Abstract
Proximal point algorithm is a type of method widely used in solving optimization problems and some practical problems such as machine learning in recent years. In this paper, a framework of accelerated proximal point algorithm is presented for convex minimization with linear constraints. The algorithm can be seen as an extension to G u ¨ ler’s methods for unconstrained optimization and linear programming problems. We prove that the sequence generated by the algorithm converges to a KKT solution of the original problem under appropriate conditions with the convergence rate of O 1 / k 2 .
Highlights
PPA) is a type of method widely used in solving optimization problems, fixed point problems, maximal monotone operator problems, and so on. e framework of the proximal point method is closely related to many algorithms
In recent years, combining the idea of the proximal point method or the proximal terms with some existing algorithms shows that it can improve the performance of the original algorithms in a certain extent. e main step of the PPA is to compute a subproblem consisting of proximal point operator
A framework of accelerated PPA is presented for constrained convex optimization. e algorithm can be seen as an extension to Gu€ler’s methods for unconstrained optimization and linear programming problems
Summary
In the literature of Birge et al [12], the authors further generalized Gu€ler’s accelerated proximal point method to unconstrained nonsmooth convex optimization problems. Much effort has been made to accelerate various first-order methods and the convergence analysis for linearly constrained convex optimization. Ke and Ma [14] proposed an accelerated augmented Lagrangian method for solving the linearly constrained convex programming and showed its convergence rate is O(1/k2). Xu [15] proposed two accelerated methods for solving structured linearly constrained convex programming and discussed their convergence rate under different conditions. Zhang et al [16] applied the proximal method of multipliers for equality constrained optimization problems and proved that, under linear independence constraint qualification and the second-order sufficiency optimality condition, it is linearly convergent.
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