The Chebyshev Center Cutting Plane Methods For Nonsmooth Constrained Optimization With Arbitrary Initial Points

Nonsmooth optimization is an important branch of optimization research,which is widely used in computer vision,signal processing,data mining,engineering design,machine learning and other practical fields.The research difficulty of nonsmooth optimization is significantly greater than that of traditional smooth optimization,since nonsmooth function has poor analytical property and no continuous differentiability.One of the core contents of nonsmooth optimization research is to design numerical solution methods with good theoretical properties,fast convergence rate and stable numerical effect.The Chebyshev center cutting plane method is an effective method to solve nonsmooth optimization,which is an improved form of the classical cutting plane method and has excellent theoretical properties and numerical performance.In this thesis,two Chebyshev center cutting plane methods with arbitrary initial points are proposed for nonsmooth optimization problems with inequality constraints,which further generalize and improve the existing research results of this kind of method,and have important research significance and application value.In chapter 3,a Chebyshev center cutting plane method with arbitrary initial points is proposed by introducing an improvement function which can deal with infeasible initial points.Firstly,the improvement function is used to transform the constrained nonsmooth optimization problem into an unconstrained nonsmooth optimization problem,and then the cutting plane model of the problem is established.Secondly,based on the idea of proximal Chebyshev center cutting plane method,the quadratic programming subproblem of generating search direction is constructed.Thirdly,combined with the idea of phase I-phase II feasible direction method,the feasible descent search conditions are designed to generate new iteration points.The method proposed in this chapter can start from any initial point.If the initial point is not feasible,the first phase aims to increase the feasibility of the iteration points.Once a feasible point is obtained,the algorithm will automatically enter the second phase and execute the feasible point algorithm.Finally,the theoretical properties of the algorithm are analyzed,and the global convergence is proved.In chapter 4,a generalized Chebyshev center cutting plane method with arbitrary initial points is proposed by introducing a more general improvement function.This method is a further extension of the one in Chapter 3,in which the new iteration point is obtained by a more generalized search direction subproblem.The global convergence of the proposed algorithm is analyzed and demonstrated under the theoretical framework similar to that in the previous chapter.In chapter 5,preliminary numerical experiments are carried out on the two algorithms,and the results show that the proposed methods are stable and effective numerically.