Convergence Of Proximal Point Methods For Non-convex Optimization

Rockafellar published two papers in 1976,one of which is the proximal point method of maximal monotone operators,the other is the application of the proximal point method to the primal optimality of inequality constrained convex optimization,the optimality of dual problem,and the minimax optimality.Among them,the proximal point method for optimality of dual problem is the famous augmented Lagrange method.The proximal point method for the maximal monotone operator corresponding to minimax optimality is the proximal method of multipliers.These methods have achieved great success in solving various structural convex optimization problems,but the convergence analysis of proximal methods of multipliers for nonconvex optimization problems and augmented Lagrange methods for complicated nonconvex matrix optimization is still blank.These theoretical problems will be studied in this dissertation.The results obtained in this paper can be summarized as follows:1.In Chapter 2,Rockafellar's proximal method of multipliers is applied to equality constrained optimization problems,in which the subproblem has better properties than augmented Lagrange method.It is proved that the convergence rate of the proximal method of multipliers for equality constrained optimization problems is linear under linear independent constraint specifications and second-order sufficient optimality conditions,with ratio constant proportional to 1/c where c is a penalty parameter exceeding the threshold of c*>0.Moreover,when the parameter c is increased to+? the convergence rate of the proximal method of multipliers is superlinear.2.In Chapter 3,we analyze the rate of convergence of the proximal method of multipliers for non-convex nonlinear programming problems.First,we prove,under the strict complementarity condition,that the rate of convergence of the proximal method of multipliers is linear and the ratio constant is proportional to 1/c when the ratio ?(?0,?0)-(?,?)?/c is small enough,which implies that the rate of convergence of the proximal method of multipliers is superlinear when the parameter c increases to+?.Second,we prove that,without strict complementarity condition,the rate of convergence of the proximal method of multipliers is proportional to 1/c when c exceeds a threshold.3.In Chapter 4,The proximal method of multipliers was proposed by Rockafellar(1976)is applied for solving the nonlinear semidefinite programming problem.It is proved that,under the linear independence constraint qualification and the strong second-order sufficiency optimality condition,the rate of convergence of the proximal method of multipliers,for a nonlinear programming problem,is linear and the ratio constant is proportional to 1/c,where c is the penalty parameter that exceeds a threshold c*>0.Moreover,the rate of convergence of the proximal method of multipliers is superlinear when the parameter c increases to+?.4.In Chapter 5,we propose two basic assumptions,under which the rate of convergence of the augmented Lagrange method for a class of composite optimization problems is estimated.We analyze the rate of local convergence of the augmented Lagrangian method for a nonlinear semidefinite nuclear norm composite optimization problem by verifying these two basic assumptions.Without requiring strict complementarity,we prove that,under the constraint nondegeneracy condition and the strong second order sufficient condition,the rate of convergence is linear and the ratio constant is proportional to 1/c,where c is the penalty parameter that exceeds a threshold c>0.The analysis is based on variational analysis about the proximal mapping of the nuclear norm and the projection operator onto the cone of positively semidefinite symmetric matrices.