Solution of Generalized Geometric Programs

Abstract

SUMMARYA cutting plane algorithm for the solution of generalized geometric programs with bounded variables is described and then illustrated by the detailed solution of a small numerical example. Convergence of this algorithm to a Kuhn-Tucker point of the program is assured if an initial feasible solution is available to initiate the algorithm. An algorithm for determining a feasible solution to a set of generalized posynomial inequalities which may be used to find a global minimum to the program as well as test for consistency of the constraint set, is also presented. Finally an application in optimal engineering design with seven variables and fourteen nonlinear inequality constraints is formulated and solved.

Disciplined Convex Programming

“…The most common choice of the norm in (1) is certainly the l2 or Euclidean norm, f ( x ) A I~A X -bl12 = Jw (3) In this case, (1) is easily recognized as a least squares problem, which as its name implies is often presented in an equivalent quadratic form, minimize f ( x )~ = IIAx --1 1 : = CZl (aTx -bi)2 (4) This problem has an analytical solution x = (ATA)-'ATb, which can be computed using a Cholesky factorization of ATA, or more accurately using a QR or SVD factorization of A [76]. A number of software packages to solve least squares problems are readily available; e.g., [I, 1061.…”

“…Several other classes of CPs have been identified recently as standard forms. These include semidefinite programs (SDPs) [155], second-order cone programs (SOCPs) [104], and geometric programs (GPs) [48,3,138,52,921. The work we present her applies to all of these special cases as well as to the general class of CPs.…”

Tractable approximate robust geometric programming

“…This transformation does not in any way change the problem data, which are the same for the posynomial form and convex form problems. Geometric programming has been used in various fields since the late 1960s; early applications of geometric programming can be found in the books Avriel (1980), Duffin et al (1967), Zener (1971) and the survey papers Ecker (1980), Peterson (1976), Boyd et al (2007). More recent applications can be found in various fields including circuit design Chen et al 2000;Dawson et al 2001;Daems et al 2003;Hershenson 2002;Hershenson et al 2001;Mohan et al 2000;Sapatnekar 1996;Singh et al 2005;Sapatnekar et al 1993;Young et al 2001), chemical process control (Wall et al 1986), environment quality control (Greenberg 1995), resource allocation in communication systems (Dutta and Rama 1992), information theory (Chiang and Boyd 2004;Karlof and Chang 1997), power control of wireless communication networks (Kandukuri and Boyd 2002;O'Neill et al 2006), queue proportional scheduling in fading broadcast channels (Seong et al 2006), and statistics (Mazumdar and Jefferson 1983).…”

“…Algorithms for solving geometric programs appeared in the late 1960s, and research on this topic continued until the early 1990s; see, e.g., Avriel et al (1975), Rajpogal and Bricker (1990). A huge improvement in computational efficiency was achieved in 1994, when Nesterov and Nemirovsky developed provably efficient interior-point methods for many nonlinear convex optimization problems, including GPs (Nesterov and Nemirovsky 1994).…”

Concurrent logic restructuring and placement for timing closure

“…d ij =τ i +r×(c net +c load )+r net ×c load ( 1 ) where d ij is the delay from pin i to pin j as shown in Figure 1, τ i is the intrinsic delay of pin i, r is driver resistance, r net and c net are the resistance and capacitance for the net, and c load is the summation of the input capacitances of the fanouts. Arrival times are calculated from the primary inputs to the primary outputs in the topological order.…”