Parallelizing SCIP-SDP via the UG framework Tristan Gally joint work with Marc E. Pfetsch, Chuen-Teck See, and Yuji Shinano SFB 805 Control of Uncertainty in Load-Carrying Structures in Mechanical Engineering January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 1
Mixed-Integer Semidefinite Programming ◮ Mixed-integer semidefinite program MISDP b T y inf m � A i y i − A 0 � 0, s.t. i =1 y i ∈ Z ∀ i ∈ I for symmetric matrices ( A i ) i ≤ m . ◮ Linear constraints, bounds, multiple blocks possible within SDP-constraint. January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 2
Contents Applications Solution Approaches SCIP-SDP Parallelization Numerical Results Conclusion January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 3
Contents Applications Solution Approaches SCIP-SDP Parallelization Numerical Results Conclusion January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 4
Truss Topology Design v i ∈ R d : i = 1, ... , n � � ◮ n nodes V = ◮ n f free nodes V f ⊂ V ◮ m possible bars E ⊆ {{ v i , v j } : i � = j } , | E | = m ◮ Force f ∈ R d f for d f = d · n f ground structure 3x3 January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 5
Truss Topology Design v i ∈ R d : i = 1, ... , n � � ◮ n nodes V = ◮ Cross-sectional areas x ∈ R m + for bars that minimize the volume ◮ n f free nodes V f ⊂ V while creating a “stable” truss ◮ m possible bars ◮ Stability is measured by the E ⊆ {{ v i , v j } : i � = j } , | E | = m 2 f T u with node compliance 1 ◮ Force f ∈ R d f for d f = d · n f displacements u ground structure 3x3 optimal structure January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 5
Truss Topology Design ◮ E : set of possible bars TTD-SDP [Ben-Tal, Nemirovski 1997] ◮ ℓ e : length of bar e � min ℓ e x e ◮ x : cross-sectional areas e ∈ E ◮ f : external force � f T � 2 c max � 0 s.t. f A ( x ) ◮ c max : upper bound on compliance x e ≥ 0 ∀ e ∈ E ◮ A e : bar stiffness matrices with stiffness matrix A ( x ) = � A e ℓ e x e . e ∈ E January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 6
Truss Topology Design ◮ In practice, we won’t be able to produce/buy bars of any desired size. ⇒ Only allow cross-sectional areas from a discrete set A . January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 7
Truss Topology Design ◮ In practice, we won’t be able to produce/buy bars of any desired size. ⇒ Only allow cross-sectional areas from a discrete set A . TTD-MISDP [Koˇ cvara 2010, Mars 2013] � � a x a min ℓ e e e ∈ E a ∈A � f T � 2 c max � 0, s.t. f A ( x ) � x a e ≤ 1 ∀ e ∈ E , a ∈A x a e ∈ { 0, 1 } ∀ e ∈ E , a ∈ A , a x a where A ( x ) = � � A e ℓ e e . e ∈ E a ∈A January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 7
Compressed Sensing ◮ Task: find sparsest solution to underdetermined system of linear equations, i.e. a solution of ℓ 0 -Minimization min � x � 0 s.t. Ax = b x ∈ R n where � x � 0 := | supp ( x ) | . January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 8
Compressed Sensing ◮ Task: find sparsest solution to underdetermined system of linear equations, i.e. a solution of ℓ 0 -Minimization min � x � 0 s.t. Ax = b x ∈ R n where � x � 0 := | supp ( x ) | . ◮ Under certain conditions on A , this is equivalent to ℓ 1 -Minimization min � x � 1 s.t. Ax = b x ∈ R n January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 8
Compressed Sensing One such condition is the (asymmetric) restricted isometry property (RIP): α 2 k � x � 2 2 ≤ � Ax � 2 2 ≤ β 2 k � x � 2 ∀ x : � x � 0 ≤ k 2 January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 9
Compressed Sensing One such condition is the (asymmetric) restricted isometry property (RIP): α 2 k � x � 2 2 ≤ � Ax � 2 2 ≤ β 2 k � x � 2 ∀ x : � x � 0 ≤ k 2 Theorem [Foucart, Lai 2008] If Ax = b has a solution x with � x � 0 ≤ k and the RIP of order 2 k holds for √ β 2 2 k 2 − 3 ≈ 2.6569, < 4 α 2 2 k then x is the unique solution of both the ℓ 0 - and the ℓ 1 -optimization problem. January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 9
