Semidefinite Programming Relaxation vs Polyhedral Homotopy ...

Semidefinite Programming Relaxation vs Polyhedral Homotopy Methodfor Problems Involving PolynomialsWorkshop on Advances in OptimizationTokyo Institute of Technology, April 19-21, 2007Masakazu KojimaNumerical resultsTokyo Institute of TechnologyWorkshop on Advances in Optimization, April 19-21, 2007 – p.1/24

Contents1. PHoMpara — Parallel implementation of the polyhedralhomotopy method ([1] Gunji-Kim-Fujisawa-Kojima ’06)2. SparsePOP — Matlab implementation of SDP relaxationfor sparse POPs ([2] Waki-Kim-Kojima-Muramatsu ’05)3. Numerical comparison between the SDP relaxation andthe polyhedral homotopy method([1]+[2]+[3] Mevissen-Kojima-Nie-Takayama)4. Concluding remarksSDP = Semidefinite Program or ProgrammingPOP = Polynomial Optimization ProblemWorkshop on Advances in Optimization, April 19-21, 2007 – p.2/24

Contents1. PHoMpara — Parallel implementation of the polyhedralhomotopy method ([1] Gunji-Kim-Fujisawa-Kojima ’06)2. SparsePOP — Matlab implementation of SDP relaxationfor sparse POPs ([2] Waki-Kim-Kojima-Muramatsu ’05)3. Numerical comparison between the SDP relaxation andthe polyhedral homotopy method([1]+[2]+[3] Mevissen-Kojima-Nie-Takayama)4. Concluding remarksSDP = Semidefinite Program or ProgrammingPOP = Polynomial Optimization ProblemWorkshop on Advances in Optimization, April 19-21, 2007 – p.3/24

The polyhedral homotopy methodImplementation on a single CPU:PHCpack [Verschelde]HOM4PS [Li-Li-Gao]PHoM [Gunji-Kim-Kojima-Takeda-Fujisawa-Mizutani]Workshop on Advances in Optimization, April 19-21, 2007 – p.4/24

The polyhedral homotopy methodImplementation on a single CPU:PHCpack [Verschelde]HOM4PS [Li-Li-Gao]PHoM [Gunji-Kim-Kojima-Takeda-Fujisawa-Mizutani]Suitable for parallel computation — all isolated solutionscan be computed independently in parallel.PHoMpara [Gunji, Kim, Fujisawa and Kojima] — NextLeykin, Verschelde and ZhuangWorkshop on Advances in Optimization, April 19-21, 2007 – p.4/24

Numerical results: Hardware — PC cluster (AMD Athlon 2.0GHz)Problem cpu time speedup(#sol) #CPUs in second rationoon-10 1 62,672 1.0(59,029) 40 1,797 34.9eco-14 1 22,653 1.0(4,096) 40 626 36.2Workshop on Advances in Optimization, April 19-21, 2007 – p.5/24

Numerical results: Hardware — PC cluster (AMD Athlon 2.0GHz)Problem cpu time speedup(#sol) #CPUs in second rationoon-10 1 62,672 1.0(59,029) 40 1,797 34.9eco-14 1 22,653 1.0(4,096) 40 626 36.2Problem #CPUs in second speedupnoon-12 40 49,458(531,417)eco-16 40 12,051(16,384)Workshop on Advances in Optimization, April 19-21, 2007 – p.5/24

Contents1. PHoMpara — Parallel implementation of the polyhedralhomotopy method ([1] Gunji-Kim-Fujisawa-Kojima ’06)2. SparsePOP — Matlab implementation of SDP relaxationfor sparse POPs ([2] Waki-Kim-Kojima-Muramatsu ’05)3. Numerical comparison between the SDP relaxation andthe polyhedral homotopy method([1]+[2]+[3] Mevissen-Kojima-Nie-Takayama)4. Concluding remarksSDP = Semidefinite Program or ProgrammingPOP = Polynomial Optimization ProblemSparsePOP (Waki-Kim-Kojima-Muramatsu ’06) = Lasserre’sSDP relaxation ’01 + “structured sparsity” — c-sparsityWorkshop on Advances in Optimization, April 19-21, 2007 – p.6/24

