A Trust-rejion Method For Bound-constrained Derivative-free Optimization Problems

Optimization theroy and method, which makes research on how to ifnd the optimal solution a-mong many feasible plans, is a very popular and useful subject. Optimization technology is wide?ly applied in many fields such as defense, industrial and agircultural production, transportation,financial, trade, management, scientific reach. With the development of the computers，optimiza?tion theroy and method is playing an increasing role in practical application. Hence, more andmore optimization problems have been solved.Derivative-free optimization algorithm is one of the most commonly used optimizationmethod，a class of method that was developed when the gradient information of the objective func?tion is not clear. These methods do not require explicit objective function gradient information,so the application is used more widely and becomes a major class of methods to solve many prob?lems.In [1], Conn, Scheinberg and Vicente constructed some basic methods of deirvative-freeoptimization. The basic idea of this method is to use some of the sample points to establish poly?nomial interpolation model for objective functions and then solve the objective function by modelanalysis.The problem is named constrained optimization if the vairables are subject to someconditions. In this paper, the main problem we solve is a class of nonliner equations that witha box constraint. By constructing the value function, the paper will transform nonlinear equationssystems into bounded constrained optimization problem. In the thesis, we construct a correspond?ing afifne scaling matrix according to the optimal conditions so as to overcome the difficult of boxconstrain. By starting from the optimal conditions, we establish a class of quadratic interpolationmodel for the objective function to propose a derivative-free optimization algorithms based ontrust region methods.The trust region method is a common and feasible method for both unconstrained optimiza?tion and constrained optimization. The basic steps of trust region are as follows: First, we settrust-region radius and ifnd the quadratic approximation of the object function. And then we geta tiral step produced by minimizing the quadratic approximation in the trust region. The trust re?gion method assures the global convergence of the algorithm, and it does not require the Hessian(or its approximation) to be indeifnite. In this paper we mainly transfer the nonlinear equationsystem into the least squares problem, by this way we build polynomial interpolation models foreach function in the objective function on the A—poised set. By constructing the afifne scal?ing matirx, the algoirthms are designed to take the advantages of the problem structure and ifndthe next itcrauon poim throughout the aiffnc scaling irusi region method for bound-constraincd nonlinear equations system, we establish a deirvative-free optimization algorithms based on trustregion methods. And by detailed analysis, we provide a complete theory proves and verify theglobal convergence of the algorithm. Meanwhile we also use computer programming to the algo?rithm for numerical expeirments. The numerical results of the proposed algorithm indicate to beeffective.Finally, the last chapter concludes the main results of this paper and proposes some furtherresearch directions.