Motivation Convergence Numerics Extensions Projected primal-dual splitting for solving constrained convex optimization 1 L. M. Brice˜ no-Arias Universidad T´ ecnica Federico Santa Mar´ ıa LAWOC 2018, Quito 5 September 2018 1 Joint work with Sergio L´ opez Rivera, USM LAWOC 2018 1/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions 1 Motivation Stationary Mean Field Games 2 Algorithm and convergence 3 Numerical experiences Stationary Mean Field Games Numerical simulation 4 Extensions and acceleration LAWOC 2018 2/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Motivation: Mean Field Games (MFG) Introduced by Lasry-Lions (2006) : Given T > 0 and ν ≥ 0 − ∂ t u T − ν ∆ u T + 1 2 |∇ u T | 2 = V ( x, m T ) in T 2 × [0 , T ] in T 2 × [0 , T ] � � ∂ t m T − ν ∆ m T − div m T ∇ u T = 0 in T 2 . m T ( x, 0) = m 0 ( x ) , u T ( x, T ) = Φ( x, m T ( x, T )) , The first is a Hamilton-Jacobi equation backward in time. It characterizes the value of an optimal control problem solved by “a typical small” player and whose cost function V (and final cost Φ) depends on the density of other players at each t ∈ [0 , T ] (at time T ). The second is a Fokker-Plank equation forward in time. It models the evolution of initial density m 0 at the Nash Equilibrium. LAWOC 2018 3/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Motivation: Stationary MFG It is interpreted as the limit behavior of the rescaled MFG when T → ∞ (see Cardaliaguet et. al 2012). SMFG − ν ∆ u ( x ) + 1 2 |∇ u ( x ) | 2 + λ = V ( x, m ( x )) in T 2 in T 2 � � − ν ∆ m ( x ) − div m ( x ) ∇ u ( x ) = 0 � � 0 ≤ m, T 2 m ( x ) dx = 1 , T 2 u ( x ) dx = 0 . Well-posedness is studied in the case of smooth and weak solutions: Lasry, Lions (2006-07), Cirant (2015-16), Gomes, Patrizi,Voskanyan (2014), Gomes, Mitaki (2015)... LAWOC 2018 4/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Motivation: Discretized SMFG If ν > 0, V ( x, · ) is increasing and we suppose that the sta- tionary system admits a unique classical solution, in Achdou, Camilli & Capuzzo Dolcetta (2013) the convergence of a dis- cretized SMFG to the unique solution to the stationary sys- tem as h → 0 is proved ( L p some p < 2). To solve the discretized system, Newton’s method can be used (see Achdou, Camilli & Capuzzo Dolcetta, 2013 and Cacace & Camilli, 2016) if the initial guess is close enough to the solution. The performance of Newton’s method depends heavily on the values of ν : for small values of ν the convergence is much slower and cannot be guaranteed in general since m h can become negative. LAWOC 2018 5/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Motivation: Discretized SMFG is FOC of ( P h ) 2 Discrete optimization problem ( P h ) N h − 1 � � � ˆ h 2 inf b ( m i,j , w i,j ) + F ( x i,j , m i,j ) ( m,w ) ∈M h ×W h i,j =1 � − ν (∆ h m ) i,j + (div h w ) i,j = 0 , ∀ 0 ≤ i, j, ≤ N h − 1 s.t. h 2 � i,j m i,j = 1 . h > 0, N h = 1 /h , M h = R N h × N h , W h = R 4( N h × N h ) . ∆ h : M h → M h , div h : W h → M h are linear ( A , B ) Define | w | 2 2 m , if m> 0 , w ∈ K , � m ˆ 0 , if ( m,w )=(0 , 0) , V ( x,m ′ ) dm ′ . b : ( m,w ) �→ F : ( x,m ) �→ 0 + ∞ , otherwise . K := R + × R − × R + × R − ( w = − mD h u ) 2 Joint work with F.J. Silva (U. Limoges) y D. Kalise (I. College) LAWOC 2018 6/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Motivation: ( P h )’s structure Assume V ( x, · ) increasing ( F ( x, · ) convex). f : m �→ � i,j F ( x i,j , m i,j ) it is convex and smooth. i,j ˆ g : ( m, w ) �→ � b ( m i,j , w i,j ) is convex, l.s.c., and nonsmooth. We recall − ∆ h = A and div h = B . Reformulation of ( P h ) ( P ) inf f ( m ) + g ( m, w ) ( m,w ) ∈M h ×W h s.t. L ( m, w ) = (0 , 1) , � A where � B L := h 2 1 ⊤ 0 LAWOC 2018 7/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Convex non-differentiable optimization problem Problem (P) minimize f ( x ) + g ( x ) + h ( Lx ) . x ∈ R N f : R N → R is differentiable and ∇ f is β − 1 − Lipschitz. g ∈ Γ 0 ( R N ) and h ∈ Γ 0 ( R M ) (l.s.c. convex proper). L is a M × N real matrix. The set of solutions is nonempty. ri(dom h ) ∩ L (ri(dom g )) � = ∅ . � 0 , if x = b otherwise , , b ∈ R M Important case: h = ι { b } : x �→ + ∞ , minimize f ( x ) + g ( x ) . Lx = b LAWOC 2018 8/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Classic approach: ADMM An equivalent formulation is Lx = y f ( x ) + g ( x ) + h ( y ) min from which we define the Augmented Lagrangian ( γ > 0): L γ ( x, y, u ) = f ( x ) + g ( x ) + h ( y ) + u · ( Lx − y ) + γ 2 � Lx − y � 2 . Under qualification conditions x solves (P) iff ( x, Lx, u ) is a saddle point of L γ . From an alternating minimization-maximization method we obtain the classical Alternating Direction method of Multipliers (ADMM) (Gabay-Mercier 80’s): LAWOC 2018 9/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Classic approach: ADMM x k +1 = argmin x L γ ( x, y k , u k ) y k +1 = argmin y L γ ( x k +1 , y, u k ) u k +1 = u k + γ ( Lx k +1 − y k +1 ) . LAWOC 2018 10/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Classic approach: ADMM f ( x ) + g ( x ) + u k · Lx + γ x k +1 = argmin x � 2 � Lx − y k � 2 � h ( y ) − u k · y + γ y k +1 = argmin y � 2 � Lx k +1 − y � 2 � u k +1 = u k + γ ( Lx k +1 − y k +1 ) . LAWOC 2018 10/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Classic approach: ADMM f ( x ) + g ( x ) + u k · Lx + γ x k +1 = argmin x � 2 � Lx − y k � 2 � h ( y ) − u k · y + γ y k +1 = argmin y � 2 � Lx k +1 − y � 2 � u k +1 = u k + γ ( Lx k +1 − y k +1 ) . In the case when h = ι { b } for some b ∈ R M , we have f ( x ) + g ( x ) + u k · Lx + γ x k +1 = argmin x � 2 � Lx − b � 2 � u k +1 = u k + γ ( Lx k +1 − b ) . LAWOC 2018 10/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Drawbacks ADMM The primal iterates ( x k ) k ∈ N do not satisfy the constraints ( Lx k � = b ). Moreover, the first step it is not easy in general (involves L and f + g ). It can be solved efficiently only in specific instances: f + g quadratic, L ⊤ L = α Id. Idea: Try to split the influence of L , f and g in the first step. LAWOC 2018 11/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Drawbacks ADMM The primal iterates ( x k ) k ∈ N do not satisfy the constraints ( Lx k � = b ). Moreover, the first step it is not easy in general (involves L and f + g ). It can be solved efficiently only in specific instances: f + g quadratic, L ⊤ L = α Id. Idea: Try to split the influence of L , f and g in the first step. Given x ∈ R N , prox f x is the unique solution to f ( y ) + 1 2 � x − y � 2 . minimize y ∈ R N Several functions f have an explicit or efficently computable prox f . Examples: � · � 1 , ι C (prox ι C = P C ), d C , etc... LAWOC 2018 11/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Other approaches Problem (P) x ∈ R N f ( x ) + g ( x ) + h ( Lx ) . min Combettes-Pesquet (2012) Let 0 < γ < ( � L � + β ) − 1 , x 0 ∈ R N and u 0 ∈ R M and iterate 1 = prox γg ( x k − γ ( ∇ f ( x k ) + L ⊤ u k )) p k 2 = prox γh ∗ ( u k + γLx k ) p k x k +1 = p k 1 ) − L ⊤ u k − ∇ f ( x k )) 1 − γ ( L ⊤ p k 2 + ∇ f ( p k u k +1 = p k 2 + γ ( Lp k 1 − Lx k ) . In the case f = 0, is the method proposed in BA-Combettes (2011). LAWOC 2018 12/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
Motivation Convergence Numerics Extensions Other approaches Problem (P) case h = ι { b } Lx = b f ( x ) + g ( x ) . min Combettes-Pesquet (2012) Let 0 < γ < ( � L � + β ) − 1 , x 0 ∈ R N and u 0 ∈ R M and iterate 1 = prox γg ( x k − γ ( ∇ f ( x k ) + L ⊤ u k )) p k 2 = u k + γ Lx k − b p k � � x k +1 = p k 1 ) − L ⊤ u k − ∇ f ( x k )) 1 − γ ( L ⊤ p k 2 + ∇ f ( p k u k +1 = p k 2 + γ ( Lp k 1 − Lx k ) . prox γh ∗ = Id − γ prox h/γ ◦ (Id /γ ) = Id − γb . Also the influences of L , f and g have been split, but primal iterates do not satisfy the constraints. The method does not exploit cocoercivity of ∇ f . LAWOC 2018 12/ 31 L. M. Brice˜ no-AriasUniversidad T´ ecnica Federico Santa Mar´ ıa
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