POP min. f 0 (x) s.t. f j (x) ≥ 0 or = 0 (j = 1,...,m).Example:f 0 (x) = ∑ nk=1 (−x2 k )f j (x) = 1 − x 2 j − 2x 2 j+1 − x 2 n (j = 1,...,n − 1).Hf 0 (x) : the n × n Hessian mat. of f 0 (x),Jf ∗ (x) : the m × n Jacob. mat. of f ∗ (x) = (f 1 (x),...,f m (x)) T ,R : the csp matrix, the n × n density pattern matrix ofI + Hf 0 (x) + Jf ∗ (x) T Jf ∗ (x) (no cancellation in ’+’).[Jf ∗ (x) T Jf ∗ (x)] ij ≠ 0 iff x i and x j are in a common constraint.Workshop on Advances in Optimization, April 19-21, 2007 – p.7/24

POP min. f 0 (x) s.t. f j (x) ≥ 0 or = 0 (j = 1,...,m).Example:f 0 (x) = ∑ nk=1 (−x2 k )f j (x) = 1 − x 2 j − 2x 2 j+1 − x 2 n (j = 1,...,n − 1).Hf 0 (x) : the n × n Hessian mat. of f 0 (x),Jf ∗ (x) : the m × n Jacob. mat. of f ∗ (x) = (f 1 (x),...,f m (x)) T ,R : the csp matrix, the n × n density pattern matrix ofI + Hf 0 (x) + Jf ∗ (x) T Jf ∗ (x) (no cancellation in ’+’).[Jf ∗ (x) T Jf ∗ (x)] ij ≠ 0 iff x i and x j are in a common constraint.Example with n = 6:⎛⎞⋆ ⋆ 0 0 0 ⋆⋆ ⋆ ⋆ 0 0 ⋆the csp matrix R =0 ⋆ ⋆ ⋆ 0 ⋆0 0 ⋆ ⋆ ⋆ ⋆⎜⎟⎝ 0 0 0 ⋆ ⋆ ⋆ ⎠⋆ ⋆ ⋆ ⋆ ⋆ ⋆Workshop on Advances in Optimization, April 19-21, 2007 – p.7/24

POP min. f 0 (x) s.t. f j (x) ≥ 0 or = 0 (j = 1,...,m).Example:f 0 (x) = ∑ nk=1 (−x2 k )f j (x) = 1 − x 2 j − 2x 2 j+1 − x 2 n (j = 1,...,n − 1).Hf 0 (x) : the n × n Hessian mat. of f 0 (x),Jf ∗ (x) : the m × n Jacob. mat. of f ∗ (x) = (f 1 (x),...,f m (x)) T ,R : the csp matrix, the n × n density pattern matrix ofI + Hf 0 (x) + Jf ∗ (x) T Jf ∗ (x) (no cancellation in ’+’).[Jf ∗ (x) T Jf ∗ (x)] ij ≠ 0 iff x i and x j are in a common constraint.Workshop on Advances in Optimization, April 19-21, 2007 – p.8/24

POP min. f 0 (x) s.t. f j (x) ≥ 0 or = 0 (j = 1,...,m).Example:f 0 (x) = ∑ nk=1 (−x2 k ) — — — c-sparsef j (x) = 1 − x 2 j − 2x 2 j+1 − x 2 n (j = 1,...,n − 1).Hf 0 (x) : the n × n Hessian mat. of f 0 (x),Jf ∗ (x) : the m × n Jacob. mat. of f ∗ (x) = (f 1 (x),...,f m (x)) T ,R : the csp matrix, the n × n density pattern matrix ofI + Hf 0 (x) + Jf ∗ (x) T Jf ∗ (x) (no cancellation in ’+’).[Jf ∗ (x) T Jf ∗ (x)] ij ≠ 0 iff x i and x j are in a common constraint.POP : c-sparse (correlatively sparse) ⇔ The n × n csp matrixR = (R ij ) allows a symbolic sparse Cholesky factorization (undera row & col. ordering like a symmetric min. deg. ordering).Workshop on Advances in Optimization, April 19-21, 2007 – p.8/24

Sparse (SDP) relaxation = Lasserre (2001) + c-sparsityPOP min. f 0 (x) s.t. f j (x) ≥ 0 or = 0 (j = 1,...,m), c-sparse.⇓A sequence of c-sparse SDP relaxation problems dependingon the relaxation order r= 1, 2,...;Workshop on Advances in Optimization, April 19-21, 2007 – p.9/24

