Posynomial Programming Research Articles

Convex relaxation for solving posynomial programs

Convex underestimation techniques for nonlinear functions are an essential part of global optimization. These techniques usually involve the addition of new variables and constraints. In the case of posynomial functions \({x_1^{\alpha _1 } x_2^{\alpha _2 }\ldots x_n^{\alpha _n } ,}\) logarithmic transformations (Maranas and Floudas, Comput. Chem. Eng. 21:351–370, 1997) are typically used. This study develops an effective method for finding a tight relaxation of a posynomial function by introducing variables yj and positive parameters βj, for all αj > 0, such that \({y_j =x_j^{-\beta _j }}\) . By specifying βj carefully, we can find a tighter underestimation than the current methods.

Convexity of Products of Univariate Functions and Convexification Transformations for Geometric Programming

We investigate the characteristics that have to be possessed by a functional mapping f:ℝ↦ℝ so that it is suitable to be employed in a variable transformation of the type x→f(y) in the convexification of posynomials. We study first the bilinear product of univariate functions f1(y1), f2(y2) and, based on convexity analysis, we derive sufficient conditions for these two functions so that ℱ2(y1,y2)=f1(y1)f2(y2) is convex for all (y1,y2) in some box domain. We then prove that these conditions suffice for the general case of products of univariate functions; that is, they are sufficient conditions for every fi(yi), i=1,2,…,n, so as ℱn(y1,y2,…,yn)=∏i=1nfi(yi) to be convex. In order to address the transformation of variables that are exponentiated to some power κ≠1, we investigate under which further conditions would the function (f)κ be also suitable. The results provide rigorous reasoning on why transformations that have already appeared in the literature, like the exponential or reciprocal, work properly in convexifying posynomial programs. Furthermore, a useful contribution is in devising other transformation schemes that have the potential to work better with a particular formulation. Finally, the results can be used to infer the convexity of multivariate functions that can be expressed as products of univariate factors, through conditions on these factors on an individual basis.

Formulating the mixed integer fractional posynomial programming

The mixed integer fractional posynomial programming (MIFPP) problem arises from the summation minimization of several quotient terms appearing in the objective function subject to given constraints. All decision variables of the problem could be binary, integer, and/or continuous with (without) absolute value functions. This paper addresses the problem of providing an optimization approach, which could certainly obtain a solution as close as possible to a global optimum. In order to solve the complex MIFPP problem, several strategies are used including linear programming relaxation, modified goal programming, logarithmic piecewise function (LPF) and so on. In addition, the proposed model solves the undefined problem of the logarithm of zero or negative in the LPF. This significantly improved the utility of LPF in real applications. Finally, an illustrative example is included to demonstrate the solution procedure of the proposed model.

Design of microelectromechanical systems for variability via chance-constrained optimization

In this paper we describe a robust design method for microelectromechanical systems (MEMS) subject to inherent geometric and material uncertainties. We propose to formulate the robust MEMS design with variability problem as a specific optimization problem, namely chance-constrained posynomial programming, in which the statistical variations of both the process parameters and design variables can be easily incorporated. Using such constrained optimization approach, automated robust design can be obtained with very suboptimal design cost, and parametric yield of each specification can be guaranteed. Finally, the method is illustrated by considering the example of a simple crab-leg resonator.

A modified goal programming model for piecewise linear functions

Piecewise linear function (PLF) is an important technique for solving polynomial and/or posynomial programming problems since the problems can be approximately represented by the PLF. The PLF can also be solved using the goal programming (GP) technique by adding appropriate linearization constraints. This paper proposes a modified GP technique to solve PLF with n terms. The proposed method requires only one additional constraint, which is more efficient than some well-known methods such as those proposed by Charnes and Cooper's, and Li. Furthermore, the proposed model (PM) can easily be applied to general polynomial and/or posynomial programming problems.

Controlled dual perturbations for posynomial programs

With geometric programs getting larger and larger, the dual based solution methods which optimize a less complicated problem than the original problem become more important. However there are two major difficulties related to the development of a dual method. One is the non-differentiability of the dual objective function over its feasible region and the other one is the conversion of a dual solution to a primal solution. In this paper we address these issues for the posynomial programs. A well-controlled dual perturbation method which guarantees ϵ-optimal primal dual solution pair is proposed to overcome the nondifferentiability difficulty and a simple linear programming method is introduced to generate a quick dual-to-primal conversion.

Non-standard posynomial geometric programs

Non-standard posynomial geometric programs are non-linear programs that can be transformed into posynomial geometric programs. In this paper, new classes of nonstandard geometric programs are introduced. We focus on minimax posynomial programs and fractional posynomial programs. A complete duality theory is developed for each class.

Geometric Programming: Methods, Computations and Applications

In this paper, we give an applications-oriented survey of geometric programming. This important class of nonlinear programming problems has been intensively studied over the past decade and has played a crucial role in an extremely broad range of applications. The focus of the paper will be on posynomial programs and the slightly more general class of signomial programs. The approach used in studying posynomial programs has been generalized to a much broader class of optimization problems, and an excellent survey of that generalized theory recently appeared in a paper by E. L. Peterson [SIAM Rev., 18 (1976), pp. 1–51]. We do not consider that generalized theory here. It is our intent to provide the reader with an understanding of the basic results in posynomial and signomial programming that have played a crucial role in applications. After reviewing these basic results, we consider computational methods and applications. Three specific applications are considered in detail, and a bibliography indicating ...

Reversed geometric programming: A branch-and-bound method involving linear subproblems

The paper proposes a branch-and-bound method to find the global solution of general polynomial programs. The problem is first transformed into a reversed posynomial program. The procedure, which is a combination of a previously developed branch-and-bound method and of a well-known cutting plane algorithm, only requires the solution of linear subproblems.

