Two Variants Of The Forward-backward Splitting Algorithm For Solving Convex Optimization Problem

Convex minimization problem are a class of relatively important problems in the field of optimization, and each of the convex optimization problem can be reformulate into a minimization problem, witch objective function is the sum of a smooth function f and a non-smooth one g. The forward-backward splitting algorithm is a basis, among all known iterative schemes. However, the backward-forward splitting algorithm was deformed to improve the rate of convergence. In this thesis, we mainly studies two variants of the forward-backward splitting algorithm for solving the convex optimization problem and make use of practical problems. Main content as follow:In the first chapter, the brief introduction of the convex optimization problems and iterative schemes is given. It also presents their values and their state-of-the-art de-velopments. The importance of the minimization problem of the sum f+g was briefly introduced. Some variants of the forward-backward splitting algorithm introduced for solv-ing the problem. In addition, some basic knowledge of needs for the research be listed in this chapter.For solving the problem of minimizing the sum f+g, the second chapter is gives a variant of the forward-backward splitting method and the corresponding algorithm of adaptive selection the Lipschitz constant in each iteratives. The convergence rate of two algorithm is proves, and demos by numerical experiments.For solving the problem of minimizing the sum f+g, the third chapter is gives another variant of the forward-backward splitting algorithm and proves the convergence when f is continuous, but don’t suppose Lipschitz continuity. Meanwhile, when the parameter sequence have a positive Lower, the0(1/k)-rate of convergence is proved, especially if f is strongly convex, then the convergence is R-linear.