Dual Theory And Solution Of The Generalized Minimum Information Entropy Problem

Nonlinear programming which is considered to be recreation of the finite dimension optimization classical theory is a very important and active branch of opsearch. Characteristic of nonlinear programming lies in: allowance of complex constraints on the optimization it studies, deep analysis of optimality and duality, emphasis on summarizing theorem and establishment of feasible algorithm. With the development of new informatics technology, nonlinear programming is more and more widely used in the research of optimal design, scientific management, systematic identification, physical and mathematical variation, etc which promote its development.Elegant dual theory in linear programming has encouraged many researchers applying dual concept in solving nonlinear programming problem. At the beginning of their research, they could transform some nonlinear programming into relevant dual, such transformation made some outcome almost similar to linear programming. But if they wanted meaningful results, they had to consider convex programming. In the late 60's integrated theory came up, one of its ways was based on conjugate function theory introduced by Fenchel, which Rochafellar had developed and improved in a series of his books.In [6], Rockafellar had deduced the Fenchel theory, but some nonlinear programming could not get the dual by Fenchel theory directly and dual theory could not be applied in solving some problem. Zhu had deduced generalized Fenchel's duality theorem in [8],and further applied it to Minimum Discrimination Information(MDI) problem. More general quadratic constraints and entropy constrains of the density programming will be dealt with in this article. The two problems are widely applied in information theory, statistical mechanics, insurance business and game theory. In the progress of deducing the dual of these two problems using the knowledge of convex programming and generalized Fenchel theory, you can find Lagrange dual of the two programming by Lagrange dual in nonlinear programming in this article. As not much profound mathematical theory has involved, it could be much widely applied.Normally, we try to get the dual form of mathematical programming to change primary programming, which is difficult to be solved into its dual programming, which is easily to be solved. After we had got the dual form of two programming, we found their form were very simply because they had only nonnegative and linear constraints. Aiming at more general nonlinear optimizations subject to linear inequality constraints, this article proposes curvilinear path trust region algorithms. Trust region methods has drawn our attention increasingly as one approximating method guarantees general convergence. Its basic step is as follows: First, we set trust-region radius, and then we get a trial step produced by constraint quadratical model. Zhuhas studied unconstrained optimal problem by combining optimal path and modified path with nonmonotonic trust region methods in [9], and the use of l2 norm by trust region method in [1] in seeking the solution of unconstrained optimization has formed simple approximate trust region path, which enlightens us to solve nonlinear programming problem by curvilinear path. In general trust region method, a trial point is accepted as a new iterate and the procedure is repeated if the true reduction achieved by the objective function at this point is comparable with the reduction predicted by the quadratic model. Otherwise, the trust region radius is reduced and a new trial point is selected. It is possible that the region subproblem need to be resolved many times before obtaining an acceptable step, and hence the total computation for completing one iteration might be expensive. This article combines approximate trust region path and nonmonotonic backtracking strategy to solve nonlinear optimization subject to linear inequality constraints, that is, we use approximate trust region path to get the research direction by minimuming quadratical model in the trust region by employing...