as the main duality theorem is proved in this paper by using a dual perturbation approach and two simple geometric inequalities. As a by-product, the insight provided by this constructive proof establishes a numerically precise dual based solution procedure for quadratic geometric programs.

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Zusammenfassung Die quadratische geometrische Optimierung wurde von Hough und Goforth eingeführt und stellt eine Erweiterung der posynomialen geometrischen Optimierung dar. Jefferson und Scott definierten unter Verwendung der verallgemeinerten geometrischen Optimierung ein duales Programm und leiteten Dualitätsbeziehungen ab, u. a. die Übereinstimmung der Optimalwerte des primalen und dualen Programms. Letzteres Resultat wird in der vorliegenden Arbeit unter Verwendung eines dualen Störproblems und zweier einfachen geometrischen Ungleichungen hergeleitet. Gleichzeitig macht die Beweismethode es möglich, ein Verfahren zur Lösung quadratischer geometrischer Optimierungsprobleme anzugeben.

     

Fuzzy正项几何规划的直接算法

将Fuzzy正项几何规划化的一变量有上、下界限制的Fuzzy正项几何规划,利用Fuzzy几何不等式,又将该变量有上、下界限制的Fuzzy正项几何规划化为单项Fuzzy正项几何规划,提出基于Fuzzy值集割平面法的两种直接算法,并通过一个数值例证实该方法的有效性。     

On solving a primal geometric program by partial dual optimization

The several published methods for mapping a dual solution estimate to a primal solution estimate in posynomial geometric programming provide no criteria for deciding how much deviation from primal feasibility, or discrepancy between the primal and dual objective function values, should be permitted before the primal solution estimate is accepted by the designer. This paper presents a new and simple dual-to-primal conversion method that uses the cost coefficients to provide a sound economic criterion for determining when to accept a primal solution estimate. The primal solution estimate generated is the exact solution to a modified primal obtained from the given primal by modifying the cost coefficients, with the exponent matrix left unchanged. The method is shown to have desirable properties when coupled with a convergent dual algorithm.     

Posynomial geometric programming with parametric uncertainty

Geometric programming provides a powerful tool for solving nonlinear problems where nonlinear relations can be well presented by exponential or power function. This paper develops a procedure to derive the lower and upper bounds of the objective of the posynomial geometric programming problem when the cost and constraint parameters are uncertain. The imprecise parameters are represented by ranges, instead of single values. An imprecise geometric program is transformed to a family of conventional geometric programs to calculate the objective value. The derived result is also in a range, where the objective value would appear. The ability of calculating the bounds of the objective value developed in this paper might help lead to more realistic modeling efforts in engineering design areas.     

A modified concave simplex algorithm for geometric programming

An algorithm for solving posynomial geometric programs is presented. The algorithm uses a modification of the concave simplex method to solve the dual program which has a nondifferentiable objective function. The method permits simultaneous changes in certain blocks of dual variables. A convergence proof follows from the convergence proof of the concave simplex method. Some computational results on problems with up to forty degrees of difficulty are included.     

Solving Posynomial Geometric Programming Problems via Generalized Linear Programming

This paper revisits an efficient procedure for solving posynomial geometric programming (GP) problems, which was initially developed by Avriel et al. The procedure, which used the concept of condensation, was embedded within an algorithm for the more general (signomial) GP problem. It is shown here that a computationally equivalent dual-based algorithm may be independently derived based on some more recent work where the GP primal-dual pair was reformulated as a set of inexact linear programs. The constraint structure of the reformulation provides insight into why the algorithm is successful in avoiding all of the computational problems traditionally associated with dual-based algorithms. Test results indicate that the algorithm can be used to successfully solve large-scale geometric programming problems on a desktop computer.     

Sensitivity analysis in posynomial geometric programming

Sensitivity analysis results for general parametric posynomial geometric programs are obtained by utilizing recent results from nonlinear programming. Duality theory of geometric programming is exploited to relate the sensitivity results derived for primal and dual geometric programs. The computational aspects of sensitivity calculations are also considered.This work was part of the doctoral dissertation completed in the Department of Operations Research, George Washington University, Washington, DC. The author would like to express his gratitude to the thesis advisor, Prof. A. V. Fiacco, for overall guidance and stimulating discussions which inspired the development of this research work.     

