三个可分离算子凸优化的平行分裂增广Lagrange乘子法-XV-.PDFVIP

XV - 1 凸凸凸优优优化化化和和和单单单调调调变变变分分分不不不等等等式式式的的的收收收缩缩缩算算算法法法 第十五讲: 三个可分离算子凸优化 的平行分裂增广Lagrange 乘子法 Parallel splitting augmented Lagrangian methods for convex optimization with three separable operators 南京大学数学系 何炳生 hebma@nju.edu.cn The context of this lecture is based on the publication [6] XV - 2 1 Convex Optimization with 3 separable operators We consider the linearly constrained convex optimization with 3 separable operators: min {θ (x)+ θ (y)+ θ (z) | Ax + By + Cz = b, x ∈ X , y ∈ Y , z ∈ Z} (1.1) where θ (x) : ℜ → ℜ, θ (y) : ℜ → ℜ, θ (z) : ℜ → ℜ are convex; A ∈ ℜ , B ∈ ℜ , C ∈ ℜ , b ∈ ℜ ; X ⊂ ℜ , Y ⊂ ℜ and Z ⊂ ℜ are given convex set. Let n = n + n + n . This optimization problem is equivalent to find (x , y , z , λ ) ∈ Ω, such that   θ (x) − θ (x ) + (x − x ) (−A λ ) ≥ 0,      θ (y) − θ (y ) + (y − y ) (−B λ ) ≥ 0, ∀ (x, y, z, λ) ∈ Ω, (1.2)   θ (z) − θ (z ) + (z − z ) (−C λ ) ≥ 0,     (λ − λ ) (Ax + By + Cz − b) ≥ 0, where Ω = X × Y × Z × ℜ . XV - 3 By denoting    x   −A λ      x    y   −B λ   