Sparse (SDP) relaxation = Lasserre (2001) + c-sparsityPOP min. f 0 (x) s.t. f j (x) ≥ 0 or = 0 (j = 1,...,m), c-sparse.⇓A sequence of c-sparse SDP relaxation problems dependingon the relaxation order r= 1, 2,...;(a) Under a moderate assumption,opt. sol. of SDP → opt sol. of POP as r → ∞.(b) r = ⌈“the max. deg. of poly. in POP”/2⌉+0 ∼ 3 is usuallylarge enough to attain opt sol. of POP in practice.(c) Such an r is unknown in theory except ∃ special cases.(d) The size of SDP increases rapidly as r → ∞.Workshop on Advances in Optimization, April 19-21, 2007 – p.9/24

Contents1. PHoMpara — Parallel implementation of the polyhedralhomotopy method ([1] Gunji-Kim-Fujisawa-Kojima ’06)2. SparsePOP — Matlab implementation of SDP relaxationfor sparse POPs ([2] Waki-Kim-Kojima-Muramatsu ’05)3. Numerical comparison between the SDP relaxation andthe polyhedral homotopy method([1]+[2]+[3] Mevissen-Kojima-Nie-Takayama)4. Concluding remarksSDP = Semidefinite Program or ProgrammingPOP = Polynomial Optimization ProblemWorkshop on Advances in Optimization, April 19-21, 2007 – p.10/24

A POP alkyl from globalibmin − 6.3x 5 x 8 + 5.04x 2 + 0.35x 3 + x 4 + 3.36x 6sub.to − 0.820x 2 + x 5 − 0.820x 6 = 0,0.98x 4 − x 7 (0.01x 5 x 10 + x 4 ) = 0, −x 2 x 9 + 10x 3 + x 6 = 0,x 5 x 12 − x 2 (1.12 + 0.132x 9 − 0.0067x 2 9) = 0,x 8 x 13 − 0.01x 9 (1.098 − 0.038x 9 ) − 0.325x 7 = 0.574,x 10 x 14 + 22.2x 11 = 35.82, x 1 x 11 − 3x 8 = −1.33,lbd i ≤ x i ≤ ubd i (i = 1, 2,...,14).14 variables, 7 poly. equality constraints with deg. 3.Workshop on Advances in Optimization, April 19-21, 2007 – p.11/24

A POP alkyl from globalibmin − 6.3x 5 x 8 + 5.04x 2 + 0.35x 3 + x 4 + 3.36x 6sub.to − 0.820x 2 + x 5 − 0.820x 6 = 0,0.98x 4 − x 7 (0.01x 5 x 10 + x 4 ) = 0, −x 2 x 9 + 10x 3 + x 6 = 0,x 5 x 12 − x 2 (1.12 + 0.132x 9 − 0.0067x 2 9) = 0,x 8 x 13 − 0.01x 9 (1.098 − 0.038x 9 ) − 0.325x 7 = 0.574,x 10 x 14 + 22.2x 11 = 35.82, x 1 x 11 − 3x 8 = −1.33,lbd i ≤ x i ≤ ubd i (i = 1, 2,...,14).14 variables, 7 poly. equality constraints with deg. 3.SparseDense (Lasserre)r ǫ obj ǫ feas cpu ǫ obj ǫ feas cpu2 1.0e-02 7.1e-01 1.8 7.2e-3 4.3e-2 14.43 5.6e-10 2.0e-08 23.0 out of memoryǫ obj = approx.opt.val. − lower bound for opt.val.ǫ feas = the maximum error in the equality constraintsWorkshop on Advances in Optimization, April 19-21, 2007 – p.11/24

Systems of polynomial equationsIs the (sparse) SDP relaxation useful to solve systems ofpolynomial equations?The answer depends on:how sparse the system of polynomial equations is,the maximum degree of polynomials.Workshop on Advances in Optimization, April 19-21, 2007 – p.12/24

Systems of polynomial equationsIs the (sparse) SDP relaxation useful to solve systems ofpolynomial equations?The answer depends on:how sparse the system of polynomial equations is,the maximum degree of polynomials.2 types of systems of polynomial equations(a) Benchmark test problems from Verschelde’s homepage;Katsura, cyclic — not c-sparse(b) Systems of polynomials arising from discretization of anODE and a DAE (Differential Algebraic Equations)— c-sparseWorkshop on Advances in Optimization, April 19-21, 2007 – p.12/24

