Abstract
This paper attempts to obtain an approximate optimum of the geometric programming (GP) problem. Generally speaking, until now, a lot of algorithms of GP have been proposed to obtain the optimum and the optimal solution and those methods need fairly much works to obtain ones.
On the other hand, there is a case that the true optimum or the true solution are not neccessarily needed, but only the approximate ones are needed. For the example, the combinatorial problem is one of the examples, where the approximate optimum of an alternative problem is used to estimate whether the alternative one is selected or not. As another example, when the problem has multi-local solutions, we need to compare each local solution to the others. As the above examples, if the approximate optimum can be easily obtained, the calculation volume to select the optimal alternative or solution will fairly decrease.
In this paper, we introduce some parameters to the primal problem to recast it into the set of the smaller-scale problems with the zero degree of difficulty. The degree of difficulty is the term to measure the calculation volume to solve the dual problem. The parameters are used to share the given problem into several ones that each of them has not all the polynomial terms and only the part of the polynomials in the given original problem. The shared, if possible, problem is easily solved by treating only the linear equations, because the degree of difficulty of the problem is zero. Moreover the sufficient conditions for the feasibility of the shared problem and the relation between the original problem and the shared ones are shown. From the optimums of the shared problems, the lower band for the optimum of the original problem is shown. To illustlate the results, some numerical examples are calculated.
