Fast ADMM for Sum of Squares Programs Using Partial Orthogonality

Transcription

1 Fast ADMM for Sum of Squares Programs Using Partial Orthogonality Antonis Papachristodoulou Department of Engineering Science University of Oxford with Yang Zheng (U Oxford) and Giovanni Fantuzzi (Imperial CL)

2 OULINE 1 Introduction SOS programs, SDPs and partial orthogonality 3 A fast ADMM algorithm 4 Numerical examples 5 Conclusion

3 OULINE 1 Introduction SOS programs, SDPs and partial orthogonality 3 A fast ADMM algorithm 4 Numerical examples 5 Conclusion

4 5. Conclusion Polynomial Optimisation Is px ( ) = px α 0? α Unconstrained polynomial optimisation α min p( x) x n max x n γ s.t. px ( ) γ 0 p( x) γ x Other areas: graph partition, matrix copositivity test, etc.

5 Analysis in Dynamical Systems Lyapunov: x = f( x), f(0) = 0, f Lipschitz, n xt ( ) D x, 0 D If V : D R cont. differentiable s.t. V(0) = 0, and V( x) > 0 in D\ {0} V( x) V ( x) = f( x) > 0 in D\{0} x then x = 0 is asymptotically stable. V(x) n R Checking if p( x) 0 is NP-hard when deg( p) 4, p [ x].

6 Positive Polynomials and Sum of Squares Shor: px ( ) is Sum of Squares px ( ) is positive semi-definite m = qi ( x) px i = 1 px ( ) ( ) 0 Worst-case polynomial NP-hard when deg( p) 4 time complexity Can be solved using semidefinite programming (SDP) (Parrilo), which can be setup and solved using SOSOOLS: Can also search for unknown coefficients of p( x) so that px ( ) is SOS P., Anderson, Valmorbida, Prajna, Seiler, Parrilo, 013

7 SOSOOLS Formulates and solves the equivalent semidefinite programme (SDP) SOSOOLS SOS program SOSP solution SDP SDP solution SeDuMi SDP3 CSDP SDPNAL SDPA SOSOOLS

8 x Van der Pol Oscillator = x 1 x = x (1 x ) x 1 1

9 Sum of Squares and Semidefinite Programming m ( ) = qi ( x) = vd( x) Xvd( x) i = 1 px Proposition: p is SOS X 0, v ( x) monomials of degree d deg( p) Example: p( x, x ) = 5x + x x x x x x x X v d ( x) x q q q x = x q q q x xx q q q xx i i Expand out and match coefficients: q = 5, q =, q + q = 1, q = 1, q = Require that X 0

10 What restricts SOS methods? Computation m ( ) i ( ) d( ) d( ), i = 1 px = q x = v x Xv x x { α n α } d v ( x) = x d n+ d 1 x x x x x x x, of size. d d = 1 n 1 1 n Reduce the size of v d (x)? Newton polytope, diagonal inconsistency, symmetry properties, and facial reduction Alternative formulation (DSOS/SDSOS), leading to linear programs (LPs) or second-order-cone programs (SOCPs) Solving the SDPs by more scalable first-order methods (FOMs) at the cost of reduced accuracy. n

11 OULINE 1 Introduction SOS programs, SDPs and partial orthogonality 3 A fast ADMM algorithm 4 Numerical examples 5 Conclusion

12 5. Conclusion Formalism t sx ( ) g( x) ug( x) is SOS = 0 i = 1 iff s( x) = v ( x) Xv ( x) = X, v ( x) v ( x), X 0 d d d d 1 if β + γ = α Indicator matrices ( Bα ) βγ, = 0 otherwise hen v ( x) v ( x) B x = d d α t α And therefore g0, α ug i i, α x = Bα, X x α i = 1 α t Coefficient Matching Conditions g0, α ug i i, α = Bα, X i = 1 i i α α α

13 5. Conclusion Our strategy min wu s. t. sx ( ) = g( x) ug( x), 0 min wu t t n i i Bα X + ug i i, α = g0, α, α d i = 1 i = 1 s Σ[ x] X 0 s. t.,,, Size of the SDP Partial orthogonality in the resulting Semidefinite Programme A fast first order (ADMM) algorithm that exploits the partial orthogonality trades accuracy for scalability. Implemented in CDCS-sos, downloadable

14 5. Conclusion Standard SDP formulation A Notation min wu Standard SDP formulation t n α + i i, α = 0, α, α d i = 1 s. t. B, X ug g,, X 0 g1,1 gt,1 vec( B1 ) =, A = 1 g1, m g t, m vec( Bm) min ξ c ξ s. t. Aξ = b, ξ K A A A R m ( t+ N ) [ 1 ], m b [ g,, g ] R, 0,1 0, m t+ N c [ w,0,,0] R, ξ K t+ N [ u, vec( X) ] R, R t S +

15 5. Conclusion Standard SDP formulation min ξ c ξ s. t. Aξ = b, ξ K A g1,1 gt,1 vec( B1 ) =, A = 1 g1, m g t, m vec( Bm) m A [ A A ] R + 1 ( t N ) 1 he density of nonzero elements in is O. n A d AA is diagonal. he m m matrix AA is of "diagonal plus low rank" form.