Katsura n system of polynomial equations; n = 8 case0 = −x 1 + 2x 2 9 + 2x 2 8 + 2x 2 7 + · · · + 2x 2 2 + x 2 1,0 = −x 2 + 2x 9 x 8 + 2x 8 x 7 + 2x 7 x 6 + · · · + 2x 3 x 2 + 2x 2 x 1 ,· · · · · · not c-sparse0 = −x 8 + 2x 9 x 2 + 2x 8 x 1 + 2x 7 x 2 + 2x 6 x 3 + 2x 5 x 4 ,1 = 2x 9 + 2x 8 + 2x 7 + 2x 6 + 2x 5 + 2x 4 + 2x 3 + 2x 2 + x 1 .Workshop on Advances in Optimization, April 19-21, 2007 – p.13/24

Katsura n system of polynomial equations; n = 8 case0 = −x 1 + 2x 2 9 + 2x 2 8 + 2x 2 7 + · · · + 2x 2 2 + x 2 1,0 = −x 2 + 2x 9 x 8 + 2x 8 x 7 + 2x 7 x 6 + · · · + 2x 3 x 2 + 2x 2 x 1 ,· · · · · · not c-sparse0 = −x 8 + 2x 9 x 2 + 2x 8 x 1 + 2x 7 x 2 + 2x 6 x 3 + 2x 5 x 4 ,1 = 2x 9 + 2x 8 + 2x 7 + 2x 6 + 2x 5 + 2x 4 + 2x 3 + 2x 2 + x 1 .Numerical results on SparsePOP (WKKM 2004)n obj.funct. relax. order r cpu∑8 xi ↑ 1 0.08∑8 x2i ↓ 2 7.1∑11 xi ↑ 1 0.14∑11 x2i ↓ 2 101.3A formulation∑in terms of a POPmax ni=1 x i or min ∑ ni=1 x2 isub.to Katsura n system , −5 ≤ x i ≤ 5 (i = 1,...,n).Different objective functions ⇒ different solutions.Workshop on Advances in Optimization, April 19-21, 2007 – p.13/24

Katsura n system of polynomial equations; n = 8 case0 = −x 1 + 2x 2 9 + 2x 2 8 + 2x 2 7 + · · · + 2x 2 2 + x 2 1,0 = −x 2 + 2x 9 x 8 + 2x 8 x 7 + 2x 7 x 6 + · · · + 2x 3 x 2 + 2x 2 x 1 ,· · · · · · not c-sparse0 = −x 8 + 2x 9 x 2 + 2x 8 x 1 + 2x 7 x 2 + 2x 6 x 3 + 2x 5 x 4 ,1 = 2x 9 + 2x 8 + 2x 7 + 2x 6 + 2x 5 + 2x 4 + 2x 3 + 2x 2 + x 1 .Numerical results on SparsePOP (WKKM 2004)n obj.funct. relax. order r cpu∑8 xi ↑ 1 0.08∑8 x2i ↓ 2 7.1∑11 xi ↑ 1 0.14∑11 x2i ↓ 2 101.3Workshop on Advances in Optimization, April 19-21, 2007 – p.14/24

Katsura n system of polynomial equations; n = 8 case0 = −x 1 + 2x 2 9 + 2x 2 8 + 2x 2 7 + · · · + 2x 2 2 + x 2 1,0 = −x 2 + 2x 9 x 8 + 2x 8 x 7 + 2x 7 x 6 + · · · + 2x 3 x 2 + 2x 2 x 1 ,· · · · · · not c-sparse0 = −x 8 + 2x 9 x 2 + 2x 8 x 1 + 2x 7 x 2 + 2x 6 x 3 + 2x 5 x 4 ,1 = 2x 9 + 2x 8 + 2x 7 + 2x 6 + 2x 5 + 2x 4 + 2x 3 + 2x 2 + x 1 .Numerical results on SparsePOP (WKKM 2004)n obj.funct. relax. order r cpu∑8 xi ↑ 1 0.08∑8 x2i ↓ 2 7.1∑11 xi ↑ 1 0.14∑11 x2i ↓ 2 101.3Numerical results on HOM4PS (Li-Li-Gao 2002)n cpu sec. #solutions8 1.9 25611 209.1 2048Workshop on Advances in Optimization, April 19-21, 2007 – p.14/24

