Spatial Branch-and-Cut for QCQP with Complex Bounded Variables Chen Chen Alper Atamt¨ urk Shmuel S. Oren January 4, 2016 Aussois 2016 SBC for QCQP – 1
Introduction (Nonconvex) Complex Bounded QCQP Why Complex? Spatial Branch-and-Cut Experiments Conclusion Introduction Aussois 2016 SBC for QCQP – 2
(Nonconvex) Complex Bounded QCQP Introduction (Nonconvex) Complex Bounded QCQP min x ∗ Q 0 x + Re ( c ∗ 0 x ) + b 0 Why Complex? s.t. x ∗ Q i x + Re ( c ∗ i x ) + b i ≤ 0 , i = 1 , ..., m Spatial Branch-and-Cut Experiments ℓ ≤ x ≤ u Conclusion x ∈ C n Aussois 2016 SBC for QCQP – 3
(Nonconvex) Complex Bounded QCQP Introduction (Nonconvex) Complex Bounded QCQP min x ∗ Q 0 x + Re ( c ∗ 0 x ) + b 0 Why Complex? s.t. x ∗ Q i x + Re ( c ∗ i x ) + b i ≤ 0 , i = 1 , ..., m Spatial Branch-and-Cut Experiments ℓ ≤ x ≤ u Conclusion x ∈ C n Applications: AC Optimal Power Flow � Signal Processing e.g. [Waldspurger, D’Aspremont, Mallat � 2015] Control Theory [Ben Tal, Nemirovski, Roos 2003] � Aussois 2016 SBC for QCQP – 3
Why Complex? Introduction Real QCQP to complex QCQP: set imaginary components to zero (Nonconvex) Complex Complex to real by defining 2x real variables: x := w + ıt . Bounded QCQP Why Complex? We use the complex structure to exploit the angle valid inequality: Spatial Branch-and-Cut Experiments Conclusion L ij ( w i w j + t i t j ) ≤ t i w j − w i t j ≤ U ij ( w i w j + t i t j ) Aussois 2016 SBC for QCQP – 4
Why Complex? Introduction Real QCQP to complex QCQP: set imaginary components to zero (Nonconvex) Complex Complex to real by defining 2x real variables: x := w + ıt . Bounded QCQP Why Complex? We use the complex structure to exploit the angle valid inequality: Spatial Branch-and-Cut Experiments Conclusion L ij ( w i w j + t i t j ) ≤ t i w j − w i t j ≤ U ij ( w i w j + t i t j ) It has a clear interpretation in polar coordinates: � � Im ( x i ) θ i := arctan Re ( x i ) arctan( L ij ) ≤ θ i − θ j ≤ arctan( U ij ) Aussois 2016 SBC for QCQP – 4
Introduction Spatial Branch-and-Cut Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Spatial Branch-and-Cut Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion Aussois 2016 SBC for QCQP – 5
Overview Introduction Spatial branching, e.g. bound bisection: Spatial Branch-and-Cut Overview ( L + U ) / 2 ≤ x ∨ ( L + U ) / 2 ≥ x SDP Relaxation A Valid Nonconvex Set Requires a lower bound — we shall use a SDP relaxation. Valid Inequalities Valid Inequalities cont. Our contributions: Properties of VIs Branching Rules Valid inequalities to strengthen the relaxation � Violation-Based Branching Branching rules � Bound tightening Experiments Bound tightening procedures � Conclusion Aussois 2016 SBC for QCQP – 6
SDP Relaxation Introduction Standard approach of [Shor 1987], [Lovasz & Schrijver 1991] (or Spatial Branch-and-Cut moment relaxation ala Lasserre): Overview SDP Relaxation min � Q 0 X � + Re ( c ∗ 0 x ) + b 0 (1) A Valid Nonconvex Set Valid Inequalities s.t. � Q i , X � + Re ( c ∗ i x ) + b i ≤ 0 , i = 1 , ..., m (2) Valid Inequalities cont. Properties of VIs ℓ ≤ x ≤ u (3) � 1 Branching Rules � x ∗ Violation-Based Y := � 0 , Branching (4) x X Bound tightening Experiments where Y is Hermitian. X = xx ∗ iff Y has rank one or zero. Conclusion Aussois 2016 SBC for QCQP – 7
A Valid Nonconvex Set Let X := W + ıT . Every bounded complex QCQP has the Introduction Spatial Branch-and-Cut following: Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities L ii ≤ W ii ≤ U ii , (5) Valid Inequalities cont. L jj ≤ W jj ≤ U jj , (6) Properties of VIs Branching Rules L ij W ij ≤ T ij ≤ U ij W ij , (7) Violation-Based Branching W ii W jj = W 2 ij + T 2 ij . (8) Bound tightening Experiments Conclusion (5)-(6) follow directly from boundedness assumption � (7) is the angle constraint (can be shown) � (8) comes from X = xx ∗ � This gives us bounds for every complex entry of X! Aussois 2016 SBC for QCQP – 8
Valid Inequalities The convex hull of (5)-(8) is given by the SDP relaxation ( X � 0 , Introduction Spatial Branch-and-Cut (5)-(7)) and Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities π 0 + π 1 W ii + π 2 W jj + π 3 W ij + π 4 T ij ≥ Valid Inequalities cont. U jj W ii + U ii W jj − U ii U jj , Properties of VIs Branching Rules π 0 + π 1 W ii + π 2 W jj + π 3 W ij + π 4 T ij ≥ Violation-Based Branching L jj W ii + L ii W jj − L ii L jj , Bound tightening Experiments Conclusion Aussois 2016 SBC for QCQP – 9
