A Bundle Method For A Class Of Nonconvex And Nonsmooth Optimization Problems And Application

Constrained optimization problems have great significance in optimization theory and wide range of applications in many fields such as engineering,national defence and economics.A key scheme of solving constrained optimization problems is to transform them into unconstrained ones,such as penalty function method,Lagrange dual method and phase ?-phase ? method.When the objective or constraint function in the problem has a special structure,the phase ?phase ? method can maximize the computational and theoretical advantages brought by this structure.The resulting unconstrained optimization problem which is transformed from the original nonsmooth constrained problem is nonsmooth,and it is hard to deal with a nonsmooth unconstrained problem.The bundle method is considered to be one of the most effective and stable algorithms for solving nonsmooth optimization problems.According to the characteristics of different problems,a variety of bundle methods have been developed,and they have been widely applied to various classical unconstrained optimization problems,and practical problems in diverse fields such as economy,engineering,and control theory.This dissertation mainly studies a bundle method for solving a class of nonconvex and nonsmooth constrained optimization problems,including the redistributed bundle method for nonsmooth constrained optimization problems,the proximal bundle method for solving nonsmooth composite constrained optimization problems,and the nonsmooth proximal bundle method with the application in H? output feedback control problem.The main content of this dissertation can be summarized as follows.1.Chapter 3 proposes a redistributed bundle method for a class of nonconvex and nonsmooth constrained optimization problems,in which the objective and constraint are nonconvex and nonsmooth lower-C2 functions.By using the improvement function,solving the original constrained problem is equal to solving a series of unconstrained problems,and the new objective function reserves the lower-C2 property.Then by using the properties of lower-C2 function,convexified parameter and term are introduced to improve convexity of the objective function of the subproblem,and a corresponding bundle method is designed.The stability of parameters and the local convergence of the method are given and numerical examples verify the effectiveness of this redistributed bundle method.2.Chapter 4 considers nonconvex and nonsmooth optimization problems with a composite constraint and develops a backtracking bundle method.By utilizing the composite structure of the constraint and improvement function,a local model and a quadratic subproblem are constructed with a concise form whose objective function changes along iterations and has similarly composite structure with the constraint,hence there are two layered cycles in the proposed method.The outer cycle updates the iteration for proximal centre.When iterations go into inner cycle,the bundle set only store information of those trail steps generated based on current proximal centre.In addition,aggregation technique is also used to control the size of bundle information produced from previous steps which makes the size of subproblem small.Under mild assumption,the global convergence of this method from an arbitrary feasible initial point is proved.However,it is difficult to get a feasible initial point sometimes,hence an infeasible bundle method combing the penalty term is proposed and the global convergence of this method is presented.In the numerical experiments,an adaptive subprogram is proposed to find the feasible iteration as soon as possible.Numerical results demonstrate the effectiveness of this method.3.Chapter 5 studies H? output feedback control problem of linear time-invariant systems and proposes a nonsmooth optimization method based on the bundle technique.First,the measurement of system performance and internal stability of close-loop system are quantified by H?norm of closed-loop transfer function and a stabilization channel,and this problem is formulated as a nonconvex and nonsmooth semi-infinite constrained optimization problem.Then by utilizing the relationship between maximum eigenvalue and maximum singular value,the problem is transformed into a constrained optimization problem which is described by the maximum eigenvalue.The nonsmooth constrained problem is transformed into a nonsmooth semi-infinite unconstrained problem and solved.Finally,the convergence to a critical point from an arbitrary feasible initial point is proved and some benchmarks are computed to demonstrate the effectiveness of this method.