Fast Algorithms for SDPs derived from the Kalman-Yakubovich-Popov Lemma

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1 Fast Algorithms for SDPs derived from the Kalman-Yakubovich-Popov Lemma Venkataramanan (Ragu) Balakrishnan School of ECE, Purdue University 8 September 2003 European Union RTN Summer School on Multi-Agent Control Hamilton Institute

2 Fast Algorithms for SDPs derived from the Kalman-Yakubovich-Popov Lemma Venkataramanan (Ragu) Balakrishnan School of ECE, Purdue University 8 September 2003 European Union RTN Summer School on Multi-Agent Control Hamilton Institute Joint work with Lieven Vandenberghe, UCLA Anders Hansson and Ragnar Wallin, Linkoping University

3 FAST ALGORITHMS FOR KYP SDPS 1 Outline A brief introduction to Semidefinite Programming (SDP)

4 FAST ALGORITHMS FOR KYP SDPS 1 Outline A brief introduction to Semidefinite Programming (SDP) Focus: LMIs from the Kalman-Yakubovich-Popov Lemma

5 FAST ALGORITHMS FOR KYP SDPS 1 Outline A brief introduction to Semidefinite Programming (SDP) Focus: LMIs from the Kalman-Yakubovich-Popov Lemma Fast algorithms for SDPs from KYP Lemma

6 FAST ALGORITHMS FOR KYP SDPS 2 Convex optimization of the form: Semidefinite Programming (SDP) minimize c T x subject to F 0 + x 1 F x p F p 0 F 0, F 1,..., F p are given symmetric matrices, c is a vector, x is the vector of optimization variables

7 FAST ALGORITHMS FOR KYP SDPS 2 Convex optimization of the form: Semidefinite Programming (SDP) minimize c T x subject to F 0 + x 1 F x p F p 0 F 0, F 1,..., F p are given symmetric matrices, c is a vector, x is the vector of optimization variables F (x) = F 0 + x 1 F x p F p 0 called an LMI F 0 means F is positive semidefinite, that is u T F u 0 for all vectors u LMIs are nonlinear, but convex constraints: If F (x) 0 and F (y) 0, then F (λx + (1 λ)y) = λf (x) + (1 λ)f (y) 0 for all λ [0, 1]

8 FAST ALGORITHMS FOR KYP SDPS 3 SDP vs. LP SDP: minimize c T x subject to F 0 + x 1 F x p F p 0 F 0, F 1,..., F p are given symmetric matrices, c is a vector, x is the vector of optimization variables LP: minimize c T x subject to a T i x b i, i = 1,..., N Same linear objective Linear matrix inequality constraint instead of linear scalar inequalities

9 FAST ALGORITHMS FOR KYP SDPS 4 More on LMIs Matrices as variables: Example: Lyapunov inequality A is given, P = P T is the variable A T P + P A 0 Can write it as an LMI in the entries of P Better to leave LMIs in a condensed form saves notation leads to more efficient computation

10 FAST ALGORITHMS FOR KYP SDPS 4 More on LMIs Matrices as variables Multiple LMIs F (1) (x) 0,..., F (N) (x) 0 same as single LMI diag(f (1) (x),..., F (N) (x)) 0

11 FAST ALGORITHMS FOR KYP SDPS 5 LMI examples Many standard constraints can be written as LMIs Linear constraints Ax + b > 0 (componentwise) Can be rewritten as an LMI using diagonal matrices

12 FAST ALGORITHMS FOR KYP SDPS 5 LMI examples Many standard constraints can be written as LMIs Linear constraints Quadratic constraints: Inequality (Ax + b) T (Ax + b) + c T x + d < 0 is equivalent to the LMI [ I Ax + b (Ax + b) T (c T x + d) ] 0

13 FAST ALGORITHMS FOR KYP SDPS 5 LMI examples Many standard constraints can be written as LMIs Linear constraints Quadratic constraints Trace constraints: Inequality P = P T, A T P + P A 0, TrP 1 is an LMI

14 FAST ALGORITHMS FOR KYP SDPS 5 LMI examples Many standard constraints can be written as LMIs Linear constraints Quadratic constraints Trace constraints Norm constraints: Inequality σ max (A) < 1 is equivalent to LMI [ I A A T I ] 0

15 FAST ALGORITHMS FOR KYP SDPS 5 LMI examples Many standard constraints can be written as LMIs Linear constraints Quadratic constraints Trace constraints Norm constraints... mixtures of these constraints and many more

