Semidefinite Programming Relaxation vs Polyhedral Homotopy ...

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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 = <strong>Semidefinite</strong> Program or <strong>Programming</strong>POP = Polynomial Optimization ProblemWorkshop on Advances in Optimization, April 19-21, 2007 – p.10/24
Page 1 and 2: Semidefinite Programming Relaxation
Page 3 and 4: Contents1. PHoMpara — Parallel im
Page 5 and 6: The polyhedral homotopy methodImple
Page 7 and 8: Numerical results: Hardware — PC
Page 9 and 10: POP min. f 0 (x) s.t. f j (x) ≥ 0
Page 11 and 12: POP min. f 0 (x) s.t. f j (x) ≥ 0
Page 13: Sparse (SDP) relaxation = Lasserre
Page 17 and 18: A POP alkyl from globalibmin − 6.
Page 19 and 20: Systems of polynomial equationsIs t
Page 21 and 22: Katsura n system of polynomial equa
Page 23 and 24: Katsura n system of polynomial equa
Page 25 and 26: cyclic n system of polynomial equat
Page 27 and 28: Discretization of Mimura’s ODE wi
Page 33 and 34: Discretization of DAE with 3 unknow
Page 35 and 36: 21.81.61.41.210.80.60.40.200 0.2 0.
Page 37 and 38: Contents1. PHoMpara — Parallel im
Page 39 and 40: Some essential differences between
Page 41 and 42: Some essential differences between

Semidefinite Programming Relaxation vs Polyhedral Homotopy ...

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