Asymmetric Proximal Point Algorithms with Moving Proximal Centers Deren Han (handeren@njnu.edu.cn) School of Mathematical Sciences, Nanjing Normal University Nanjing, 210023, China. September 2, 2014 Deren Han (NJNU) Asymmetric PPA September 2, 2014 1 / 26
Outline Introduction 1 APPA with Moving Proximal Centers 2 Worst-case Rate of Convergence 3 Linear Convergence 4 Two Concrete Algorithms 5 The Saddle Point Problem Multiblock Decomposible Convex Optimization Conclusion 6 Deren Han (NJNU) Asymmetric PPA September 2, 2014 2 / 26
VI ( x − x ∗ ) T F ( x ∗ ) ≥ 0 , ∀ x ∈ Ω , (1) Ω : a nonempty, closed and convex set in R N ; F : a continuous and monotone mapping defined on R N . Deren Han (NJNU) Asymmetric PPA September 2, 2014 3 / 26
VI ( x − x ∗ ) T F ( x ∗ ) ≥ 0 , ∀ x ∈ Ω , (1) Ω : a nonempty, closed and convex set in R N ; F : a continuous and monotone mapping defined on R N . Examples : Equations, Complementarity Problems, Constrained Optimization Problems, Saddle Point Problems, etc. Deren Han (NJNU) Asymmetric PPA September 2, 2014 3 / 26
Proximal Point Algorithms Classical PPA: � � F ( x k + 1 ) + 1 ( x k + 1 − x k ) ( x − x k + 1 ) T ≥ 0 , ∀ x ∈ Ω , (2) c k c k is the proximal parameter and x k is the proximal center. 1 Deren Han (NJNU) Asymmetric PPA September 2, 2014 4 / 26
Proximal Point Algorithms Classical PPA: � � F ( x k + 1 ) + 1 ( x k + 1 − x k ) ( x − x k + 1 ) T ≥ 0 , ∀ x ∈ Ω , (2) c k c k is the proximal parameter and x k is the proximal center. 1 A General Version: F ( x k + 1 ) + M k ( x k + 1 − x k ) ( x − x k + 1 ) T � � ≥ 0 , ∀ x ∈ Ω , (3) where the metric proximal parameter M k ∈ R n × n is positive definite and symmetric. Deren Han (NJNU) Asymmetric PPA September 2, 2014 4 / 26
Proximal Point Algorithms Classical PPA: � � F ( x k + 1 ) + 1 ( x k + 1 − x k ) ( x − x k + 1 ) T ≥ 0 , ∀ x ∈ Ω , (2) c k c k is the proximal parameter and x k is the proximal center. 1 A General Version: F ( x k + 1 ) + M k ( x k + 1 − x k ) ( x − x k + 1 ) T � � ≥ 0 , ∀ x ∈ Ω , (3) where the metric proximal parameter M k ∈ R n × n is positive definite and symmetric. M k := 1 / c k M , c k F ( x k + 1 ) + M ( x k + 1 − x k ) ( x − x k + 1 ) T � � ≥ 0 , ∀ x ∈ Ω . (4) Deren Han (NJNU) Asymmetric PPA September 2, 2014 4 / 26
Role of M in Algorithms: Make the subproblems easier: Preconditioner: T. Pock and A. Chambolle [1]. Decomposable of the subproblems. [1] T. Pock and A. Chambolle. Diagonal preconditioning for first order primal-dual algorithms in convex optimization , IEEE Inter. Con. Comput. Vis., 2011, pp. 1762-1769. Deren Han (NJNU) Asymmetric PPA September 2, 2014 5 / 26
A simple example The saddle point problem: v ∈V Φ( u , v ) := f ( u ) + v T Au − g ( v ) . min u ∈U max (5) Deren Han (NJNU) Asymmetric PPA September 2, 2014 6 / 26
