Lower bounds for polynomials using geometric programming

We make use of a result of Hurwitz and Reznick, and a consequence of this result due to Fidalgo and Kovacec, to determine a new sufficient condition for a polynomial $f\in\mathbb{R}[X_1,...,X_n]$ of even degree to be a sum of squares. This result generalizes a result of Lasserre and a result of Fidalgo and Kovacec, and it also generalizes the improvements of these results given in [6]. We apply this result to obtain a new lower bound $f_{gp}$ for $f$, and we explain how $f_{gp}$ can be computed using geometric programming. The lower bound $f_{gp}$ is generally not as good as the lower bound $f_{sos}$ introduced by Lasserre and Parrilo and Sturmfels, which is computed using semidefinite programming, but a run time comparison shows that, in practice, the computation of $f_{gp}$ is much faster. The computation is simplest when the highest degree term of $f$ has the form $\sum_{i=1}^n a_iX_i^{2d}$, $a_i>0$, $i=1,...,n$. The lower bounds for $f$ established in [6] are obtained by evaluating the objective function of the geometric program at the appropriate feasible points

An Unconstrained Q - G Programming Problem and its Application

In practice there exist many methods to solve unconstrained, constrained and mixed quadratic and geometric programming problems. In this paper an attempt is made to develop unconstraint Q – G programming problem by combining quadratic programming problem and geometric programming problem. This model is solved using the technique of geometric programming problem. A hypothetical example is considered to illustrate the model. Keywords: Unconstrained Quadratic Programming problems, Geometric Programming problem, Primal and Dual problem, Orthogonality and normality conditions

Duality and Geometric Programming

Two main problems arise from the use of the Transcendental Logarithmic form: 1. For practical and estimation purposes, the authors take the approximating function as the true function and include any possible source of error in the error term of the regression equation. This implies that there is no way of telling whether the results are affected by stochastic or approximation error. 2. The Cobb-Douglas and the CES production function have the property of "self duality", i.e., both the production and the cost forms are members of the same family of functional forms. This makes irrelevant the choice of representation of the technology by the production or cost functions. The Transcendental Logarithmic Form when taken as, the true form for the primal (dual) problem and then taken again as the true form of the dual (primal), makes one of the selections arbitrary since the form is not self-dual. This point is treated by Burgess [9] who shows with empirical results the consequences of choosing the cost or the production Transcendental Logarithmic form as a representation of the underlying technology. This paper is addressed to the possible solution of these two problems while still being able to work with more general production functions. We propose for the consideration of the economists interested in the Theory of Production, the Geometric Programming (GP) method of solving cost minimization problems which is extensively used in engineering. The similarities observed in both fields also indicate the possible benefits of closer communication among them. In the coming sections, we give an introduction to GP and illustrate with examples using the Cobb-Douglas, CES, and a more general explicit production function