Geometric Programming Based On Trust Region Algorithm

Geometric programming is a special class of nonlinear programming of optimization method. Its object function and constrained functions are generalizedmultivariate polynomials, that is, the algebra sum of the product of the power of variables or its equivalent forms.as a special class of programming, it includes linear programming, quadratic and rational fraction programming. Geometric programming possesses lots of characteristics in form and property. Many methods have been proposed based on exploiting the characteristics of geometric programming. Over the last two decades, trust region methods have attracted much attention on the basis of their remarkable numerical reliability in conjunction with excellent global convergence properties.They have been proven to be very effective and robust techniques. Applying the trust region method to the geometric programming is the porpose of this paper, then present better and faster algorithm for the geometric programming.The creationary achievements of this paper can be summed up into the following three aspects:1)For the unconstrained optimization, we propose a nonmonotonic BFGS-trust-region algorithm, applying nonmonotonic algorithm to solve the problem of trust-region, people have acquired large achievements, the key of this paper is to propose a new BFGS formula. The advantage of this algorithm is that the subproblem of trust-region method ensures the update matrix is positive, that is the subproblem is a strictly convex quadratic programming, we combine correlation to prove the algorithm possesses global convergence under suitable conditions.2) For the positive type geometric programming,from the dual of the positive type geometric programming, Present three methods as follows:i) Present a method of integrating the trust region algorithm and the interior point method to solve the geometric programming, trust region algorithm has the peculiarity of robust fit and reliability, integrate the trust region and interior point to construct an algorithm for positive type geometric programming. The method not only reduce the iteration step (reduce calculations) but also get over the big degrees of difficulty. it has proved the validity of the algorithm. Under suitable conditions, it is proved that any accumulation point of the sequence generated by the algorithm is the optimal point.ii) Integrate the gradient projection algorithm and the interior point algorithm to solve the geometric program, and construct a new algorithm for positive type geometric program, and discuss the convergence, the algorithm has merit of simple configuration, small account quantity and strong stability.iii) using projected and trust region interior point method, and it proved that the method is globally convergent under certain condition.3) For the unconstrained generalized geometric programming, on the basis of making full use of the characteristics of unconstrained generalized geometric programming, we establish a new algorithm, nonmonotonic trust region algorithm via the conjugate path for solving (GGP). First, A new type of condensation problem is presented; second, a particular conjugate path is constructed for the problem, along which we get the approximate solution of the problem by nonmonotonic trust region algorithm, and further prove that the algorithm has global convergence and quadratic convergence properties.