Two New Customized Proximal Point Algorithms Without Relaxation for Linearly Constrained Convex Optimization

Improved Lagrangian-PPA based prediction correction method for linearly constrained convex optimization

“…where r and s are the parameters in (15). With this preparation, we are able to prove the following lemma which is key to our proposed method.…”

“…In [17], Ma et al proposed a class of customized proximal point algorithms for linearly constrained convex optimization problems, which contained several existing customized proximal point algorithms. A new customized proximal point algorithm for linearly constrained convex optimization problem has been proposed in [15], which do not involve relaxation step.…”

“…Lagrangian-PPA based contraction method. In this section, we construct a contraction method based on the predictor (15). The sequence {u k } generated by the contraction methods in [7,8,10,12] is Fejér monotone with respect to the solution set Ω * .…”

“…and H is a symmetric positive definite matrix. For given u k and the predictor ũk generated by (15), we let u k+1 (α) be an α-dependent new iterate defined by…”

“…Lemma 3.2. For given u k = (x k , λ k ), let ũk be generated by (15) and u k+1 (α) be defined by (23). Then for any α ≥ 0, we have…”