On the computational utility of posynomial geometric programming solution methods

Ten codes or code variants were used to solve the five equivalent posynomial GP problem formulations. Four of these codes were general NLP codes; six were specialized GP codes. A total of forty-two test problems was solved with up to twenty randomly generated starting points per problem. The convex primal formulation is shown to be intrinsically easiest to solve. The general purpose GRG code called OPT appears to be the most efficient code for GP problem solution. The reputed superiority of the specialized GP codes GGP and GPKTC appears to be largely due to the fact that these codes solve the convex primal formulation. The dual approaches are only likely to be competitive for small degree of difficulty, tightly constrained problems.     

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.     

A modified reduced gradient method for dual posynomial programming

In this paper, we present a method for solving the dual of a posynomial geometric program based on modifications of the reduced gradient method. The modifications are necessary because of the numerical difficulties associated with the nondifferentiability of the dual objective function. Some preliminary numerical results are included that compare the proposed method with the modified concave simplex method of Beck and Ecker of Ref. 1.This research was supported in part by the National Science Foundation, Grant No. MCS75-09443-A01.The authors sincerely thank P. Vandeputte for his computing assistance. We also wish to thank T. Magnanti and unknown referees for helpful criticism.     

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 class of rank-two ellipsoid algorithms for convex programming

In this paper, we develop a method for constructing minimum volume ellipsoids containing a wedge-shaped subset of a given ellipsoid. This construction yields a class of ellipsoid algorithms for convex programming that use rank-two update formulae. Research supported by the National Science Foundation, Grant No. MC582-01790.     

Nonlinear programming via an exact penalty function: Global analysis

In this paper we motivate and describe an algorithm to solve the nonlinear programming problem. The method is based on an exact penalty function and possesses both global and superlinear convergence properties. We establish the global qualities here (the superlinear nature is proven in [7]). The numerical implementation techniques are briefly discussed and preliminary numerical results are given.This work is supported in part by NSERC Grant No. A8639 and the U.S. Dept. of Energy.     

A simplex algorithm for piecewise-linear programming I: Derivation and proof

The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. This three-part paper develops and analyzes a general, computationally practical simplex algorithm for piecewiselinear programming.Part I derives and justifies the essential steps of the algorithm, by extension from the simplex method for linear programming in bounded variables. The proof employs familiar finite-termination arguments and established piecewise-linear duality theory.Part II considers the relaxation of technical assumptions pertaining to finiteness, feasibility and nondegeneracy of piecewise-linear programs. Degeneracy is found to have broader consequences than in the linear case, and the standard techniques for prevention of cycling are extended accordingly.Part III analyzes the computational requirements of piecewise-linear programming. The direct approach embodied in the piecewise-linear simplex algorithm is shown to be inherently more efficient than indirect approaches that rely on transformation of piecewise-linear programs to equivalent linear programs. A concluding section surveys the many applications of piecewise-linear programming in linear programming,l ₁ estimation, goal programming, interval programming, and nonlinear optimization.This research has been supported in part by the National Science Foundation under grant MCS-8217261.     

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.This author's research was supported in part by the Center for the Study of Environmental Policy, The Pennsylvania State University.     

An equivalence between two algorithms for quadratic programming

In this paper, we demonstrate that the Van de Panne—Whinston symmetric simplex method when applied to a certain implicit formulation of a quadratic program generates the same sequence of primal feasible vectors as does the Von Hohenbalken simplicial decomposition algorithm specialized to the same program. Such an equivalence of the two algorithms extends earlier results for a least-distance program due to Cottle—Djang.This research was prepared as part of the activities of the Management Sciences Research Group, Carnegie-Mellon University, under Contract N00014-75-C-0621 NR 047-048 with the Office of Naval Research.     

A numerically stable dual method for solving strictly convex quadratic programs

An efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point. Its implementation utilizes the Cholesky and QR factorizations and procedures for updating them. The performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code (the principal motivation for the development of the algorithm). These computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available. The algorithm is also compared theoretically to the modified-simplex type dual methods of Lemke and Van de Panne and Whinston and it is illustrated by a numerical example. This research was supported in part by the Army Research Office under Grant No. DAAG 29-77-G-0114 and in part by the National Science Foundation under Grant No. MCS-6006065.     

I/O-efficient algorithms for computing planar geometric spanners

We present I/O-efficient algorithms for computing planar Steiner spanners for point sets and sets of polygonal obstacles in the plane.     

Second order optimality conditions for generalized semi-infinite programming problems

We consider generalized semi-infinite programming problems. Second order necessary and sufficient conditionsfor local optimality are given. The conditions are derived under assumptions such that the feasible set can be described by means of a finite number of optimal value functions. Since we do not require a strict complementary condition for the local reduction these functions are only of class C¹ A sufficient condition for optimality is proven under much weaker assumptions.     

