Quadratic reformulation techniques for 0-1 quadratic programs

Transcription

1 OSE SEMINAR 2014 Quadratic reformulation techniques for 0-1 quadratic programs Ray Pörn CENTER OF EXCELLENCE IN OPTIMIZATION AND SYSTEMS ENGINEERING ÅBO AKADEMI UNIVERSITY ÅBO NOVEMBER 14th 2014

2 2 Structure of the presentation Joint work with PhD student Otto Nissfolk and prof. Tapio Westerlund. Background Convexification and some examples Quadratic and semidefinite programming Quadratic Convex Reformulation (QCR method) Nondiagonal perturbation (NDQCR) Numerical experiments

3 3 0-1 Quadratic Program (QP) A standard 0-1 QP has the form: Q, A, B are matrices and q, a, b are vectors of appropriate dimensions. Some applications include: Max-Cut of a graph (unconstrained) Knapsack problems (inequality constrained) Graph bipartitioning Task allocation Quadratic assignment problems Coulomb glass Gray-scale pattern problems, taixxc instances from QAPLIB

4 4 Convexity The following are equivalent (Q = Q T ): The quadratic function f x = x T Qx is convex on R n. The matrix Q is positive semidefinite (psd, Q 0). All eigenvalues of Q are non-negative (λ i 0). A sufficient condition for convexity: A diagonally dominant matris is psd. Definition: A matrix Q is diagonally dominant if Q ii Q ij i i j

5 5 Convexification of 0-1 QPs Basic approach: If Q is indefinite, add sufficient large quadratic terms to the diagonal and subtract the same amount from the linear terms. Recall that: x i 0,1 x i 2 = x i Example 1 f x = x T x = x x 1 x 2 + 2x 2 2 f x = x T x = xt x 2 3 T x = 3x x 1 x 2 + 5x 2 2 2x 1 3x 2 Indefinite Positive semidefinite

6 6 Convexification of 0-1 QPs Example 2: a) Diagonal dominance, b) Minimum eigenvalue, c) Best diagonal min x T Qx s. t. x 0,1 4 Q = eig(q) = a) Diagonal dominance Q = q = eig( Q) = min x T Qx q T x s. t. x [0,1] 4 optimal value = b) Minimum eigenvalue Q = q = eig( Q) = min x T Qx q T x optimal value = s. t. x [0,1] 4

7 Convexification of 0-1 QPs c) The best diagonal. The QCR method allows computation of the diagonal that gives the highest optimal value of the relaxation. Q = q = min x T Qx q T x optimal value = s. t. x [0,1] 4 eig( Q) = min x T Qx optimal value = 3 s. t. x 0,1 4 Bounding:

8 Semidefinite relaxation of 0-1 QPs 8 Relaxation into a positive semidefinite matrix variable X = xx T X xx T 0 1 xt x X 0 A quadratic expression in x is linear in X: x T Qx = Q X = i j Q ij X ij Binary condition: x i 0,1 x i 2 x i = 0 X ii = x i Semidefinite relaxation: min s. t. Q X + q T x Ax = a Bx b diag X = x 1 x T x X 0

9 Deriving the dual problem 9 Lagrangian relaxation of 0-1 QP: f x, λ, μ, δ = x T Qx + q T x + λ T Ax a + μ T Bx b + = x T (Q + Diag δ n i=1 Q q c δ i (x i 2 x i ) )x + (q + A T λ + B T T μ δ) x λ T a μ T b Lagrangian dual problem: sup inf x T Qx + q T x + δ, λ, μ x R n c which equals a semidefinite program max s. t. t 1 t + c 2 qt q Q δ R n, λ R m k, μ R +

10 The primal and dual 10 min s. t. Q X + q T x Ax = a Bx b diag X = x 1 x T x X 0 max s. t. t 1 t + c 2 qt q Q δ R n, λ R m k, μ R + Solution give optimal values on the multipliers : δ, λ, μ. These are used to construct the best diagonal perturbation of matrix Q according to Q = Q + Diag δ.

11 11 Strengthening Inclusion of constraints may improve bounding quality. There are many ways to include or construct quadratic constraints. 1) Add new redundant quadratic constraints (some examples) x i x j 0, x i x j x i + x j 1, x i x j x i, x i x j x j 2) Combine and multiply existing linear constraints (some examples) p T x = s x i p T x = x i s i p T x = s (1 x i )p T x = (1 x i )s i p T x = s r T x = t pt xr T x = st x T pr T x = st Ax = a Ax a 2 = 0 x T A T Ax = a T a

12 12 Our strengthening Original 0-1 QP Strengthened SDP relaxation Multipliers from all quadratic constraints are used to convexify the objective function so that the lower bound becomes as high as possible. Multipliers: δ R n, α R, S, T, U, V 0

13 13 Convexified 0-1 QP problem

14 14 NDQCR method

15 15 NDQCR versus QCR Example 3:

16 16 NDQCR versus QCR Best reformulation strategy (ii) Multipliers Matrices Convexified QP

17 17 Boolean least squares The problem is to identify a binary signal x 0,1 n from a collection of noisy measurements.

18 18 Coulomb glass problem Given n sites in the plane. k of these sites are filled with electrons. Find the configuration that has minimal energy. Energy = Coulomb interaction + site specific energy Variables: x 0,1 n x i = 0, if site i is empty 1, if site i is filled min s. t. x T Qx + q T x e T x = k x 0,1 n Q ij = 0, if i = j 1, 2r ij if i j r ij

19 19 NDQCR on Coulomb glass problems (n=50, n=100) k = n 2 in all experiments Constraints X ij 0 and X ij x i + x j 1 are included for indeces corresponding to the p% largest elements of Q. Even a small fraction of nondiagonal elements has a large impact on the total solution time. 2% - 10% non-diagonal elements result in fastest solution times.

20 20 The taixxc instances from QAPLIB These instances are a special type of QAP problem where the flow matrix F is binary and rank-1. F = bb T where b 0,1 n. The objective function of QAP can be rewritten and simplified using the binary rank-1 property: trace DXFX T = trace DXbb T X T = trace DXb Xb T = trace Dyy T = trace y T Dy = y T Dy

21 21 NDQCR (Y 0) versus QCR on tai36c p =11 QCR NDQCR CPU time 433 s 15 s SDP gap 24 % 4 %

22 22 NDQCR on problem tai64c

23 23 Conclusions A technique for non-diagonal perturbation was presented. Non-diagonal perturbation was obtained from squared norm constraints and a set of redundant RLT inequalities. Gives tight bounding and fast solution for small to medium sized problems. Full application becomes impossible for large problems. The inclusion of just a few RLT inequalities may also have large impact on the solution time and bounding quality. Future work: Construct reasonable fast heuristic procedures to find a good set of inequalities to include.

24 24 Some references on quadratic reformulation 24

25 25 THANK YOU FOR YOUR ATTENTION 25