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.

On finding a point in the relative interior of a polyhedral set and degeneracy in geometric programming

AbstractGeometric programming is now a well-established branch of optimization theory which has its origin in the analysis of posynomial programs. Geometric programming transforms a mathematical program with nonlinear objective function and nonlinear inequality constraints into a dual problem with nonlinear objective function and linear constraints. Although the dual problem is potentially simpler to solve, there are certain computational difficulties to be overcome. The gradient of the dual objective function is not defined for components whose values are zero. Moreover, certain dual variables may be constrained to be zero (geometric programming degeneracy).To resolve these problems, a means to find a solution in the relative interior of a set of linear equalities and inequalities is developed. It is then applied to the analysis of dual geometric programs.

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.

COMPUTATIONAL ASPECTS OF GEOMETRIC PROGRAMMING 3. SOME PRIMAL AND DUAL ALGORITHMS FOR POSYNOMIAL AND SIGNOMIAL GEOMETRIC PROGRAMS

In this paper we consider algorithmic and computational aspects of selected methods for posynomial and signomial programming problems. In particular, we consider the modified concave simplex method for dual posynomial programs and discuss some recent developments suggesting a new dual algorithm. We also consider linear programming based algorithms for primal posynomial programs. Finally, we discuss a recently developed branch and bound method for signomial programming.

OPTIMAL SELECTION OF STEAM TURBINE EXHAUST ANNULUS AND CONDENSER SIZES BY GEOMETRIC PROGRAMMING

Abstract In this paper, a posynomial programming model is developed for the optimal selection of steam turbine exhaust annulus and condenser sizes. After formulating the general model, a numerical example is presented and sensitivity analysis is considered. A simple formula expressing the optimal cost of the condensing system as a function of selected parameters is obtained.

Reducing Reversed Posynomial Programs

In this paper we study degeneracy in reversed posynomial programming. The word “degeneracy” was used in the original development of geometric (prototype posynomial) programming to describe a program in which a term will approach zero in an optimal sequence. A generalization of the original meaning to include reversed constraints is given. Corresponding to each degenerate reversed program, a unique reduced form is determined from the matrix of exponents by dropping specified terms and constraints. It is shown that, subject to a constraint qualification, the reduced and original programs have equal infima. A condition for boundedness of maximization problems subject to prototype constraints is given. Also, sufficient conditions are presented which guarantee that an optimal solution will be achieved and will occur at a point at which all variables are strictly positive.

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).

Reversed Geometric Programs Treated by Harmonic Means

A is a (generalized) polynomial with arbitrary real exponents, but positive coefficients and positive independent variables. Each posynomial program in which a posynomial is to be minimized subject to only inequality posynomial constraints is termed a program. The study of each reversed program is reduced to the study of a corresponding family of (prototype) geometric (namely, posynomial programs in which a posynomial is to be minimized subject to only upper-bound inequality posynomial constraints). This reduction comes from using the classical arithmeticharmonic mean inequality to invert each lower-bound inequality constraint into an equivalent robust family of conservatively approximating upper-bound inequality constraints. The resulting families of programs are then studied with the aid of the *Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213 Partially supported by Research Grant DA-AROD-31-124-71-G17 **Northwestern University, Evanston, Illinois 6O2O1 Partially supported by a Summer Fellowship Grant from Northwestern University. techniques of (prototype) programming. This approach has important computational features not possessed by other approaches, and it can easily be applied to the even larger class of we11-posed algebraic (namely, programs involving real-valued functions that are generated solely by addition, subtraction, multiplication, division, and the extraction of roots).

Geometric programming: Duality in quadratic programming and [formula omitted]p-approximation III (degenerate programs)

The duality theory of geometric programming as developed by Duffin, Peterson, and Zener is based on abstract properties shared by certain classical inequalities, such as Cauchy's arithmetic-geometric mean inequality and Hölder's inequality. Inequalities with these abstract properties have been termed “geometric inequalities.” In this sequence of papers, a new geometric inequality is established and used to extend the “refined duality theory” for “posynomial” geometric programs. This extended duality theory treats both “quadratically constrained quadratic programs” and “ l p -constrained l p -approximation (regression) problems” through a rather novel and unified formulation of these two classes of programs. This work generalizes some of the work of others on (linearly-constrained) quadratic programs and provides a new explicit formulation of duality for constrained approximation problems. Duality theories have been developed for a larger class of programs, namely all convex programs, but those theories (when applied to the programs considered here) are not nearly as strong as the theory developed here. This theory has virtually all of the desirable features of its analog for posynomial programs, and its proof provides useful computational procedures.

Geometric Programming: Duality in Quadratic Programming and $l_p $-Approximation II (Canonical Programs)

The duality theory of geometric programming as developed by Duflin, Peterson and Zener [7] is based on abstract properties shared by certain classical inequalities, such as Cauchy’s arithmetic-geometric mean inequality and Hölder’s inequality. Inequalities with these abstract properties have been termed “geometric inequalities” [7, p. 195]. In this sequence of papers [15], [16], [17] a new geometric inequality is established and used to extend the “refined duality theory” for “posynomial” geometric programs [6] and [7, Chap. VI]. This extended duality theory treats both “quadratically-constrained quadratic programs” and “$l_p $-constrained $l_p $-approximation (regression) problems” through a rather novel and unified formulation of these two classes of programs. This work generalizes some of the work of others on (linearly-constrained) quadratic programs and provides a new explicit formulation of duality for constrained approximation problems. Duality theories have been developed for a large class of programs, namely all convex programs, but those theories (when applied to the programs considered here) are not nearly as strong as the theory developed here. This theory has virtually all of the desirable features [7, p. 11] of its analogue for posynomial programs, and its proof provides useful computational procedures.