sufficient conditions for exact sdp reformulations of qcqpsalw1/files/qcqp_informs.pdf · 2019. 11....

Sufficient Conditions for Exact SDPReformulations of QCQPs

Alex L. Wang Fatma Kılınç-Karzan

Carnegie Mellon University

INFORMS 2019

Wang, Kılınç-Karzan Sufficient Conditions for Exact SDP Reformulations of QCQPs 1 / 19

1 Introduction: SDP Relaxations of QCQP

2 Symmetries in quadratic forms

3 A dual object

4 Conclusion


Quadratically Constrained Quadratic Programs (QCQP)

• q0, q1, . . . , qm : Rn → R be (possibly nonconvex!) quadraticfunctions

qi(x) = x>Aix+ 2b>i x+ ci

• Want to find

Opt := infx∈Rn

q0(x)

∣∣∣∣∣∣∣q1(x) ≤ 0...qm(x) ≤ 0


The QCQP Epigraph

Opt = infx∈Rn

{q0(x) | qi(x) ≤ 0, ∀i ∈ [m]}

= infx,t

{t

∣∣∣∣ q0(x) ≤ tqi(x) ≤ 0, ∀i ∈ [m]

}=: inf

x,t

{t∣∣∣ (x, t) ∈ E }

= infx,t{t | (x, t) ∈ conv(E)}


The standard SDP relaxation of QCQP

Standard (Shor) SDP relaxation

Opt = infx,X

〈A0, X〉+ 2b>0 x+ c0

∣∣∣∣∣∣∣∣∣〈A1, X〉+ 2b>1 x+ c1 ≤ 0...〈Am, X〉+ 2b>mx+ c1 ≤ 0

X = xx>

≥ inf

x,X

〈A0, X〉+ 2b>0 x+ c0

∣∣∣∣∣∣∣∣∣〈A1, X〉+ 2b>1 x+ c1 ≤ 0...〈Am, X〉+ 2b>mx+ c1 ≤ 0

X � xx>

=: OptSDP


SDP epigraph

OptSDP = infx,X

〈A0, X〉+ 2b>0 x+ c0

∣∣∣∣∣∣∣∣∣〈A1, X〉+ 2b>1 x+ c1 ≤ 0...〈Am, X〉+ 2b>mx+ c1 ≤ 0X � xx>

= inf

x,X, t

t∣∣∣∣∣∣∣〈A0, X〉+ 2b>0 x+ c0 ≤ t〈Ai, X〉+ 2b>i x+ ci ≤ 0, ∀i ∈ [m]X � xx>

• Let ESDP be the projection of the epigraph onto (x, t) variables


The story so far

• QCQP is NP hard, but we can solve SDP relaxation instead• What are sufficient conditions for

• SDP tightness: Opt = OptSDP?• Convex hull result: conv(E) = ESDP?


Related work

• SDP reformulations for QCQPs with a single constraint[Fradkov and Yakubovich, 1979], [W, Kılınç-Karzan, 2019]

• SDP reformulations for variants of TRS[Sturm and Zhang, 2003], [Burer, 2015], [Yang, Anstreicher, andBurer, 2018]

• SDP reformulations for Diagonal QCQPs[Burer and Ye, 2018]

• Quadratic Matrix Programming[Beck, 2007], [Beck, Drori, and Teboulle, 2012]


Outline

• Analyze a parameter k which captures “amount of symmetry”in a given QCQP• Under additional “polyhedrality” assumption,k ≥ aff dim({bi}) + 1 =⇒ SDP tightness and convex hullresult




3 A dual object

4 Conclusion


Quadratic eigenvalue multiplicity

Definition

Let 1 ≤ k ≤ n be the largest integer such that for eachi = 0, . . . ,m, the matrix Ai ∈ Sn has the following block form

Ai = Ai ⊗ Ik =

Ai

Ai. . .Ai

where Ai ∈ Sn/k


Quadratic eigenvalue multiplicity

• Ai = Ai ⊗ Ik• Suppose n = 4

x21 + x2

2 + x23 + x2

4

1

11

1

k = 4

(x1 − x2)2 + (x3 − x4)2

1 −1−1 1

1 −1−1 1

k = 2


Our first result

Theorem

Suppose primal feasibility and dual strict feasibility. If Γ ispolyhedral and

k ≥ aff dim({b(γ) | γ ∈ F}) + 1

for every semidefinite face F of Γ of affine dimension at mostm− 1, then

conv(E) = ESDP and Opt = OptSDP .