16 Partial Orthogonality g1,1 gt,1 vec( B1 ) t g0, α ug i i, α = Bα, A1 =, A = i = 1 g1, m g t, m vec( Bm) AA 1 1 AA 1. D. Bertsimas, R. M. Freund, and X. A. Sun, An accelerated first order method for solving sos relaxations of unconstrained polynomial optimization problems, Optimization Methods and Software, vol. 8, no. 3, pp , D. Henrion and J. Malick, Projection methods in conic optimization, in Handbook on Semidefinite, Conic and Polynomial Optimization. Springer, 01, pp

17 OULINE 1 Introduction SOS programs, SDPs and partial orthogonality 3 A fast ADMM algorithm 4 Numerical examples 5 Conclusion

18 ADMM for the homogeneous self-dual embedding 3. ADMM for the Homogenous Self dual Embedding min ξ s.t. min c ξ wu Aξ = b ξ K t n α + i i, α = 0, α, α d i = 1 s. t. B, X ug g,, X 0 max y s.t. by Ay z c z + = K KK conditions Primal feasibility Aξ = b, ξ K Dual feasibility A y + z = c, z K Zero-duality gap c ξ b y = 0 z 0 A c ξ s = A 0 b y κ c b 0 τ

19 ADMM for the Homogeneous self-dual embedding 3. ADMM for the Homogenous Self dual Embedding z 0 A c ξ s = A 0 b y κ c b 0 τ τ, κ are two non-negative and complementary variables Notational simplicity z ξ 0 A c m v s, u y, Q A 0 b, C = K R R κ τ c b 0 + Feasibility problem find ( uv, ) s.t. v = Qu ( uv, ) C C

20 ADMM for the Homogeneous self-dual embedding find ( uv, ) s.t. v = Qu, ( u, v) C C ADMM steps (similar to solver SCS [1]) k+ 1 1 k k ˆ ( ) ( ) u = I + Q u + v k+ 1 k+ 1 k u = P( uˆ v ) C v = v uˆ + u k+ 1 k k+ 1 k+ 1 Affine projection Conic projection 0 A c Q A 0 b c b 0 Affine projection takes advantage of diagonal plus low rank for efficient computation [1] O Donoghue, B., Chu, E., Parikh, N. and Boyd, S. (016). Conic optimization via operator splitting and homogeneous self-dual embedding. Journal of Optimization heory and Applications, 169(3),

21 ADMM for the Homogeneous self-dual embedding k+ 1 1 k k ˆ ( ) ( ) u = I + Q u + v k+ 1 k+ 1 k u = P( uˆ v ) C v = v uˆ + u k+ 1 k k+ 1 k+ 1 Affine projection i i i In affine projection, one needs to invert/factorize m m I + AA R Recall that AA is "diagonal plus low rank" ( I + AA ) = ( P + AA ) = P P A( I + A P A) A P t t I + A P A R 1 1 Since in typical SOS programs t m, partial orthogonality can provide a speed-up in the affine projection.

22 OULINE 1 Introduction SOS programs, SDPs and partial orthogonality 3 A fast ADMM algorithm 4 Numerical examples 5 Conclusion

23 CDCS: Cone Decomposition Conic Solver 4. CDCS: Cone Decomposition Conic Solver An open source MALAB solver for partially decomposable conic programs; CDCS supports constraints on the following cones: Free variables non-negative orthant second-order cone the positive semidefinite cone. Input-output format is aligned with SeDuMi; SOS module can be invoked with option sos Works with latest YALMIP release. Syntax: [x,y,z,info] = cdcs(at,b,c,k,opts); Download from

24 5. Conclusion Numerical Examples Implemented in CDCS, invoked by the option sos Compared to SCS and SeDuMi, SDPA, SDP3 Stopping condition for CDCS and SCS: ε = 10-4, 000 iterations min ( xx + x x x 3 x x ) x i j i j j i j 1 < i j n n s. t. x 1 i i = 1

25 5. Conclusion Numerical Examples min ( xx + x x x 3 x x ) x i j i j j i j 1 < i j n n s. t. x 1 i i = 1

26 5. Conclusion Lyapunov function construction Randomly generated polynomial dynamical systems of degree three with a locally asymptotically stable equilibrium at the origin.

27 5. Conclusion Nuclear Receptor Signalling ested our algorithm on a model of nuclear receptor signalling. he model has 37 states, and a cubic vector field ested local stability in the ball with radius 0.01 Constructing a quadratic Lyapunov function Dimension of the resulting SDP: Cone size: K.s = [ ], K.f = 1406; Number of constraints: , the size of A:, (5535 is the number of variables) Interior point solvers First-order solvers SeDuMi SDP3 SDPA CSDP SCS CDCS Failed Failed Failed Failed 151 s 57.3 s Failed: Out of memory Numerical results on a PC with.8 GHz Intel Core i7 and 8 GB of RAM

28 OULINE 1 Introduction SOS programs, SDPs and partial orthogonality 3 A fast ADMM algorithm 4 Numerical examples 5 Conclusion

29 5. Conclusion Conclusions SOS programs have found in a wide range of applications, e.g. polynomial optimization problems and nonlinear systems analysis questions. One fundamental challenge is the poor scalability of SOS programs to large instances Fortunately, the resulting SDPs are highly sparse and structured, (partial orthogonality in this talk) We have developed a fast ADMM algorithm to solve this class of SDPs at the cost of reduced accuracy. CDCS: Download from

30 hank you for your attention! Q & A 1. Zheng, Y., Fantuzzi, G., & Papachristodoulou, A. (017). Fast ADMM for sum of squares programs using partial orthogonality. arxiv preprint arxiv: Zheng, Yang, Giovanni Fantuzzi, and Antonis Papachristodoulou. "Exploiting Sparsity in the Coefficient Matching Conditions in Sum of Squares Programming using ADMM." IEEE Control Systems Letters (017). Paper hb03.3