cyclic n system of polynomial equations: n = 5 case0 = x 1 + x 2 + x 3 + x 4 + x 5 ,0 = x 1 x 2 + x 2 x 3 + x 3 x 4 + x 4 x 5 + x 5 x 1 , not c-sparse0 = x 1 x 2 x 3 + x 2 x 3 x 4 + x 3 x 4 x 5 + x 4 x 5 x 1 + x 5 x 1 x 2 ,0 = x 1 x 2 x 3 x 4 + x 2 x 3 x 4 x 5 + x 3 x 4 x 5 x 1 + x 4 x 5 x 1 x 2 + x 5 x 1 x 2 x 3 ,0 = −1 + x 1 x 2 x 3 x 4 x 5 .Numerical results on SparsePOP: obj.funct.+lbd, ubd on x in obj.funct. relax. order r cpu5 ∑ x i ↑ 3 1.836 ∑ x i ↑ 4 753.2Workshop on Advances in Optimization, April 19-21, 2007 – p.15/24

cyclic n system of polynomial equations: n = 5 case0 = x 1 + x 2 + x 3 + x 4 + x 5 ,0 = x 1 x 2 + x 2 x 3 + x 3 x 4 + x 4 x 5 + x 5 x 1 , not c-sparse0 = x 1 x 2 x 3 + x 2 x 3 x 4 + x 3 x 4 x 5 + x 4 x 5 x 1 + x 5 x 1 x 2 ,0 = x 1 x 2 x 3 x 4 + x 2 x 3 x 4 x 5 + x 3 x 4 x 5 x 1 + x 4 x 5 x 1 x 2 + x 5 x 1 x 2 x 3 ,0 = −1 + x 1 x 2 x 3 x 4 x 5 .Numerical results on SparsePOP: obj.funct.+lbd, ubd on x in obj.funct. relax. order r cpu5 ∑ x i ↑ 3 1.836 ∑ x i ↑ 4 753.2Numerical results on HOM4PS (Li-Li-Gao)n cpu sec.#solutions5 0.1 706 0.2 156Workshop on Advances in Optimization, April 19-21, 2007 – p.15/24

Discretization of Mimura’s ODE with 2 unknowns u, v : [0, 5] → Ru xx = −(20/9)(35 + 16u − u 2 )u + 20uv,v xx = (1/4)((1 + (2/5)v)v − uv),u x (0) = u x (5) = v x (0) = v x (5) = 0,Discretize:x i = i∆x (i = 0, 1, 2,... ), u x (x i ) ≈ (u(x i+1 ) − u(x i−1 ))/(2∆x).Workshop on Advances in Optimization, April 19-21, 2007 – p.16/24

Discretization of Mimura’s ODE with 2 unknowns u, v : [0, 5] → Ru xx = −(20/9)(35 + 16u − u 2 )u + 20uv,v xx = (1/4)((1 + (2/5)v)v − uv),u x (0) = u x (5) = v x (0) = v x (5) = 0,Discretize:x i = i∆x (i = 0, 1, 2,... ), u x (x i ) ≈ (u(x i+1 ) − u(x i−1 ))/(2∆x).Discretized system of polynomials with ∆x = 1:f 1 (u,v) = 76.8u 1 + u 3 + 35.6u 2 1 − 20.0u 1 v 1 − 2.22u 3 2,f 2 (u,v) = −1.25v 1 + v 2 + 0.25u 1 v 1 − 0.1v1,2f 3 (u,v) = u 1 + 75.8u 2 + u 3 + 35.6u 2 2 − 20.0u 2v 2 − 2.22u 3 2 ,f 4 (u,v) = v 1 − 2.25v 2 + v 3 + 0.25u 2 v 2 − 0.1v2,2f 5 (u,v) = u 2 + 75.8u 3 + u 4 + 35.6u 2 3 − 20.0u 3 v 3 − 2.22u 2 3,f 6 (u,v) = v 2 − 2.25v 3 + v 4 + 0.25u 3 v 3 − 0.1v3,2f 7 (u,v) = u 3 + 76.8u 4 + 35.6u 2 4 − 20.0u 4 v 4 − 2.22u 3 4,f 8 (u,v) = v 3 − 1.25v 4 + 0.25u 4 v 4 − 0.1v4.2Here u i = u(x i ), v i = v(x i ) (i = 0, 1, 2, 3, 4, 5),u 0 = u 1 , u 5 = u 4 , v 0 = v 1 and v 5 = v 4 .⇒ c-sparseWorkshop on Advances in Optimization, April 19-21, 2007 – p.16/24