A simple example The saddle point problem: v ∈V Φ( u , v ) := f ( u ) + v T Au − g ( v ) . min u ∈U max (5) PDHG scheme: u k − τ A T v k , u k + 1 ˆ = arg min u ∈U f ( u ) + 1 u k + 1 u k + 1 � 2 , = 2 τ � u − ˆ u k + 1 + θ ( u k + 1 − u k ) , u k + 1 (6) ¯ = v k + σ A ¯ v k + 1 u k + 1 , ˆ = v k + 1 arg min v ∈V g ( v ) + 1 v k + 1 � 2 = 2 σ � v − ˆ Deren Han (NJNU) Asymmetric PPA September 2, 2014 6 / 26
A simple example The saddle point problem: v ∈V Φ( u , v ) := f ( u ) + v T Au − g ( v ) . min u ∈U max (5) PDHG scheme: u k − τ A T v k , u k + 1 ˆ = arg min u ∈U f ( u ) + 1 u k + 1 u k + 1 � 2 , = 2 τ � u − ˆ u k + 1 + θ ( u k + 1 − u k ) , u k + 1 (6) ¯ = v k + σ A ¯ v k + 1 u k + 1 , ˆ = v k + 1 arg min v ∈V g ( v ) + 1 v k + 1 � 2 = 2 σ � v − ˆ PPA point of view 1 − A T τ I m . M := (7) 1 − θ A σ I n Deren Han (NJNU) Asymmetric PPA September 2, 2014 6 / 26
Difficulty in Analysis: Can not bound the progress of consecutive iterations using M -norm; Deren Han (NJNU) Asymmetric PPA September 2, 2014 7 / 26
Difficulty in Analysis: Can not bound the progress of consecutive iterations using M -norm; Remedies: Correction steps; Deren Han (NJNU) Asymmetric PPA September 2, 2014 7 / 26
Difficulty in Analysis: Can not bound the progress of consecutive iterations using M -norm; Remedies: Correction steps; Our Motivation: Schemes without corrections. Deren Han (NJNU) Asymmetric PPA September 2, 2014 7 / 26
Definitions Let F ( · ) be a mapping from R N into R N . Then, F ( · ) is said to be Monotone: if ( F ( x ) − F ( y )) T ( x − y ) ≥ 0 ∀ x , y ∈ R N ; Strongly monotone with modulus µ if ( F ( x ) − F ( y )) T ( x − y ) ≥ µ � x − y � 2 ∀ x , y ∈ R N . Lipschitz continuous: if ∀ x , y ∈ R N ; � F ( x ) − F ( y ) � ≤ L � x − y � Co-coercive with modulus σ > 0 if ( F ( x ) − F ( y )) T ( x − y ) ≥ σ � F ( x ) − F ( y ) � 2 ∀ x , y ∈ R N . (8) Deren Han (NJNU) Asymmetric PPA September 2, 2014 8 / 26
Reformulation M s ≡ 1 2 ( M + M T ) : The symmetric part of M . Define C := M − 1 / 2 ( M − M s ) M − 1 / 2 ; s s K ≡ M 1 / 2 Ω = { M 1 / 2 x : x ∈ Ω } ; s s K ∗ ≡ M 1 / 2 Ω ∗ = { M 1 / 2 x ∗ : x ∗ ∈ Ω ∗ } ; s s F M ( y ) ≡ M − 1 / 2 F ( M − 1 / 2 ˜ y ) , ∀ y ∈ K , ; s s For any x ∈ Ω , we define y := M 1 / 2 x . Thus, s y ′ ≡ M 1 / 2 x ′ , y k + 1 ≡ M 1 / 2 x k + 1 , and y k ≡ M 1 / 2 x k . (9) s s s Then, a vector x ∗ is a solution of the variational inequality (1) if and only if y ∗ := M 1 / 2 x ∗ is a solution of the variational inequality of s finding y ∈ K such that ( y ′ − y ∗ ) T ˜ F M ( y ∗ ) ≥ 0 , ∀ y ′ ∈ K . (10) The solution set of (10) is thus given by K ∗ . Deren Han (NJNU) Asymmetric PPA September 2, 2014 9 / 26
