Study On Numerical Algorithms For Nonsmooth Semi-infinite Programming Problems

Semi-infinite programming(SIP)problems have numerous applications in civil engineering,electronic circuit design,portfolio problem,robot trajectory planning,vibrating membrane problem,minimal cost control of air pollution.In the past few decades,scholars have conducted in-depth research on the constraint qualifications,optimality conditions,duality theory and numerical algorithms for semi-infinite programming.Most numerical algorithms for solving semi-infinite problems assume that the objective function and constraint function are continuously differentiable.Therefore,how to solve nonsmooth semi-infinite programming problems is a topic worthy of study.This dissertation is devoted to study on numerical algorithms for nonsmooth semi-infinite programming problems,including a discretization algorithm for nonsmooth convex semi-infinite programming problems based on bundle methods,a bundle method for solving nonsmooth convex semi-infinite programming with inexact information and a feasible proximal bundle algorithm with convexification for nonsmooth nonconvex semi-infinite programming.The main results of this dissertation can be summarized as follows:Chapter 3 presents a discretization algorithm for solving a class of nonsmooth convex semiinfinite programming problems that is based on a bundle method.Instead of employing the inexact calculation to evaluate the lower level problem,we apply a discretization scheme.The discretized problem is constantly changing in the iterative process,and its constraints can be regarded as a nonsmooth constraint.The discretized problems can be rewritten as unconstrained problems by using the improvement functions.In particular,the number of constraints of the subproblem is independent of the number of constraints of the discretized problem.Therefore,the method proposed in this chapter overcomes the difficulty of classical discretization method.We apply a refinement step which can be used to reduce the cost of the evaluations for the constraint functions during iteration.Our algorithm allows flexible control the number of constraints of the subproblems via the aggregation technique.Under the Slater constraint qualifcation(SCQ),the obtained solution converges to the solution of original SIP.The computational results show the good performance of the new algorithm.Chapter 4 mainly studies the numerical method to solve the nonsmooth convex semi-infinite problems with inexact information.The new algorithm still uses the discretization method to approximate the lower level problem.Although the objective function and the constraint function are convex with respect to x,it is impossible to guarantee that the cutting plane model is smaller than the original function in the presence of noise,so it is necessary to introduce new conditions to judge whether the noise is too large.In addition,we apply a parametrized improvement function which can be used to control the speed with which feasibility is achieved.In our convergence analysis,we show that every accumulation point of our iteration points is feasible for original problem under the SCQ.The computational results show that the new algorithm has better performance than the bundle method to solve the nonsmooth convex constrained problem with inexact information.Chapter 5 presents a feasible bundle method with convexification for solving nonsmooth nonconvex SIP problems.We derive finite upper bound problems for the SIP by replacing the infinite number of constraints with a finite number of convex upper constraints.Based on the idea of the locally convexification,we define the augmented functions for the objective function and the upper bound constraint functions respectively.To analyse the convergence of the algorithm,we construct an upper envelop model associated with the cutting plane.Under the SCQ,our algorithm can generate a feasible original point and a feasible solution.Under the EMFCQ,our algorithm converges to an approximate KKT point of the original problem.Under slightly stronger assumptions,every accumulation point of the serious iterate sequence is a local approximate solution of the original SIP problem.Preliminary computational results on solving nonconvex nonsmooth constrained optimization problems and nonconvex nonsmooth semi-infinite optimization problems are reported to demonstrate the good performance of the new algorithms.