On geometric programming and complementary slackness

When the terms in a convex primal geometric programming (GP) problem are multiplied by slack variables whose values must be at least unity, the invariance conditions may be solved as constraints in a linear programming (LP) problem in logarithmically transformed variables. The number of transformed slack variables included in the optimal LP basis equals the degree of difficulty of the GP problem, and complementary slackness conditions indicate required changes in associated GP dual variables. A simple, efficient search procedure is used to generate a sequence of improving primal feasible solutions without requiring the use of the GP dual objective function. The solution procedure appears particularly advantageous when solving very large geometric programming problems, because only the right-hand constants in a system of linear equations change at each iteration.The influence of J. G. Ecker, the writer's teacher, is present throughout this paper. Two anonymous referees and the Associate Editor made very helpful suggestions. Dean Richard W. Barsness provided generous support for this work.     

The complementary unboundedness of dual feasible solution sets in convex programming

F.E. Clark has shown that if at least one of the feasible solution sets for a pair of dual linear programming problems is nonempty then at least one of them is both nonempty and unbounded. Subsequently, M. Avriel and A.C. Williams have obtained the same result in the more general context of (prototype posynomial) geometric programming. In this paper we show that the same result is actually false in the even more general context of convex programming — unless a certain regularity condition is satisfied.We also show that the regularity condition is so weak that it is automatically satisfied in linear programming (prototype posynomial) geometric programming, quadratic programming (with either linear or quadratic constraints),l _p -regression analysis, optimal location, roadway network analysis, and chemical equilibrium analysis. Moreover, we develop an equivalent regularity condition for each of the usual formulations of duality.Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-73-2516.     

Geometric programming with signomials

The difference of twoposynomials (namely, polynomials with arbitrary real exponents, but positive coefficients and positive independent variables) is termed asignomial.Each signomial program (in which a signomial is to be either minimized or maximized subject to signomial constraints) is transformed into an equivalent posynomial program in which a posynomial is to be minimized subject only to inequality posynomial constraints. The resulting class of posynomial programs is substantially larger than the class of (prototype)geometric programs (namely, posynomial programs in which a posynomial is to be minimized subject only to upper-bound inequality posynomial constraints). However, much of the (prototype) geometric programming theory is generalized by studying theequilibrium solutions to thereversed geometric programs in this larger class. Actually, some of this theory is new even when specialized to the class of prototype geometric programs. On the other hand, all of it can indirectly, but easily, be applied to the much larger class of well-posedalgebraic programs (namely, programs involving real-valued functions that are generated solely by addition, subtraction, multiplication, division, and the extraction of roots).This research was partially supported by Research Grant No. DA-AROD-31-124-6680 from the Army Research Office, Durham, North Carolina, and by a Summer Fellowship Grant from Northwestern University.     

Second order algorithms for the posynomial geometric programming dual,part I: Analysis

In this paper we analyse algorithms for the geometric dual of posynomial programming problems, that make explicit use of second order information. Out of two possible approaches to the problem, it is shown that one is almost always superior. Interestingly enough, it is the second, inferior approach that has dominated the geometric programming literature.This work was partially supported by the National Research Council of Canada, Grant A3552 and National Science Foundation Grant ENG78-21615.Earlier versions of this paper were presented at the Optimization Days Conference in Montreal (May 1976) and the TIMS meeting in Athens (July 1977).     

Inventory model of deteriorated items with a constraint: A geometric programming approach

An inventory model for deteriorating items is built-up with limited storage space. Here, demand rate for the items is finite, items deteriorate at constant rates and are replenished instantaneously. Following EOQ model, the problem is formulated with and without truncation on the deterioration term and ultimately is converted to the minimization of a signomial expression with a posynomial constraint. It is solved by modified geometric programming (MGP) method and non-linear programming (NLP) method. The problem is supported by numerical examples. The results from two versions of the model (with and without truncation) and two methods (i.e. MGP and NLP) are compared.