| Nonconvex optimization problems involve many fields in practical applications,such as image processing,power systems,compressed sensing and machine learning.Thus,the study of nonconvex optimization problems has aroused the interest and enthusiasm of many scholars.However,with the advent of the era of big data,large-scale problems are increasing.In the face of huge databases,how to solve large-scale nonconvex optimization problems has a strong theoretical and application background,and the research is of great significance.In this dissertation,we propose splitting algorithms for two-block optimization problems with general linear constraints and for two-block indivisible optimization problems with equality linear constraints.The main work and results are summarized below.First,studies a class of general linear equality and inequality constraint nonconvex two-block optimization,direct augmented Lagrangian processing of equality and inequality constraints,learns from the alternating direction method of multipliers(ADMM)-sequence quadratic optimization(SQO)algorithm,and linearizes the augmented Lagrangian term of inequality constraint,by splitting and reducing the dimension of the iterative subproblem,gets two small quadratic optimization(QO)subproblems,the descent direction of the augmented Lagrangian function is thus generated.Furthermore,with the help of Armijo line search,new iteration points are generated,designed a new ADMM-SQO algorithm for the discussed problem.Under general assumptions,obtain the convergence result of the algorithm.Finally,we conduct numerical experiments to verify the numerical effectiveness of the proposed algorithm.Second,a class of two nonconvex nonsmooth nonseparable optimization problems with equality constraints is studied.Based on the Peaceman-Rachford(PR)split algorithm,combining linearization,regularization and Armijo line search technique,two linear regularization PR split algorithms for the problem discussed are proposed.Using the idea of the PR splitting algorithm decomposes the subproblem associated with the augmented Lagrangian method into two small-scale subproblems.In order to facilitate the solution of the justmentioned subproblems and make them have good theoretical properties,the smooth terms in the objective functions are linearized,and then the regularization terms are added to each subproblem respectively.Under usual conditions,the global convergence and the iteration complexity of the two proposed methods are proved.Finally,numerical experiments show that the two proposed algorithms are effective. |
