The convergence of stationary iterations with indefinite splitting

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1 The convergence of stationary iterations with indefinite splitting Michael C. Ferris Joint work with: Tom Rutherford and Andy Wathen University of Wisconsin, Madison 6th International Conference on Complementarity Problems Berlin, Germany August 7, 2014 Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

2 The problems VI(F, C): x C, F (x ), x x 0, x C MOPEC: x, y: x i solves min x i K i (x i,y) θ(x i, x i, y), i y solves VI (F (x, ), C) A 1 A 1,2 A 1,p E 1 A 2,1 A A p-1,p E p-1 A p,1 A p,p-1 A p E p F 1 F p-1 F p D Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

3 Strongly Convex Nash Equilibria min x 1 0 min x 2 0 No solution for θ 1: 1 2 x 1 2 θx 1 x 2 4x 1 s.t. x 1 + x x 2 2 x 1 x 2 3x 2 x 1 (x 2 ) = (θx 2 + 4) +, x 2 (x 1 ) = (x 1 + 3) + Solution 4 3 θ < 1: x 1 = 4+3θ 1 θ, x 2 = x Solution θ 4 3 : x 1 = 0, x 1 = 3 Jacobi works provided θ < 1, but theory fails Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

4 The Issues This is not the optimality conditions of a single optimization problem: 1 1 θ x 1 4 x p 1 1 p x 2 3 x 2 The matrix A in general is never diagonally dominant except in trivial cases Iterations based on succesive inversion of local blocks (or successive optimization of local strategies) can converge. We establish sufficient conditions which guarantee convergence of block Jacobi and block Gauss-Seidel iterations for such matrices. Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

5 Iteration with Indefinite Splitting Ax = b Splitting A = P N naturally leads to a stationary iteration of the form x 0 arbitrary, Px k+1 = Nx k + b, k = 0, 1,... This iteration may or may not converge; simply applicable sufficient conditions for convergence are particularly valuable. Most well-known such conditions are diagonal dominance: if the preconditioner is P = diag(a) (leading to Jacobi iteration) or P is the lower triangular part of A (leading to Gauss-Seidel iteration), then convergence is guaranteed if the strict diagonal dominance condition a i,i > a i,j, i = 1,..., n (1) j=1,...,n,j i is satisfied by A = {a i,j, i, j = 1,..., n}. Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

6 Weaker diagonal dominance conditions For irreducible matrices, it is well documented that the weaker condition a i,i a i,j, i = 1,..., n (2) j=1,...,n,j i is also sufficient provided strict inequality holds for at least one row index, i The condition (1) or (2) also guarantees that A R n n is invertible, so a unique solution exists. Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

7 The Setting We focus on matrices of the form A 1 A 1,2 A 1,p E 1. A 2,1 A A = Ap 1,p E p 1 A p,1 A p,p 1 A p E p F 1 F p 1 F p D (3) where A i = [ Qi B T i B i 0 ], i = 1,..., p (4) with Q i = Q T i R n i n i positive definite and B i R m i n i of full rank m i < n i for each i (m i > 0). These conditions guarantee that each A i is invertible. The submatrix D R s s, s 0 must be symmetric and invertible (unless s = 0). Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

8 Existing Block Theory For the blocked matrix (3) a result of Feingold and Varga (1962) applies: If A is block irreducible and ( A 1 i 2 ) 1 E i 2 + A i,j 2, i = 1,..., p (5) and ( D 1 2 ) 1 j=1,...,p,j i j=1,...,p,j i F i 2 (6) with strict inequality in (6) or for at least one index, i, in (5), then A is invertible (existence and uniqueness) Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

9 Relation to Iteration Before considering these conditions in more detail, consider a block Jacobi or block Gauss-Seidel iteration based on the splitting with P = A 1 A 2... A p D or P = A 1 A 2,1 A A p,1 A p,p-1 A p F 1 F p-1 F p D Asymptotic convergence of the corresponding stationary (or simple) iteration will be guaranteed for any starting vector if all of the eigenvalues, λ, of I P 1 A lie strictly inside the unit disc. Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

10 The link Such eigenvalues satisfy (I P 1 A)x = λx, x 0 or equivalently (A + (λ 1)P)x = 0, x 0. In the case of block Jacobi, asymptotic convergence will be guaranteed if there does not exist any λ with λ 1 such that the matrix λa 1 A 1,2 A 1,p E 1. A 2,1 λa A(λ) = A + (λ 1)P = Ap 1,p E p 1 A p,1 A p,p 1 λa p E p F 1 F p 1 F p λd is singular. But ( (λa i ) 1 2 ) 1 = λ ( A 1 i 2 ) 1 ( A 1 i 2 ) 1 whenever λ 1 with a identical argument holding for D. Hence satisfaction of the conditions (5),(6) not only guarantees invertibility of A, but also guarantees covergence of the block Jacobi iteration. Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