16 FAST ALGORITHMS FOR KYP SDPS 6 SDP applications Systems and control (quite well-known)

17 FAST ALGORITHMS FOR KYP SDPS 6 SDP applications Systems and control (quite well-known) Circuit design

18 FAST ALGORITHMS FOR KYP SDPS 6 SDP applications Systems and control (quite well-known) Circuit design Nonconvex optimization

19 FAST ALGORITHMS FOR KYP SDPS 6 SDP applications Systems and control (quite well-known) Circuit design Nonconvex optimization... many others

20 FAST ALGORITHMS FOR KYP SDPS 7 Outline A brief introduction to Semidefinite Programming (SDP) Focus: LMIs from the Kalman-Yakubovich-Popov Lemma Fast algorithms for SDPs from KYP Lemma

21 FAST ALGORITHMS FOR KYP SDPS 8 Kalman-Yakubovich-Popov lemma Frequency-domain inequality, rational in frequency ω, and affine in a design vector x, expressed as [ (jωi A) 1 B I ] p [ (jωi A) ( x i M i N) 1 B I i=1 ] 0

22 FAST ALGORITHMS FOR KYP SDPS 8 Kalman-Yakubovich-Popov lemma If (A, B) is controllable, then [ (jωi A) 1 B I ] p [ (jωi A) ( x i M i N) 1 B I i=1 ] 0 hold for all ω R

23 FAST ALGORITHMS FOR KYP SDPS 8 Kalman-Yakubovich-Popov lemma If (A, B) is controllable, then [ (jωi A) 1 B I ] p [ (jωi A) ( x i M i N) 1 B I i=1 ] 0 hold for all ω R [ AP + P A P B B T P 0 is feasible (an LMI with variables P, x) ] + p x i M i N 0 i=1

24 FAST ALGORITHMS FOR KYP SDPS 9 KYP Lemma consequences Semi-infinite frequency domain inequality is exactly equivalent to LMI (no sampling) P serves as an auxiliary variable Of enormous importance for systems, control, and signal processing

25 FAST ALGORITHMS FOR KYP SDPS 10 KYP LMI applications Linear system analysis and design: w Plant z Controller Problem: Design LTI controller for LTI plant Constraints specified as frequency domain inequalities on TF from w to z Youla parametrization used to express TF from w to z! px T (jω, x) = T 1 (jω) + T 2 (jω) x i Q i (jω) T 3 (jω), i=1 KYP Lemma used to obtain LMIs in variable x

26 FAST ALGORITHMS FOR KYP SDPS 10 KYP LMI applications Linear system analysis and design Digital filter design: An FIR or more general filter design problem: Find x such that H(e jθ, x) = Xp 1 i=0 x i H i (e jθ ) satisfies frequency-domain constraints (i.e., for all θ [0, 2π]) KYP Lemma used to obtain LMIs in variable x

27 FAST ALGORITHMS FOR KYP SDPS 10 KYP LMI applications Linear system analysis and design Digital filter design Robust control analysis: Stability of interconnected systems via passivity or small-gain analysis Techniques that take advantage of uncertainty structure/nature Performance analysis via Lyapunov functions

28 FAST ALGORITHMS FOR KYP SDPS 11 Focus on: KYP SDP minimize c T x + Tr(CP ) subject to [ A T P + P A P B B T P 0 ] + p i=1 x im i N where c R p, C S n, A R n n, B R n m, M i S n+m, N S n+m

29 FAST ALGORITHMS FOR KYP SDPS 11 Focus on: KYP SDP minimize c T x + Tr(CP ) subject to [ A T P + P A P B B T P 0 ] + p i=1 x im i N where c R p, C S n, A R n n, B R n m, M i S n+m, N S n+m (Extension to multiple LMIs in multiple variables straightforward) minimize subject to c T x + P K k=1 Tr(C kp k )» A T k P k + P k A k P k B k B T k P k 0 + P p i=1 x im ki N k, k = 1,..., K.

30 FAST ALGORITHMS FOR KYP SDPS 12 Numerical solution of SDPs All SDPs are convex optimization problems: Generic algorithms will work in polynomial-time Matlab LMI Control Toolbox available Moderate size problems solved quite easily But...