Γ :=

{γ ∈ Rm

∣∣∣∣∣ γ ≥ 0, A0 +

m∑i=1

γiAi � 0

}


Our first result

Corollary

Suppose primal feasibility and dual strict feasibility. If {Ai}mi=0 arediagonal and

k ≥ min (m , 1 + |{bi 6= 0}mi=1|) ,

thenconv(E) = ESDP and Opt = OptSDP .

• When m = 1

• When b1 = b2 = · · · = bm = 0, the condition is triviallysatisfied• Ai = αiIn for all i = 0, . . . ,m and n ≥ m


Example: Swiss cheese

• Ai = αiIn for all i = 0, . . . ,m and n ≥ m• Minimizing distance to a piece of Swiss cheese

infx∈Rn

‖x‖2∣∣∣∣∣∣

inside ball constraintsoutside ball constraintslinear constraints

• If nonempty and n ≥ m, then the standard SDP relaxation istight for this QCQP




3 A dual object

4 Conclusion


Aggregation

• For γ ∈ Rm such that γ ≥ 0, define

q(γ, x) := q0(x) +

m∑i=1

γiqi(x) A(γ) := A0 +

m∑i=1

γiAi

• DefineΓ := {γ ∈ Rm : γ ≥ 0, A(γ) � 0}

A0

A1 A2 A0A1

A2


What does Γ look like?

A0

A2

A1

A0A1

A2

A0

A1

A2

A0A1

A2


Rewriting the SDP in terms of Γ

• Γ plays a crucial role in the analysis!

Theorem

Suppose primal feasibility and dual strict feasibility, then

OptSDP = minx∈Rn

supγ∈Γ

q(γ, x)

ESDP =

{(x, t) : sup

γ∈Γq(γ, x) ≤ t

}

• If Γ is polyhedral, then ESDP is defined by finitely many convexquadratics =⇒ SOC-representable




3 A dual object

4 Conclusion


Conclusion

• Analyzed• “amount of symmetry” k• the geometry of a dual object Γ

• Assuming primal feasibility and dual strict feasibility

Polyhedral Γ General Γ

SDP tightness aff dim({b(γ) | γ ∈ F}) + 1 m+ 1Convex hull result aff dim({b(γ) | γ ∈ F}) + 1 m+ 2

• Results extend to equalities!• A general framework for proving sufficient conditions• Future directions

• Are the assumptions on k sharp?• Can techniques say anything about when SDP is

approximately tight?


Thank you. Questions?

Slides and preprint available (hopefully) sooncs.cmu.edu/~alw1


cs.cmu.edu/~alw1

References I

Beck, Amir (2007). “Quadratic matrix programming”. In: SIAMJournal on Optimization 17.4, pp. 1224–1238.

Beck, Amir, Yoel Drori, and Marc Teboulle (2012). “A newsemidefinite programming relaxation scheme for a class ofquadratic matrix problems”. In: Operations Research Letters40.4, pp. 298–302.

Burer, Samuel (2015). “A gentle, geometric introduction tocopositive optimization”. In: Mathematical Programming 151.1,pp. 89–116.

Burer, Samuel and Yinyu Ye (2018). “Exact semidefiniteformulations for a class of (random and non-random) nonconvexquadratic programs”. In: Mathematical Programming, pp. 1–17.

Fradkov, Alexander L. and Vladimir A. Yakubovich (1979). “TheS-procedure and duality relations in nonconvex problems ofquadratic programming”. In: Vestn. LGU, Ser. Mat., Mekh.,Astron 6.1, pp. 101–109.


References II

Sturm, J. F. and S. Zhang (2003). “On Cones of NonnegativeQuadratic Functions”. In: Mathematics of Operations Research28.2, pp. 246–267.

Yang, Boshi, Kurt Anstreicher, and Samuel Burer (2018).“Quadratic programs with hollows”. In: MathematicalProgramming 170.2, pp. 541–553.


sufficient conditions for exact sdp reformulations of qcqpsalw1/files/qcqp_informs.pdf · 2019. 11....

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