Acyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs

Acyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs Raphael Louca & Eilyan Bitar School of Electrical and Computer Engineering American Control Conference (ACC) Chicago, July 2015

Quadratically Constrained Quadratic Program (QCQP) Consider the following QCQP: minimize x Cx s.t. x A k x b k, k = 1,..., m, x C n where C, A 1,..., A m H n and b = [b 1,..., b m ] R m. Applications: optimal power flow, max-cut, 0-1 integer programs. NP-hard in general. 2

Objective Identify a family of nearby QCQPs that: are computationally tractable, yield feasible and nearly optimal solutions for the original QCQP. Characterize nearby problems via perturbations to the problem data ( A, b, C). 3

Semidefinite Relaxation QCQP can be reformulated as a rank-constrained SDP: minimize tr(cx) s.t. A(X) b, X 0, rank(x) 1. where we define A(X) = [tr(a 1 X),..., tr(a m X)]. Semidefinite relaxation by removing the rank constraint. minimize tr(cx) s.t. A(X) b, X 0. Parametrize SDP by its data: d = (A, b, C). 4

A Family of Perturbed SDPs Perturbed SDP: minimize tr((c + C)X) s.t (A + A)(X) b, X 0. Additive perturbations on data : d = ( A, 0, C). Would like to characterize perturbations to nearby SDPs that admit solutions, which (a) are feasible for the rank-constrained SDP(d), (b) have good performance guarantees. 5

A Numerical Example Original SDP(d) minimize tr(cx) subject to A(X) b, X 0. d = (A, b, C), problem data F(d), feasible set. 6

A Numerical Example Original SDP(d) minimize tr(cx) subject to A(X) b, X 0. d = (A, b, C), problem data F(d), feasible set X, minimizer of SDP(d). 7

Perturbation-Based Approximation Goal: Identify perturbations d of the problem data d satisfying: 1. Feasibility condition: F(d + d) is a nonempty subset of F(d). 2. Recovery condition: a minimizer X of SDP(d + d) with rank(x) 1 can be computed in poly-time. 3. Performance guarantee: OPT tr(cx) OPT + ϕ(d, d), where OPT is the optimal value of the SDP(d) and ϕ(d, d) 0 uniformly as d 0. 8

A Numerical Example Perturbed SDP(d + d) minimize tr((c + C)X) subject to (A + A)(X) b, X 0. d = (A, b, C), problem data d = ( A, 0, C), perturbation on data F(d + d), perturbed feas. set 9

Talk Outline Prior work Preliminary results Main results Existence and characterization of acyclic approximations Performance guarantees Future work 10

Prior Work Alternating projections [von Neumann, 39, Grigoriadis et al. 00, Lewis et al. 08] Nuclear norm minimization [Fazel et al. 01, Recht et al. 07, Chandrasekaran 12] Randomized rounding: Max-cut: 0.8785 approx. ratio. [Goemans et al. 95] QCQPs with: (a) A 1,..., A m 0 : O(log m) approx. ratio. [Nemirovski et al. 03] (b) b 1,..., b m > 0 : data-dependent approx. ratio. [He et al. 08] 11

Graph of Semidefinite Program Definition: Given problem data d, let G(d) denote the undirected graph induced by the collective sparsity pattern of the matrices C, A 1,..., A m. Example: 12

Linear Separability Definition: The data d is off-diagonally linearly separable from the origin if for all i j, there is a line through (0, 0) such that all points in lie on one side of the line. { [C] ij, [A 1 ] ij,..., [A m ] ij } Example: 13

Guarantees Semidefinite Relaxation: minimize tr(cx) s.t. A(X) b, X 0. X H n Theorem: [Sojoudi et al. 14, Bose et al 14] If the data d = (A, b, C) are: 1. off-diagonally linearly separable from the origin and 2. the graph G(d) is acyclic, then a minimizer X satisfying rank(x) 1 can be computed in poly-time. 14

Acyclic Semidefinite Approximations Definition: The SDP(d + d) is an (α, β)-acyclic approximation of the SDP(d) if 15

Acyclic Semidefinite Approximations Definition: The SDP(d + d) is an (α, β)-acyclic approximation of the SDP(d) if 1. G(d + d) is acyclic, 16

Acyclic Semidefinite Approximations Definition: The SDP(d + d) is an (α, β)-acyclic approximation of the SDP(d) if 1. G(d + d) is acyclic, 2. d + d are off-diagonally linearly separable from the origin, 17

Distance to Infeasibility For d = (A, b, C), the set of primal infeasible problem instances is: I P = {d F(d) = }. Definition: [Renegar 94] The distance of the data d to I P is: dist IP (d) = inf{ d d π d IP }. Dual distance to infeasibility, dist ID (d), similarly defined. Can be approximated by solving an SDP. [Freund et al. 99] 19

Recovery Guarantees Theorem: Let SDP(d + d) be an (α, β)-acyclic approximation of SDP(d). If 1. C, A 1,..., A m 0 and 2. α < dist IP (d), then 3. Feasibility condition: F(d + d) is a nonempty subset of F(d). 4. Recovery condition : a minimizer X of the SDP(d + d) with rank(x ) 1 can be computed in poly-time. 20

Performance Guarantees 5. Performance guarantee: Let OPT be the optimal value of SDP(d). Any minimizer X of SDP(d + d) satisfies: where OPT tr(cx ) OPT + ϕ(d, d), ϕ(d, d) = max{ b 2, OPT}(µ + ν max{ C F + β, OPT}), µ = β dist ID (d), and ν = α dist ID (d) (dist IP (d) α). 21

Discussion of Results Error function ϕ(d, d) is monotone in (α, β) and ϕ(d, d) 0, uniformly as d 0. Good performance guarantees for: Near-acyclic and near-linearly-separable QCQPs, i.e., α, β small. Well-conditioned problems (i.e., distance to infeasibility is large). Performance guarantee is data-dependent. Condition not necessary. 22

A Numerical Example Perturbed SDP(d + d) X argmin tr((c + C)X) subject to (A + A)(X) b, X 0. Conservative sufficient condition α dist IP (d). Realized approximation ratio: tr(cx ) OPT = 1.55. 23

Conclusions Existence and characterization of a family of nearby QCQPs that are computationally tractable, yield feasible and nearly optimal solutions for the original QCQP. Performance guarantees are data-dependent. Good performance guaranteed for SDPs that are: Near-acyclic and near-linearly-separable, Well-conditioned. Can be generalized to double-sided inequality and equality constraints for special cases. 24

Future Directions Efficient algorithms to construct semidefinite acyclic approximations. A generalization to arbitrary rank-constrained SDPs: minimize tr(cx) subject to A(X) b, X 0, rank(x) r. X H n 25

Thank you! Raphael Louca e-mail: rl553 @cornell.edu 26