31 FAST ALGORITHMS FOR KYP SDPS 12 Numerical solution of SDPs All SDPs are convex optimization problems: Generic algorithms will work in polynomial-time Matlab LMI Control Toolbox available Moderate size problems solved quite easily But... KYP SDPs tend to be of very large scale Large problem sizes due to: underlying problems themselves auxiliary variable P Rest of the talk on efficient solution of KYP SDPs using convex duality

32 FAST ALGORITHMS FOR KYP SDPS 13 Rewrite SDP as Convex duality minimize c T x subject to F 0 + x 1 F x p F p S = 0 S 0

33 FAST ALGORITHMS FOR KYP SDPS 13 Convex duality Primal SDP minimize c T x subject to F 0 + x 1 F x p F p S = 0 S 0 Dual SDP maximize TrF 0 Z subject to Z 0 TrF i Z = c i, i = 1,..., m

34 FAST ALGORITHMS FOR KYP SDPS 13 Convex duality Primal SDP minimize c T x subject to F 0 + x 1 F x p F p S = 0 S 0 Dual SDP maximize TrF 0 Z subject to Z 0 TrF i Z = c i, i = 1,..., m If Z is dual feasible, then TrF 0 Z p If x is primal feasible, then c T x d Under mild conditions, p = d At optimum, S opt Z opt = F (x opt )Z opt = 0

35 FAST ALGORITHMS FOR KYP SDPS 14 Primal-dual algorithms Solve primal and dual problem together: minimize c T x + TrF 0 Z subject to F 0 + x 1 F x p F p S = 0 S 0, Z 0 TrF i Z = c i, i = 1,..., m

36 FAST ALGORITHMS FOR KYP SDPS 14 Primal-dual algorithms Solve primal and dual problem together: minimize c T x + TrF 0 Z (= TrSZ) subject to F 0 + x 1 F x p F p S = 0 S 0, Z 0 TrF i Z = c i, i = 1,..., m (Optimal value is zero!)

37 FAST ALGORITHMS FOR KYP SDPS 15 Why primal-dual algorithms? At every iteration, we have upper and lower bounds, thus guaranteed accuracy Early termination possible Other advantages at algorithmic level

38 FAST ALGORITHMS FOR KYP SDPS 16 Primal-dual algorithm outline For simplicity, suppose we have a feasible point, i.e., x, Z and S s.t. F 0 + x 1 F x p F p S = 0 S 0, Z 0 TrF i Z = c i, i = 1,..., m (More general case, with infeasible starting points, essentially the same)

39 FAST ALGORITHMS FOR KYP SDPS 16 Primal-dual algorithm outline For simplicity, suppose we have a feasible point, i.e., x, Z and S s.t. At each iteration: F 0 + x 1 F x p F p S = 0 S 0, Z 0 TrF i Z = c i, i = 1,..., m Compute product SZ. If it is small, stop Otherwise, take steps S, Z, and x such that x 1 F x p F p S = 0 TrF i Z = 0, i = 1,..., m (maintain feasibility) S + S 0, Z + Z 0

40 FAST ALGORITHMS FOR KYP SDPS 16 Primal-dual algorithm outline For simplicity, suppose we have a feasible point, i.e., x, Z and S s.t. At each iteration: F 0 + x 1 F x p F p S = 0 S 0, Z 0 TrF i Z = c i, i = 1,..., m Compute product SZ. If it is small, stop Otherwise, take steps S, Z, and x such that x 1 F x p F p S = 0 TrF i Z = 0, i = 1,..., m (maintain feasibility) S + S 0, Z + Z 0 (S + S)(Z + Z) is made smaller (address objective)

41 FAST ALGORITHMS FOR KYP SDPS 17 Solving search equations 1. x 1 F x p F p S = 0 2. TrF i Z = 0, i = 1,..., m 3. (S + S)(Z + Z) is made smaller 4. S + S 0, Z + Z 0

42 FAST ALGORITHMS FOR KYP SDPS 17 Solving search equations 1. x 1 F x p F p S = 0 (1), (2) linear equations 2. TrF i Z = 0, i = 1,..., m 3. (S + S)(Z + Z) is made smaller 4. S + S 0, Z + Z 0

43 FAST ALGORITHMS FOR KYP SDPS 17 Solving search equations 1. x 1 F x p F p S = 0 2. TrF i Z = 0, i = 1,..., m 3. (S + S)(Z + Z) is made smaller 4. S + S 0, Z + Z 0 (1), (2) linear equations (3) accomplished via Newton step, another linear equation