A Useful Lemma Lemma 1 Let the mapping G : R n → R n be co-coercive on a nonempty, closed, convex subset W in R n with modulus σ > 1 / 2 , Then, for any x , y , z ∈ W , we have ( x − y ) T ( G ( z ) − G ( y )) ≥ − ν 2 � x − z � 2 , (11) where ν is an arbitrary number satisfying 0 < 1 2 σ < ν < 1 . (12) Deren Han (NJNU) Asymmetric PPA September 2, 2014 10 / 26
Co-Coerciveness Lemma 2 [Proposition 12.5.3](Facchinei and Pang 2003) The mapping α ˜ F M − C is co-coercive over K with modulus greater than 1 / 2 if either one of the following two conditions holds: F is Lipschitz continuous on Ω with modulus L and a τ ∈ ( 0 , 1 ) 1 exists such that, for all y 1 , y 2 ∈ K , F M ( y 1 ) − M − 1 / 2 MM − 1 / 2 � α ˜ F M ( y 2 ) − α ˜ ( y 2 − y 1 ) � ≤ τ � y 2 − y 1 � ; s s F ( x ) = Qx + q , for some positive semidefinite matrix Q ≡ D + E 2 with D positive definite and E symmetric, and 0 < � I n + H � < 2 , where H ≡ D − 1 / 2 ED − 1 / 2 Deren Han (NJNU) . Asymmetric PPA September 2, 2014 11 / 26 s s
The Exact Version Deren Han (NJNU) Asymmetric PPA September 2, 2014 12 / 26
The Exact Version Remark: The exact version of APPA-MPC (“APPA-MPC-E" in short): The proximal center in x k in classical PPA is shifted to x k − α M − 1 F ( x k ) in (3.1). Deren Han (NJNU) Asymmetric PPA September 2, 2014 12 / 26
Convergence Lemma 3 The subproblem (3.1) of the APPA-MPC-E can be rewritten as ( y ′ − y k + 1 ) T [˜ F M ( y k + 1 ) + ( I + C )( y k + 1 − y k ) + α ˜ F M ( y k )] ≥ 0 , ∀ y ′ ∈ K . (13) Deren Han (NJNU) Asymmetric PPA September 2, 2014 13 / 26
Convergence Lemma 3 The subproblem (3.1) of the APPA-MPC-E can be rewritten as ( y ′ − y k + 1 ) T [˜ F M ( y k + 1 ) + ( I + C )( y k + 1 − y k ) + α ˜ F M ( y k )] ≥ 0 , ∀ y ′ ∈ K . (13) Lemma 4 Let y ∗ be an arbitrary solution point in K ∗ .. Then, we have � y k + 1 − y ∗ � 2 ≤ � y k − y ∗ � 2 − ( 1 − ν ) � y k − y k + 1 � 2 , (14) where ν is an arbitrary number satisfying (12) . Deren Han (NJNU) Asymmetric PPA September 2, 2014 13 / 26
The Inexact Version Deren Han (NJNU) Asymmetric PPA September 2, 2014 14 / 26
Convergence Lemma 5 Let { x k } be the sequence generated by APPA-MPC-I. Then, there exist a positive scalar Γ such that the following inequality holds for any y ′ ∈ K : y k + 1 ) T (˜ y k + 1 − y k )+ α ˜ ( y ′ − ¯ y k + 1 )+( I + C )(¯ F M ( y k )) ≥ − ǫ k � y ′ − ¯ y k + 1 � 2 − Γ ǫ k . F M (¯ (15) Deren Han (NJNU) Asymmetric PPA September 2, 2014 15 / 26
Convergence Lemma 5 Let { x k } be the sequence generated by APPA-MPC-I. Then, there exist a positive scalar Γ such that the following inequality holds for any y ′ ∈ K : y k + 1 ) T (˜ y k + 1 − y k )+ α ˜ ( y ′ − ¯ y k + 1 )+( I + C )(¯ F M ( y k )) ≥ − ǫ k � y ′ − ¯ y k + 1 � 2 − Γ ǫ k . F M (¯ (15) Theorem 6 The sequence { x k } generated by the APPA-MPC-I globally converges to a solution point of the variational inequality (1). Deren Han (NJNU) Asymmetric PPA September 2, 2014 15 / 26
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