Discretization of Mimura’s ODE with 2 unknowns u, v : [0, 5] → Ru xx = −(20/9)(35 + 16u − u 2 )u + 20uv,v xx = (1/4)((1 + (2/5)v)v − uv),u x (0) = u x (5) = v x (0) = v x (5) = 0,Discretize:x i = i∆x (i = 0, 1, 2,... ), u x (x i ) ≈ (u(x i+1 ) − u(x i−1 ))/(2∆x).Workshop on Advances in Optimization, April 19-21, 2007 – p.17/24

Discretization of Mimura’s ODE with 2 unknowns u, v : [0, 5] → Ru xx = −(20/9)(35 + 16u − u 2 )u + 20uv,v xx = (1/4)((1 + (2/5)v)v − uv),u x (0) = u x (5) = v x (0) = v x (5) = 0,Discretize:x i = i∆x (i = 0, 1, 2,... ), u x (x i ) ≈ (u(x i+1 ) − u(x i−1 ))/(2∆x).Numerical results on SparsePOP∆x n obj.funct. relax. order r cpu1.0 8 ∑ r i u(x i ) ↑ 3 11.30.5 18 ∑ r i u(x i ) ↑ 3 57.8Here r i ∈ (0, 1) : random numbers.Workshop on Advances in Optimization, April 19-21, 2007 – p.17/24

Discretization of Mimura’s ODE with 2 unknowns u, v : [0, 5] → Ru xx = −(20/9)(35 + 16u − u 2 )u + 20uv,v xx = (1/4)((1 + (2/5)v)v − uv),u x (0) = u x (5) = v x (0) = v x (5) = 0,Discretize:x i = i∆x (i = 0, 1, 2,... ), u x (x i ) ≈ (u(x i+1 ) − u(x i−1 ))/(2∆x).Numerical results on SparsePOP∆x n obj.funct. relax. order r cpu1.0 8 ∑ r i u(x i ) ↑ 3 11.30.5 18 ∑ r i u(x i ) ↑ 3 57.8Here r i ∈ (0, 1) : random numbers.1212101088664422∆x = 1.000 0.5 1 1.5 2 2.5 3 3.5 4 4.5 500 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5∆x = 0.5Workshop on Advances in Optimization, April 19-21, 2007 – p.17/24

Discretization of Mimura’s ODE with 2 unknowns u, v : [0, 5] → Ru xx = −(20/9)(35 + 16u − u 2 )u + 20uv,v xx = (1/4)((1 + (2/5)v)v − uv),u x (0) = u x (5) = v x (0) = v x (5) = 0,Discretize:x i = i∆x (i = 0, 1, 2,... ), u x (x i ) ≈ (u(x i+1 ) − u(x i−1 ))/(2∆x).Numerical results on SparsePOP∆x n obj.funct. relax. order r cpu1.0 8 ∑ r i u(x i ) ↑ 3 11.30.5 18 ∑ r i u(x i ) ↑ 3 57.8Here r i ∈ (0, 1) : random numbers.Workshop on Advances in Optimization, April 19-21, 2007 – p.18/24

Discretization of Mimura’s ODE with 2 unknowns u, v : [0, 5] → Ru xx = −(20/9)(35 + 16u − u 2 )u + 20uv,v xx = (1/4)((1 + (2/5)v)v − uv),u x (0) = u x (5) = v x (0) = v x (5) = 0,Discretize:x i = i∆x (i = 0, 1, 2,... ), u x (x i ) ≈ (u(x i+1 ) − u(x i−1 ))/(2∆x).Numerical results on SparsePOP∆x n obj.funct. relax. order r cpu1.0 8 ∑ r i u(x i ) ↑ 3 11.30.5 18 ∑ r i u(x i ) ↑ 3 57.8Here r i ∈ (0, 1) : random numbers.Numerical results on HOM4PS∆x n cpu sec. #solutions #real solutions1.0 8 2.2 1296 2220.5 18 167.7 10,077,696 not traced(M.vol.) (M.vol.) (M.cells=1089)Workshop on Advances in Optimization, April 19-21, 2007 – p.18/24