Current state of the art of algorithms and computer software for geometric programming

This paper attempts to consolidate over 15 years of attempts at designing algorithms for geometric programming (GP) and its extensions. The pitfalls encountered when solving GP problems and some proposed remedies are discussed in detail. A comprehensive summary of published software for the solution of GP problems is included. Also included is a numerical comparison of some of the more promising recently developed computer codes for geometric programming on a specially chosen set of GP test problems. The relative performance of these codes is measured in terms of their robustness as well as speed of computation. The performance of some general nonlinear programming (NLP) codes on the same set of test problems is also given and compared with the results for the GP codes. The paper concludes with some suggestions for future research.An earlier version of this paper was presented at the ORSA/TIMS Conference, Chicago, 1975.This work was supported in part by the National Research Council of Canada, Grant No. A-3552, Canada Council Grant No. S74-0418, and a research grant from the School of Organization and Management, Yale University. The author wishes to thank D. Himmelblau, T. Jefferson, M. Rijckaert, X. M. Martens, A. Templeman, J. J. Dinkel, G. Kochenberger, M. Ratner, L. Lasdon, and A. Jain for their cooperation in making the comparative study possible.     

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.

     

Comparison of a special-purpose algorithm with general-purpose algorithms for solving geometric programming problems

We study the performance of four general-purpose nonlinear programming algorithms and one special-purpose geometric programming algorithm when used to solve geometric programming problems. Experiments are reported which show that the special-purpose algorithm GGP often finds approximate solutions more quickly than the general-purpose algorithm GRG2, but is usually not significantly more efficient than GRG2 when greater accuracy is required. However, for some of the most difficult test problems attempted, GGP was dramatically superior to all of the other algorithms. The other algorithms are usually not as efficient as GGP or GRG2. The ellipsoid algorithm is most robust.This work was supported in part by the National Science Foundation, Grant No. MCS-81-02141.     

Nonlinear 0–1 programming: I. Linearization techniques

Any real-valued nonlinear function in 0–1 variables can be rewritten as a multilinear function. We discuss classes of lower and upper bounding linear expressions for multilinear functions in 0–1 variables. For any multilinear inequality in 0–1 variables, we define an equivalent family of linear inequalities. This family contains the well-known system of generalized covering inequalities, as well as other linear equivalents of the multilinear inequality that are more compact, i.e., of smaller cardinality. In a companion paper [7]. we discuss dominance relations between various linear equivalents of a multilinear inequality, and describe a class of algorithms for multilinear 0–1 programming based on these results. Research supported by the National Science Foundation under grant ECS7902506 and by the Office of Naval Research under contract N00014-75-C-0621 NR 047-048.     

Variable metric relaxation methods,part II: The ellipsoid method

The deepest, or least shallow, cut ellipsoid method is a polynomial (time and space) method which finds an ellipsoid, representable by polynomial space integers, such that the maximal ellipsoidal distance relaxation method using this fixed ellipsoid is polynomial: this is equivalent to finding a linear transforming such that the maximal distance relaxation method of Agmon, Motzkin and Schoenberg in this transformed space is polynomial. If perfect arithmetic is used, then the sequence of ellipsoids generated by the method converges to a set of ellipsoids, which share some of the properties of the classical Hessian at an optimum point of a function; and thus the ellipsoid method is quite analogous to a variable metric quasi-Newton method. This research was supported in part by the F.C.A.C. of Quebec, and the N.S.E.R.C. of Canada under Grant A 4152.     

Accelerated convergence for the Powell/Hestenes multiplier method

It is known that augmented Lagrangian or multiplier methods for solving constrained optimization problems can be interpreted as techniques for maximizing an augmented dual functionD _c(). For a constantc sufficiently large, by considering maximizing the augmented dual functionD _c() with respect to, it is shown that the Newton iteration for based on maximizingD _c() can be decomposed into taking a Powell/Hestenes iteration followed by a Newton-like correction. Superimposed on the original Powell/Hestenes method, a simple acceleration technique is devised to make use of information from the previous iteration. For problems with only one constraint, the acceleration technique is equivalent to replacing the second (Newton-like) part of the decomposition by a finite difference approximation. Numerical results are presented.     

Interior path following primal-dual algorithms. part I: Linear programming

We describe a primal-dual interior point algorithm for linear programming problems which requires a total of number of iterations, whereL is the input size. Each iteration updates a penalty parameter and finds the Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the logarithmic barrier function problem. The algorithm is based on the path following idea.     

Algorithmic equivalence in quadratic programming,I: A least-distance programming problem

It is demonstrated that Wolfe's algorithm for finding the point of smallest Euclidean norm in a given convex polytope generates the same sequence of feasible points as does the van de Panne-Whinstonsymmetric algorithm applied to the associated quadratic programming problem. Furthermore, it is shown how the latter algorithm may be simplified for application to problems of this type.This work was supported by the National Science Foundation, Grant No. MCS-71-03341-AO4, and by the Office of Naval Research, Contract No. N00014-75-C-0267.