| Trust region algorithm is a kind of effective algorithm for nonlinear optimization,the basic idea of the algorithms is: structure a quadratic model using the information ofthe objective function at a point, making it have a good approximation with the targetfunction near the point, then according to the minimum points of the quadratic model toproduce the next iteration points, and adjust the size of the trust region radius dependingon the degree of approximation between the quadratic model and the objective function.Trust region method is robust and has very good convergence property and manyresearchers are attracted. This article mainly focuses on the improvement of thealgorithm framework, and the theoretical analyses on the improved algorithmconvergence were conducted.The main research contents are as follows:1. On the basis of the previous work, the BFGS correction formula is improved andused to trust region algorithm for unconstrained optimization problem and an improvedBFGS-trust-region algorithm are proposed. This algorithm can guarantee the positivedefiniteness of the modified matrices, and the convergence is proved in certainconditions.2. Considering convex quadratic programming problem under the condition ofequality constraints, first the restricted problem is transformed into unconstrainedoptimization problem, then, based on the traditional trust region algorithm, a newalgorithm is given which combined line search technology. Under some suitableconditions, the global convergence is proved.3. A trust region algorithm which combines nonmonotone technique with linesearch technology for solving optimization problems is constructed. Different fromtraditional trust region method, it takes line search for the next iteration instead ofresolving the sub-problem when the trail step is not successful. This may allow aconsiderable computational saving. Global convergence is also demonstrated undercertain conditions.4. With the idea of transformation from the restricted question to unconstrainedoptimization problem, a new hybrid trust region algorithm combined with nonmonotone line search technology and improved BFGS correction formula for convex quadraticprogramming problems with equality constraints is presented based on traditional trustregion method. Its global convergence is proved under certain conditions. |
