Problem structure in semidefinite programs arising in control and signal processing

Transcription

1 Problem structure in semidefinite programs arising in control and signal processing Lieven Vandenberghe Electrical Engineering Department, UCLA Joint work with: Mehrdad Nouralishahi, Tae Roh Semidefinite Programming and Its Applications IMS, National University of Singapore January 13, 2006

2 Outline Model calibration in optical lithography. Optimization over nonnegative polynomials. Robust least squares. 1

3 Model calibration in optical lithography mask imaging system wafer PSfrag replacements source PSfrag replacements Model of the lithography process layout optical system resist/etch process simulated pattern Model calibration: adjust optical model to include resist/etching effects. 2

4 I(x) = The Hopkins model f(x v) K(v) J(u v)k(u) f(x u) du dv f(x): input image (mask); I(x): output image intensity Coherent illumination: J(u v) = 1 I(x) = K(u)f(x u)du 2 = (K f)(x) 2 Incoherent illumination: J(u v) = δ(u v) I(x) = K(u)f(x u) 2 du Other choices for J model partially coherent systems. 3

5 Discrete Hopkins model Optimal coherent approximation I(x) = f H x W f x f x is vector of input values around at x; for example, for 1-D systems, f x = (f(x N), f(x N 1),..., f(x + N)) Low-rank approximation ( r I(x) = fx H φ k φ H k k=1 ) f x = r (φ k f)(x) 2 k=1 φ 1,..., φ r are first few eigenvectors of W. 4

6 SDP formulation (for real data) minimize tr(dw ) subject to ɛi W W 0 ɛi l k a T k W a k u k, k = 1,..., m symmetry constraints on W W 0 W 0 is prior model of optical system. Measurements give m upper/lower bounds on intensity for given inputs. Objective rewards smoothness or low rank (D = I). 5

7 Symmetry constraints One-dimensional system: W S n is persymmetric, i.e., W = JW J, J = Two-dimensional system: W S n2 with W = (J n I n )W (J n I n ) = (I n J n )W (I n J n ) General constraint: W = P W P P a symmetric permutation matrix 6

8 Exploiting symmetry Suppose P is a symmetric permutation matrix (P 2 = I): P = [ V e V o ] [ I 0 0 I ] [ Ve V o ] T V e/o are eigenvectors with eigenvalue ±1. Property: if W = P W P, then W = [ V e V o ] [ W e 0 0 W o ] [ Ve V o ] T Reduces the number of variables: One-dimensional problem: 2 matrices of size n/2 Two-dimensional problem: 4 matrices of size n 2 /4 7

9 Variable upper bounds LP with upper bounds minimize subject to c T x Ax = b 0 x 1 Can introduce slack variables to convert to standard form x + s = 1, s 0. n new variables, n (sparse) equality constraints. Upper bounds are easily incorporated in IP methods (at no extra cost). 8

10 SDP with upper bounds Introduce slack variables: minimize tr(cx) subject to tr(a i X) = b i, i = 1,..., m 0 X I X + S = I, S 0 Adds n(n + 1)/2 variables and constraints. Hence, it is important to handle upper bounds directly. 9

11 Newton equations for SDP with upper bounds T1 1 1 XT where T 1, T T2 1 XT m y j A j = D j=1 tr(a i X) = d i, i = 1,..., m Find congruence that simultaneously diagonalizes T 1 and T 2 : R T T 1 R = I, R T T 2 R = diag(γ) 1 Make change of variables X = R T XR. 10

12 Transformed equations: X diag(γ) X diag(γ) + m y j à j = D j=1 tr(ãi X) = d i, i = 1,..., m Eliminate X from first equation: X = m y j (Ãj Γ) D Γ, Γ kl = 1/(1 + γ k γ l ) j=1 Solve m tr(ãi(ãj Γ)) y j = b i, j=1 i = 1,..., m Cost: comparable to SDP without upper bounds. 11

13 Example Calibration step (using simulated data) Left: mask. Right: measured intensity. W, W 0 S Sample output image at m = 2500 points. 12

14 Test input and actual output Predicted output with uncalibrated and calibrated model 13

15 Outline Model calibration in optical lithography. Optimization over nonnegative polynomials. Robust least squares.

16 A useful SDP standard form minimize subject to tr(bx) A diag(cxc T ) = b X 0 maximize subject to b T y C T diag(a T y)c B Interpretation (for A R m r with rows a T i, C Rr n ) Constraints are equivalent to tr(f i X) = b i, i = 1,..., m, where F i = C T diag(a i )C, i = 1,..., m. Matrices F i are linear combinations of r rank-one matrices. Possible for arbitrary F i if r = mn. Interesting when r mn. F i s can be dense and full-rank. 14

17 Newton equations T 1 XT 1 + C T diag(a T y)c = D A diag(c XC T ) = d Eliminate X from first equation, substitute in second: H y = g with H = A ( (CT C T ) (CT C T ) ) A T Cost if m = O(n), r = O(n): O(n 3 ) operations 15

18 SDP formulation of sums of squares x T f(t) = s (yk T q(t)) 2 k=1 r.h.s. sampled at t 1,..., t N, s k=1 ((Cy k ) (Cy k )) = k diag(cy k y T k C T ) = diag(cxc T ) where C = [ q(t 1 ) q(t 2 ) q(t N ) ] T, X = k y ky T k. Coefficients of l.h.s. from samples at t 1,..., t N : x = A diag(cxc T ) 16

