Alternating Minimization Algorithm For Minimizing The Sum Of Strongly Convex Function And Weakly Convex Function

The separable convex optimization problem is a very important kind of optimization problem,which has a very important application in image and signal processing and other practical problems.The Alternating Minimization Algorithm(AMA)proposed by Paul Tseng to solve the separable convex optimization problem whose objective function is the sum of strongly convex function and convex function,but in some practical problems,the objective function is the sum of strongly convex function and weakly convex function,so this dissertation mainly uses AMA to solve the separable convex optimization problem of "strongly + weakly".This dissertation mainly includes the following two parts.In the first part,we present AMA for solving a class of "strongly + weakly" separable convex optimization problems.Under mild assumptions,we show that the sequence generated by AMA is globally convergent to the solution of this optimization problem.Moreover,if one of the function in the optimization problem is smooth,it can be proved that the convergence rate of sequence generated by AMA is linear.In the second part,we first generalized the model in Part one,and use AMA to solve this model.Under some mild assumptions about the strongly convex modulus and the weakly convex modulus,we show that the sequence generated by AMA is globally convergent to the stationary point of this generalized model.Moreover,if one of the objective functions is smooth,the convergent rate of the sequence generated by AMA is linear.