11 Another piece Let µ i denote the smallest eigenvalue of the positive definite matrix Q i and γ i denote the smallest eigenvalue of the positive definite (Schur complement) matrix B i Q 1 i Bi T, then there are no eigenvalues of [ Qi B A i = i T ] B i 0 in the interval ( ( ) ) 1 µ i µ 2i + 4γ i µ i, µ i 2 which contains the origin. Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

12 Finally... If the matrix A given by (3),(4) is block irreducible, then it is invertible and the block Jacobi and block Gauss-Seidel iterations for a linear system Ax = b converge to x for any starting vector if { ) } 1 min ( µ 2i + 4γ i µ i µ i, µ i 2 E i 2 + A i,j 2, i = 1,..., p (7) and d j=1,...,p,j i j=1,...,p,j i F i 2 (8) with strict inequality in (8) 1 or for at least one index, i, in (7). 1 d is the absolute value of eighenvalue of D closest to origin Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

13 A Simplification If for each i = 1,..., p, γ i 2µ i then A is invertible and the block Jacobi and block Gauss-Seidel iterations for a linear system Ax = b converge to x for any starting vector if µ i E i 2 + A i,j 2, i = 1,..., p (9) and d j=1,...,p,j i j=1,...,p,j i F i 2 (10) with strict inequality in (10) or for at least one index, i, in (9). Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

14 Simple example min x 1 0.5x 2 1 θx 1 x 2 4x 1 s.t. 2x x 2 = 1 min x 2 0.5x 2 2 x 1 x 2 3x θ x p 1 = x 2 3 Solution: x 1 = 0.2, x 2 = 2.8, p 1 = 1.4θ 2.1 Jacobi works, but convergence guaranteed if θ < 3/2. Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

15 Extensions Can also prove same result for SOR schemes Can apply regularization (proximal iterations) on the constraints: for ɛ i, α i > 0 [ Qi + α A i = i I Bi T ], ɛ i I B i can be used for some subset (or indeed all) of the indices i = 1,..., p. This increases the value of µ i and γ i in the above and strengthens the theory No rates given here Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

16 Strongly convex optimization 1 min x 1 2 x 1 2 x 1 x 2 4x 1 s.t. x 1 + x 2 = 1 1 min x 2 2 x 2 2 x 1 x 2 3x α x ɛ 1 1 p 1 = x 2 3 Solution: x 1 = 1, x 2 = 2, p 1 = 7 Jacobi fails: after 4 steps back at (1, 1) T Modified Jacobi α 1, ɛ 1 = 0.1 solves in 50 steps Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

17 Extension to Inequality Case (QP iterates) Each Jacobi iterate replaces system of equations with solution of a small(er) scale Quadratic Programs: min x i x i T Q i x i + c i (x i ) T x i s.t. B i x i = b(x i ) Solution is typically found in (many) fewer iterations than unconstrained case Can use any QP solver for subproblems (and/or VI solver) Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

18 Extension to Inequality Case (Normal Map) Replace systems of equations by (normal map formulation of) complementarity problems i: Q i ((x i ) + ) B T i p + c i + x i (x i ) + = 0 B i ((x i ) + ) = b i Note this is a natural extension of the case considered above Choose active set at each iteration based on prediction from previous iteration Need to employ a regularization on the subproblem constraints Apply theory to all selections of the resulting linear systems Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

19 Extension to Inequality Case (Interior Point) Apply interior point code to solve each QP subproblem min x i x i T Q i x i + c i (x i ) T x i s.t. B i x i = b(x i ) Resulting systems to solve have form [ Qi + X 1 i W i Bi T B i 0 ] i = 1,..., p where X and W are diagonals of iterates and slacks at previous iteration Update barrier parameter after each Jacobi step Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

20 Economic Application Model is a partial equilibrium, geographic exchange model. Goods are distinguished by region of origin. There is one unit of region r goods. These goods may be consumed in region r or they may be exported. Each region solves: min f r (X, T ) s.t. F (X, T ) = 0, T j = T j, j r X,T r where f r (X, T ) is a quadratic form and F (X, T ) is linear and defines X uniquely as a function of T. F (X, T ) defines an equilibrium; here it is simply a set of equations, not a complementarity problem Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

21 Results Gauss-Seidel residuals Iteration deviation Tariff revenue region SysOpt MOPEC Note that competitive solution produces much less revenue than system optimal solution Model has non-convex objective, but each subproblem is solved globally (lindoglobal) Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

22 Conclusions MOPEC problems capture complex interactions between optimizing agents Policy implications addressable using MOPEC MOPEC available to use within the GAMS modeling system New sufficient conditions for existence, uniqueness and convergence shown in special cases Many new settings available for deployment; need for more theoretic and algorithmic enhancements Ferris (Univ. Wisconsin) MOPEC splitting ICCP / 22

Up to this point, our main theoretical tools for finding eigenvalues without using det{a λi} = 0 have been the trace and determinant formulas

Finding Eigenvalues Up to this point, our main theoretical tools for finding eigenvalues without using det{a λi} = 0 have been the trace and determinant formulas plus the facts that det{a} = λ λ λ n, Tr{A}