19 Applications Trigonometric polynomial nonnegative on (subinterval of) [0, 2π] (via discrete Fourier transform). Cosine polynomials nonnegative on (subinterval of) [0, π] (via discrete cosine transform). Real polynomials nonnegative on (subintervals of) R (via discrete polynomial transforms and orthogonal polynomials). Sum-of-squares representations of multivariate nonnegative polynomials. 17

20 Nonnegative trigonometric polynomial x 0 + 2R(x 1 e jω + + x n e jnω ) = y0 + y 1 e jω + + y n e jnω 2 r.h.s. at ω = 2πk/N, k = 0,..., N 1, where N 2n + 1: (W y) (W y) = diag(w XW H ), X = yy H W : first n + 1 columns of DFT matrix of length N. Inverse DFT maps sampled values to coefficients of l.h.s.: x = 1 N W H diag(w XW H ) (1) Hence, polynomial is nonnegative iff (1) for some X 0. 18

21 Dual interpretation: parametrization of Toeplitz matrix 1 2 2z 0 z 1 z n z 1 2z 0 z n z n z n 1 2z 0 = 1 N W H diag(r(w z))w Follows from expressions for convolution of z with a vector u: z u = z z 1 z z n z n 1 z 0 u = 1 N W H ((W z) (W u)) = 1 N W H diag(w z)w u 19

22 Example: Linear-phase Nyquist filter minimize sup ω ωs h 0 + h 1 cos ω + + h n cos nω with h 0 = 1/M, h km = 0 for positive integer k H(ω) PSfrag replacements ω (Example with n = 50, M = 5, ω s = 0.69.) 20

23 SDP formulation Semi-infinite LP minimize t subject to t h 0 + h 1 cos ω + + h n cos nω t, ω s ω π SDP formulation minimize t subject to h + te 0 = A 1 diag(c 1 X 1 C1 T ) + A 2 diag(c 2 X 2 C2 T ) h + te 0 = A 1 diag(c 1 X 3 C1 T ) + A 2 diag(c 2 X 4 C2 T ) X 1 0, X 2 0, X 3 0, X 4 0 Variables: t, h i (except for i = km), 4 matrices X i of size roughly n/2. A i, C i constructed from DCT matrices of length N n. 21

24 Nonnegative polynomial on R x 0 p 0 (t) + x 1 p 1 (t) + + x n p n (t) = (u 0 p 0 (t) + + u m p m (t)) 2 + (v 0 p 0 (t) + + v m p m (t)) 2 (p k : basis polynomial of degree k; n = 2m) r.h.s. evaluated at t 0,..., t N : (Cu) (Cu) + (Cv) (Cv) = diag(cxc T ), X = uu T + vv T where C ij = p j (t i ) If N n and t i are distinct, the sample values uniquely specify x: x = A diag(cxc T ) (2) Polynomial is nonnegative iff (2) holds for some X 0. 22

25 Discrete polynomial transform V DPT = p 0 (t 0 ) p 1 (t 0 ) p N (t 0 ) p 0 (t 1 ). p 1 (t 1 ). p N (t 1 ). p 0 (t N ) p 1 (t N ) p N (t N ), W DPT = V 1 DPT p 0, p 1,... : system of orthogonal polynomials t 0,..., t N : roots of p N+1 V DPT maps coefficients of x 0 p 0 (t) + x 1 p 1 (t) + + x N p N (t) to values at t 0,... t N W DPT maps values at t 0,..., t N to coefficients 23

26 Three-term recursion t p 0 (t) p 1 (t) p 2 (t). p N (t) = β 0 α α 0 β 1 α α 1 β β N p 0 (t) p 1 (t) p 2 (t). p N (t) + α N p N+1 (t) tp(t) = Jp(t) + α N p N+1 (t)e N Eigenvalues of J are the roots of p N+1 : t i p(t i ) = Jp(t i ) V DPT, W DPT easily computed from eigenvalue decomposition of J. 24

27 Magnitude FIR filter design Numerical example π minimize Y (ω) dω ω s subject to 1/δ p Y (ω) δ p, 0 ω ω p Y (ω) δ s, ω s ω π Y (ω) 0, 0 ω π where Y (ω) = y 0 + y 1 cos ω + + y n cos nω. Constraints result in 4 LMI constraints. Variables: y and 8 auxiliary matrix variables of size roughly n/2. 25

28 Example (n = 101) 10 0 Y (ω) 10 2 PSfrag replacements ω 26

29 Time per iteration (Matlab on 2.8GHz P4) 10 3 Sfrag replacements Time per iteration (s) SDPT3 fast implementation SDPT3-style primal-dual method with fast solution of Newton equations n 27

30 Outline Model calibration in optical lithography. Optimization over nonnegative polynomials. Robust least squares.

31 Robust least-squares minimize sup u 2 1 A(u)x b 2 2 where A(u) = Ā + U diag(du)v T Example. A is lower triangular Toeplitz with coefficients h k + u k A(u) = 1 N W H 1 diag(w 1 (h + u))w 2 W 1, W 2 : first n + 1, resp. m + 1, columns of DFT matrix 28

32 SDP formulation minimize subject to t + λ t 0 (Āx b)t 0 λi D T diag(v T x)u T (Āx b) U diag(v T x)d 0 0 Cost per iteration is dominated by constructing the matrix V ( (DT 23 U) (DT 23 U) T + (DT 22 D T ) (U T T 33 U) ) V T T ij are submatrices of scaling matrix O(n 3 ) operations (if all dimensions are of the same order) 29

33 Conclusions Some interesting types of problem structure that are easily exploited. Symmetry constraints. Upper bounds on the matrix variables. A generalization of low-rank structure. 30