Proximal Point Algorithms And Applications In Optimization

In this dissertation, we focus on adapting proximal point algorithms to solve set-valued equations, variational inequalities problems and optimization problems. It is organized as follows:In Chapter 1, we introduce proximal point algorithms' framework and current research situation, optimization problems and variational inequalities' background and current research situation, and some basic concepts and lemmas, which are used in this dissertation.In Chapter 2, we investigate set-valued equations 0 ∈ T(x) in the case when T : Rⁿ → 2^Rn is pseudomonotone in Rⁿ. Our algorithm is as followsAlgorithm 2.1Step 1. Set x⁰ ∈ Rⁿ be an initial vector.Step 2. Given x^k and β_k ∈ [β,∞)(β > 0), find (x|^)k, e^k such thatx^k + e^k∈(x|^)k + β_kT((x|^)k), (1) where e^k denotes error criterion conforming to||e^k|| ≤ η_k||x^k -(x|^)k||, (2) (3)Step 3. If (x|^)k = x^k, stop; otherwise letx^k+1= (x|^)k-e^k Return to step 2.In compare with [11], the projection step is not requirement in step 3 of Algorithm 2.1, where let x^k+1 = P_K((x|^)k — e^k), where K denotes domain of T, P_k(.) denotes the projection on K in [11]. Furthmore, we prove the superlinear convergence of Algorithm 2.1 under mild growth condition.