The Filter Line Search Method For Nonlinear Inequality Constrained Optimization Problems And The Application Of Unary Optimization

In this thesis,several efficient line search algorithms are presented for nonlinear optimiza-tion,including the optimization with nonlinear inequality constraints,the unary optimization with bounded constraints and the unary optimization with linear inequality constraints.Under some reasonable conditions,the global convergence and local convergence rate are established.Numer-ical results show that the proposed algorithm is feasible and efficient.Line search method,which requires very little computation to determine step size,is one of the basic strategies to ensure global convergence of optimization method.Fletcher and Leyffer proposed a filter method as an alternative to the traditional merit function for solving the nonlinear optimization.The underlying concept of the filter method is that the trial points are accepted if there is a sufficient decrease in the objective function or constraints violation function.The significant advantage of the filter method is that it does not need to estimate the penalty parameter which could be difficult to obtain.In chapter 2,a line search filter-SQP method for nonlinear inequality constrained optimiza-tion is presented by combining the line search method and filter method.The search direction is generated by solving the quadratic programming while a backtracking line search procedure is used to generate step size.Under mild conditions,the global convergence properties are ob-tained.Furthermore,the algorithm can overcome the Maratos effect by employing the second order correction step,so that the superlinear local convergence rate is achieved.Some numerical experiments show that the proposed algorithm is effective.In chapter 3,a line search filter-SQP method with Lagrangian function for nonlinear inequal-ity constrained optimization is proposed.The difference between this algorithm and the algorithm in chapter 2 is that the Lagrangian function value instead of the objective function value is used in the filter.By using the Lagrangian function value in the filter,it is shown that the algorithm does not suffer from the Maratos effect without a second order correction,so that local convergence rate is obtained.Some numerical experiments to show the effectiveness of the proposed algorithm.The objective function of the linear robust regression problem and the dual objective function of the entropy problem in information theory are all unary function.In chapter 4,an affine scaling algorithm in association with line search technique is proposed for solving unary optimization subject to bound constraints on variables.By introducing a scaling matrix,the unary optimization with bounded constraints is transformed to a nonlinear system of equations without bounded con-straints.The search direction is produced by the modified Newton method and the approximation of the Hessian matrix at each iteration is updated by using rank-one updates.By using the interior line search technique,each iterates generated by the line search are strictly feasible.The proof of global convergence and local convergence rates are presented under some reasonable assumptions.Numerical results show that the algorithm is feasible.In chapter 5,an affine scaling derivative-free algorithm in association with line search tech-nique is presented for unary optimization subject to linear inequality constraints.In many practice it is impractical to obtain the derivative of function,so in some cases,derivative-free algorithms are used for finding the solution.Under some reasonable conditions,the global convergence is established.Local superlinear convergence rate of the proposed algorithm is established under the strong second order sufficiency condition by using the identification function to overcome the absence of the strict complementarity assumption.