Valid Inequalities cont. The coefficients π are defined as Introduction Spatial Branch-and-Cut � Overview π 0 := − L ii L jj U ii U jj , SDP Relaxation � A Valid Nonconvex Set π 1 := − L jj U jj , Valid Inequalities � Valid Inequalities cont. π 2 := − L ii U ii , Properties of VIs U jj )1 − f ( L ij ) f ( U ij ) Branching Rules � � � � π 3 := ( L ii + U ii )( L jj + 1 + f ( L ij ) f ( U ij ) , Violation-Based Branching Bound tightening U jj ) f ( L ij ) + f ( U ij ) � � � � π 4 := ( L ii + U ii )( L jj + 1 + f ( L ij ) f ( U ij ) . Experiments Conclusion where √ 1 + x 2 − 1) /x, � ( x � = 0 f ( x ) := x = 0 . 0 , Aussois 2016 SBC for QCQP – 10
Properties of VIs Introduction If L = U then VI + SDP gives an exact solution, i.e. convergence � Spatial Branch-and-Cut Can be added in a sparse manner for sparse SDPs � Overview SDP Relaxation Can be interpreted as complex analogues of the RLT � A Valid Nonconvex Set Valid Inequalities inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion Aussois 2016 SBC for QCQP – 11
Branching Rules Consider a strategy of bisecting matrix bounds L ij , U ij . The Introduction Spatial Branch-and-Cut question then is: which i, j entry to choose? Overview SDP Relaxation A Valid Nonconvex Set Valid Inequalities Valid Inequalities cont. Properties of VIs Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion Aussois 2016 SBC for QCQP – 12
Branching Rules Consider a strategy of bisecting matrix bounds L ij , U ij . The Introduction Spatial Branch-and-Cut question then is: which i, j entry to choose? Overview SDP Relaxation Objective-based approach, e.g. reliability branching � A Valid Nonconvex Set [Achterberg, Koch, Martin, 2005]. We apply a spatial Valid Inequalities Valid Inequalities cont. branching-adapted rule of Belotti et al. as a benchmark Properties of VIs ( rb-int-br ), call it RBEB. Branching Rules Violation-Based Branching Bound tightening Experiments Conclusion Aussois 2016 SBC for QCQP – 12
Branching Rules Consider a strategy of bisecting matrix bounds L ij , U ij . The Introduction Spatial Branch-and-Cut question then is: which i, j entry to choose? Overview SDP Relaxation Objective-based approach, e.g. reliability branching � A Valid Nonconvex Set [Achterberg, Koch, Martin, 2005]. We apply a spatial Valid Inequalities Valid Inequalities cont. branching-adapted rule of Belotti et al. as a benchmark Properties of VIs ( rb-int-br ), call it RBEB. Branching Rules Violation-Based Branching Our proposal: violation-based branching. � Bound tightening Experiments Conclusion Aussois 2016 SBC for QCQP – 12
Violation-Based Branching Introduction Recall that we want a rank-one matrix in the SDP solution. Spatial Branch-and-Cut However, rank is a global measure. Overview For n > 1 a nonzero Hermitian positive semidefinite n × n matrix SDP Relaxation X has rank one iff all of its 2 × 2 principal minors are zero. A Valid Nonconvex Set Valid Inequalities Proposal: find the 2 × 2 submatrix with greatest minimum Valid Inequalities cont. eigenvalue. Properties of VIs Branching Rules Three options to branch: { i, i } , { j, j } , { i, j } . Violation-Based Branching Two proposals to select among the three: Bound tightening MVSB: strong branching (try all three) Experiments MVWB: use a worst-case approximation that is easily computable Conclusion Aussois 2016 SBC for QCQP – 13
Bound tightening Introduction Standard procedure is to minimize/maximize a variable over a Spatial Branch-and-Cut relaxation. With SDP relaxation this is expensive! Overview We propose two closed-form procedures. First, consider SDP Relaxation A Valid Nonconvex Set ax 2 + xy + c ≤ 0 , Valid Inequalities Valid Inequalities cont. ℓ ≤ y ≤ u. Properties of VIs Branching Rules Violation-Based By use of aggregating variables, many quadratic inequalities can be Branching Bound tightening put in such a form. Bounds on x can be inferred from the bounds on Experiments y . Conclusion Second, tightening based on the principle that the sum of differences around a cycle must equal to zero. Defining δ ij := x i − x j , then for some cycle of indices x , say { 1 , 2 , 3 } , we have δ 12 + δ 23 + δ 31 = 0 . The arctangents of L ij , U ij represent such differences. Aussois 2016 SBC for QCQP – 14
Introduction Spatial Branch-and-Cut Experiments Setup SDP Alone Does Not Converge In A Nutshell Experiments BoxQP Results Conclusion Aussois 2016 SBC for QCQP – 15
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