44 FAST ALGORITHMS FOR KYP SDPS 17 Solving search equations 1. x 1 F x p F p S = 0 2. TrF i Z = 0, i = 1,..., m 3. (S + S)(Z + Z) is made smaller 4. S + S 0, Z + Z 0 (1), (2) linear equations (3) accomplished via Newton step, another linear equation Solution strategy: First, eliminate S from the linear equations Next eliminate Z Solve a dense linear system in variable x Reconstruct Z and S S + S 0, Z + Z 0 ensured using line search

45 FAST ALGORITHMS FOR KYP SDPS 18 Outline A brief introduction to Semidefinite Programming (SDP) Focus: LMIs from the Kalman-Yakubovich-Popov Lemma Fast algorithms for SDPs from KYP Lemma

46 FAST ALGORITHMS FOR KYP SDPS 19 General-purpose implementation for KYP SDPs minimize c T x + Tr(CP ) subject to [ A T P + P A P B B T P 0 ] + p i=1 x im i N A R n n, B R n 1 (A, B) controllable p + n(n + 1)/2 variables

47 FAST ALGORITHMS FOR KYP SDPS 20 Primal SDP Primal and dual KYP SDPs minimize c T x + Tr(CP ) subject to [ A T P + P A P B B T P 0 ] + p i=1 x im i N Dual SDP maximize Tr(N Z) subject to AZ 11 + Z 11 A T + zb T + B z T = C TrM i Z = c i [ Z11 z Z = z T 2z n+1 ] 0

48 FAST ALGORITHMS FOR KYP SDPS 20 Primal SDP Primal and dual KYP SDPs minimize c T x + Tr(CP ) subject to [ A T P + P A P B B T P 0 ] + p i=1 x im i N Dual SDP maximize Tr(N Z) subject to AZ 11 + Z 11 A T + zb T + B z T = C TrM i Z = c i [ Z11 z Z = z T 2z n+1 (For future reference z = [ z T, z n+1 ] T ) ] 0

49 FAST ALGORITHMS FOR KYP SDPS 21 Search equations for KYP SDPs W ZW + [ A T P + P A P B B T P 0 ] + p x i M i = D i=1 A Z 11 + Z 11 A T + zb T + B z T = 0 W 0; values of W, D change at each iteration TrM i Z = 0

50 FAST ALGORITHMS FOR KYP SDPS 21 Search equations for KYP SDPs W ZW + [ A T P + P A P B B T P 0 ] + p x i M i = D i=1 A Z 11 + Z 11 A T + zb T + B z T = 0 TrM i Z = 0 W 0; values of W, D change at each iteration For convenience: [ A K(P ) = T P + P A P B B T P 0 ], M(x) = p i=1 x im i

51 FAST ALGORITHMS FOR KYP SDPS 21 Search equations for KYP SDPs W ZW + [ A T P + P A P B B T P 0 ] + p x i M i = D i=1 A Z 11 + Z 11 A T + zb T + B z T = 0 TrM i Z = 0 W 0; values of W, D change at each iteration For convenience: [ ] A K(P ) = T P + P A P B B T, M(x) = p P 0 i=1 x im i Then, K adj ( Z) = A Z 11 + Z 11 A T + zb T + B z T, M adj ( Z) = {TrM i Z}

52 FAST ALGORITHMS FOR KYP SDPS 22 Standard method of solving the search equations W ZW + K( P ) + M( x) = D K adj ( Z) = 0 M adj ( Z) = 0

53 FAST ALGORITHMS FOR KYP SDPS 22 Standard method of solving the search equations W ZW + K( P ) + M( x) = D K adj ( Z) = 0 M adj ( Z) = 0 General-purpose solvers eliminate Z from first equation: K adj (W 1 (K( P ) + M( x))w 1 ) = K adj (W 1 DW 1 ) M adj (W 1 (K( P ) + M( x))w 1 ) = M adj (W 1 DW 1 ) A dense set of linear equations in P, x Cost: At least O(n 6 )

54 FAST ALGORITHMS FOR KYP SDPS 23 Alternative method: Step 1 W ZW + K( P ) + M( x) = D K adj ( Z) = 0 M adj ( Z) = 0

55 FAST ALGORITHMS FOR KYP SDPS 23 Alternative method: Step 1 W ZW + K( P ) + M( x) = D A Z 11 + Z 11 A T + zb T + B z T = 0 M adj ( Z) = 0 Use second equation to express Z 11 in terms of z: Z 11 = n i=1 z ix i, where AX i + X i A T + Be T i + e ib T = 0 [ n Thus Z = B( z) = i=1 z ] ix i z z T 2 z n+1

