Iterative Algorithms For Two Types Of Nonconvex Optimization Problems

As an important class of optimization problems, nonconvex optimization problems are widely used in the fields of economic, biochemistry, engineering design and information technology and so on. These problems generally posses multiple local optima that are not globally optimal, and may be difficult to solve. Recently, various specialized algorithms have been proposed for globally solving these problems. This paper proposes iterative algorithms for solving two types of nonconvex optimization problems respectively. One is the profit maximization problem in economy, whose model is the generalized geometric programming, and the other is the generalized polynomial multiplicative problem with generalized polynomial constraints. The main contents of this paper are as follows:In Chapter 1, we first give optimization models that will be considered in this paper. Next, we present the significance of application background, theoretical research and related work of these optimization models. Finally, the main work of this paper is briefly introduced.In Chapter 2, a unified iterative approach is presented for the profit maximum prob-lem in economy. Firstly, the practical problem is written as the form of generalized geometric programming. Then, the problem can be transformed into convex program by some operations, and the approach can obtain satisfactory results by solving a se-quence of convex programs. At last, convergence results are obtained. It is also observed from computational results that the proposed iterative approach is energetic for solving profit-maximization problems.In Chapter 3, we consider a class of the generalized polynomial multiplicative problem with generalized polynomial constraints. By using some simple transformation techniques, the equivalent form can be gained. By the similar convexification method in Chapter 2, we transform the equivalent problem into convex program. By solving a sequence of con-vex programs, we can obtain the optimal solution of the problem. Convergence analysis is given at last. Numerical results illustrate the feasibility and efficiency of the proposed algorithm.