Compressed Sensing One such condition is the (asymmetric) restricted isometry property (RIP): α 2 k � x � 2 2 ≤ � Ax � 2 2 ≤ β 2 k � x � 2 ∀ x : � x � 0 ≤ k 2 Theorem [Foucart, Lai 2008] If Ax = b has a solution x with � x � 0 ≤ k and the RIP of order 2 k holds for √ β 2 2 k 2 − 3 ≈ 2.6569, < 4 α 2 2 k then x is the unique solution of both the ℓ 0 - and the ℓ 1 -optimization problem. But NP -hard in general to compute optimal constants . . . January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 9
Compressed Sensing The optimal constant α 2 k (and similarly β 2 k ) for α 2 k � x � 2 2 ≤ � Ax � 2 2 ≤ β 2 k � x � 2 ∀ x : � x � 0 ≤ k 2 can be computed via the following non-convex cardinality-constrained QP: RIP-QP � Ax � 2 inf 2 � x � 2 s.t. 2 = 1, � x � 0 ≤ k . January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 10
Compressed Sensing Idea: Substitute X := xx ⊤ to obtain the non-convex rank-constrained MISDP RIP-Rk1-MISDP Tr( A T AX ) min s.t. Tr( X ) = 1 − z j ≤ X jj ≤ z j ∀ j ≤ n n � z j ≤ k j =1 Rank( X ) =1 X � 0 z ∈{ 0, 1 } n January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 11
Compressed Sensing RIP-MISDP Tr( A T AX ) min s.t. Tr( X ) = 1 − z j ≤ X jj ≤ z j ∀ j ≤ n n � z j ≤ k j =1 Rank( X ) = 1 X � 0 z ∈{ 0, 1 } n Theorem [G., Pfetsch 2016] There always exists an optimal solution for (RIP-MISDP) with Rank( X ) = 1. January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 12
Further Applications ◮ Combinatorial optimization problems strengthened by semidefinite relaxations ◮ Max-cut / minimum k -partitioning ◮ Stable set ◮ Quadratic assignment problems (including TSP as special case) ◮ . . . ◮ Nonlinear / semidefinite problems with binary decisions ◮ Cardinality-constrained least squares ◮ Transmission switching problems for AC power flow ◮ Design and control of linear time-invariant systems ◮ . . . January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 13
Contents Applications Solution Approaches SCIP-SDP Parallelization Numerical Results Conclusion January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 14
LP-Based Cutting Plane Approach ◮ Solve LP in each node and enforce SDP constraint via linear cuts. ◮ Cannot use gradient cuts since SDP constraint is non-smooth in general. ◮ Use characterization u ⊤ X u ≥ 0 for all u ∈ R n . X � 0 ⇔ ◮ For eigenvector v to smallest eigenvalue of Z ⋆ := � m i − A 0 �� 0, the i =1 A i y ⋆ linear cut � m � v ⊤ � A i y i − A 0 v ≥ 0 i =1 is valid and cuts off y ∗ . January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 15
SDP-Based Branch and Bound ◮ Solve an SDP in each node of the branch-and-bound tree. ◮ Needs additional techniques to ensure constraint qualifications for validity of interior-point SDP solvers. ◮ Less efficient warmstarts available for interior-point solvers. ◮ No simplex tableau available for cutting plane generation. January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 16
Comparison of the two Approaches ◮ Cutting-plane-based approaches used by most commercial solvers for mixed-integer second-order cone (MISOCP). ◮ Outer approximation for SOCPs possible with polynomial number of cuts. (Ben-Tal/Nemirovski 2001) ◮ Outer approximation for SDPs needs exponential number of cuts. (Braun et al. 2015) ◮ SDP relaxations can still be solved up to given precision in polynomial time. January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 17
Contents Applications Solution Approaches SCIP-SDP Parallelization Numerical Results Conclusion January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 18
SCIP-SDP ◮ Plugin for SCIP to extend it to mixed-integer semidefinite programming. ◮ Supports both SDP-based branch and bound and LP-based cutting plane approach, can switch between both via parameter settings. January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 19
SCIP-SDP ◮ Plugin for SCIP to extend it to mixed-integer semidefinite programming. ◮ Supports both SDP-based branch and bound and LP-based cutting plane approach, can switch between both via parameter settings. ◮ Extends SCIP by an SDP constraint handler, SDP relaxation handler and several heuristics, propagators, file readers, . . . . ◮ Uses penalty approach in case constraint qualification fails. ◮ Includes interfaces to three different SDP solvers: MOSEK, SDPA and DSDP . ◮ Currently one of the fastest publicly available MISDP solvers. January 14, 2019 | Parallelizing SCIP-SDP via the UG framework | Tristan Gally | 19
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