56 FAST ALGORITHMS FOR KYP SDPS 23 Alternative method: Step 1 W ZW + K( P ) + M( x) = D A Z 11 + Z 11 A T + zb T + B z T = 0 M adj ( Z) = 0 Use second equation to express Z 11 in terms of z: Z 11 = n i=1 z ix i, where AX i + X i A T + Be T i + e ib T = 0 [ n Thus Z = B( z) = i=1 z ] ix i z z T 2 z n+1 Substituting in first and third equations gives W B( z)w + K( P ) + M( x) = D M adj (B( z)) = 0

57 FAST ALGORITHMS FOR KYP SDPS 24 Alternative method: Step 2 W B( z)w + K( P ) + M( x) = D M adj (B( z)) = 0 Note that G = K( P ) for some P B adj (G) = 0

58 FAST ALGORITHMS FOR KYP SDPS 24 Alternative method: Step 2 W B( z)w + K( P ) + M( x) = D M adj (B( z)) = 0 Note that G = K( P ) for some P B adj (G) = 0 Use to eliminate P : B adj (W B( z)w ) + B adj (M( x)) = B adj (D) M adj (B( z)) = 0 n + p + 1 linear equations in n + p + 1 variables z, x

59 FAST ALGORITHMS FOR KYP SDPS 25 Reduced search equations of the form Alternative method: Summary [ P11 P 12 P12 T 0 ] [ z x ] = [ q1 0 ] Cost of solving is O(n 3 ) operations (if we assume p = O(n)) From z, x, can find Z, P in O(n 3 ) operations Need to precompute X i s (O(n 4 )) P 12 is independent of current iterates and can be pre-computed, in O(n 4 ) Constructing P 11 requires constructing terms such as Tr(X i W 11 X j W 11 ) and W 11 X i W 12 (also O(n 4 )) Overall cost dominated by O(n 4 )

60 FAST ALGORITHMS FOR KYP SDPS 26 Numerical example KYP IPM SeDuMi (primal) n = p prep. time time/iter. time/iter CPU time in seconds on 2.4GHz PIV with 1GB of memory KYP-IPM: Matlab implementation of alternative method SeDuMi (primal): SeDuMi version 1.05 applied to primal problem Prep. time is time to compute matrices X i #iterations in both methods is comparable (7 15)

61 FAST ALGORITHMS FOR KYP SDPS 27 Further reduction in computation Use factorization of A to compute terms such as Tr(X i W 11 X j W 11 ) without computing X i, i.e., without explicitly solving AX i + X i A T + Be T i + e i B T = 0, i = 1,..., n Advantages: no need to store matrices X i, faster construction of reduced search equations Possible factorizations: eigenvalue decomposition, companion form,... For example, if A has distinct eigenvalues A = V diag(λ)v 1, easy to write down search equations in O(n 3 ), in terms of V and λ

62 FAST ALGORITHMS FOR KYP SDPS 28 Existence of distinct stable eigenvalues By assumption, (A, B) is controllable; hence can arbitrarily assign eigenvalues of A + BK by choosing K Choose T = [ I 0 K I ], and replace LMI by equivalent LMI ( [ T T A T P + P A P B B T P 0 ] + ) N x i M i T T T NT i=1 [ (A + BK) T P + P (A + BK) P B B T P 0 ] + N x i (T T M i T ) T T NT i=1 Conclusion: Can assume without loss of generality that A is stable with distinct eigenvalues

63 FAST ALGORITHMS FOR KYP SDPS 29 Numerical example Five randomly generated problems with p = 50, n = 100,..., 500 KYP IPM (fast) KYP IPM SeDuMi (primal) n prep. time time/iter prep. time time/iter prep. time time/iter KYP-IPM (fast) uses eigenvalue decomposition of A to construct reduced search equations Preprocessing time and time/iteration grow as O(n 3 )

64 FAST ALGORITHMS FOR KYP SDPS 30 SDPs derived from the KYP-lemma Conclusions A useful class of SDPs, widely encountered in systems, control and signal processing Difficult to solve using general-purpose software Generic solvers take O(n 6 ) computation Fast solution using interior-point methods Custom implementation based on fast solution of search equations (cost O(n 4 ) or O(n 3 ))

Semidefinite Programming Duality and Linear Time-invariant Systems

Semidefinite Programming Duality and Linear Time-invariant Systems Venkataramanan (Ragu) Balakrishnan School of ECE, Purdue University 2 July 2004 Workshop on Linear Matrix Inequalities in Control LAAS-CNRS,