Discretization of DAE with 3 unknowns y 1 ,y 2 ,y 3 : [0, 2] → Ry ′ 1 = y 3 , 0 = y 2 (1 − y 2 ), 0 = y 1 y 2 + y 3 (1 − y 2 ) − t, y 1 (0) = y 0 1.2 solutions : y(t) = (t, 1, 1) and y(t) = (y 0 1 + t 2 2, 0,t).Workshop on Advances in Optimization, April 19-21, 2007 – p.19/24

Discretization of DAE with 3 unknowns y 1 ,y 2 ,y 3 : [0, 2] → Ry ′ 1 = y 3 , 0 = y 2 (1 − y 2 ), 0 = y 1 y 2 + y 3 (1 − y 2 ) − t, y 1 (0) = y 0 1.2 solutions : y(t) = (t, 1, 1) and y(t) = (y1 0 + t 2 2, 0,t).Numerical results on SparsePOP— c-sparsey 0 1 ∆t n obj.funct. relax. order r cpu0 0.02 297 ∑ y 2 (t i ) ↑ 2 30.91 0.02 297 ∑ y 1 (t i ) ↑ 2 33.9Workshop on Advances in Optimization, April 19-21, 2007 – p.19/24

21.81.61.41.210.80.60.40.200 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Solution: y(t) = (t, 1, 1)Workshop on Advances in Optimization, April 19-21, 2007 – p.20/24

32.521.510.500 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Solution: y(t) = (y 0 1 + t 2 2, 0,t)Workshop on Advances in Optimization, April 19-21, 2007 – p.21/24

Contents1. PHoMpara — Parallel implementation of the polyhedralhomotopy method ([1] Gunji-Kim-Fujisawa-Kojima ’06)2. SparsePOP — Matlab implementation of SDP relaxationfor sparse POPs ([2] Waki-Kim-Kojima-Muramatsu ’05)3. Numerical comparison between the SDP relaxation andthe polyhedral homotopy method([1]+[2]+[3] Mevissen-Kojima-Nie-Takayama)4. Concluding remarksSDP = Semidefinite Program or ProgrammingPOP = Polynomial Optimization ProblemWorkshop on Advances in Optimization, April 19-21, 2007 – p.22/24

Some essential differences between HomotopyContinuation and (sparse) SDP Relaxation — 1:Workshop on Advances in Optimization, April 19-21, 2007 – p.23/24

Some essential differences between HomotopyContinuation and (sparse) SDP Relaxation — 1:(a) HC works on C n while SDPR on R n .(b) HC aims to compute all isolated solutions; in SDPR,computing all isolated solutions is possible but expensive.(c) SDPR can process inequalities.Workshop on Advances in Optimization, April 19-21, 2007 – p.23/24

Some essential differences between HomotopyContinuation and (sparse) SDP Relaxation — 2:Workshop on Advances in Optimization, April 19-21, 2007 – p.24/24

Some essential differences between HomotopyContinuation and (sparse) SDP Relaxation — 2:(d) SDPR is sensitive to degrees of polynomials of a POPbecause the SDP relaxed problem becomes larger rapidlyas they increase.⇒ SDPR can be applied to POPs with lower degreepolynomials such as degree ≤ 4 in practice.(e) HC fits parallel computation more than SDPR.(f) The effectiveness of sparse SDPR depends on thec-sparsity; for example, discretization of ODE, DAE,Optimal control problem and PDE.Workshop on Advances in Optimization, April 19-21, 2007 – p.24/24

Some essential differences between HomotopyContinuation and (sparse) SDP Relaxation — 2:(d) SDPR is sensitive to degrees of polynomials of a POPbecause the SDP relaxed problem becomes larger rapidlyas they increase.⇒ SDPR can be applied to POPs with lower degreepolynomials such as degree ≤ 4 in practice.(e) HC fits parallel computation more than SDPR.(f) The effectiveness of sparse SDPR depends on thec-sparsity; for example, discretization of ODE, DAE,Optimal control problem and PDE.Thank you!Workshop on Advances in Optimization, April 19-21, 2007 – p.24/24