PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.1/39

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1 PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP Michal Kočvara Institute of Information Theory and Automation Academy of Sciences of the Czech Republic and Czech Technical University kocvara@utia.cas.cz PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.1/39

2 PBM Method for convex NLP Ben-Tal, Zibulevsky, 92, 97 Combination of: (exterior) Penalty meth., (interior) Barrier meth., Method of Multipliers Problem: (CP) min {f(x) : g i(x) 0, i = 1,..., m} x R n Assume: 1. f, g i ( i = 1,..., m ) convex 2. X nonempty and compact (A1) 3. ˆx so that g i (ˆx) < 0 for all i = 1,..., m (A2) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.2/39

3 PBM Method for convex NLP ϕ possibly smooth, domϕ possibly large (ϕ 0 ) ϕ strictly convex, strictly monotone increasing and C 2 (ϕ 1 ) domϕ = (, b) with 0 < b (ϕ 2 ) ϕ(0) = 0, (ϕ 4 ) lim ϕ (t) = t b (ϕ 3 ) ϕ (0) = 1, (ϕ 5 ) lim t ϕ (t) = 0 1 ϕ(t) b ϕ (t) b PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.3/39

4 PBM Method for convex NLP Examples: ϕ r 1 (t) = { c1 1 2 t2 + c 2 t + c 3 t r c 4 log(t c 5 ) + c 6 t < r. PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.4/39

5 PBM Method for convex NLP Examples: ϕ r 1 (t) = { c1 1 2 t2 + c 2 t + c 3 t r c 4 log(t c 5 ) + c 6 t < r. ϕ r 2 (t) = { c t2 + c 2 t + c 3 t r, c 4 t c 5 + c 6 t < r, r 1, 1 PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.4/39

6 PBM Method for convex NLP Examples: ϕ r 1 (t) = { c1 1 2 t2 + c 2 t + c 3 t r c 4 log(t c 5 ) + c 6 t < r. ϕ r 2 (t) = { c t2 + c 2 t + c 3 t r, c 4 t c 5 + c 6 t < r, r 1, 1 Properties: C 2, bounded second derivative = improved behaviour of Newton s method composition of barrier branch (logarithmic/reciprocal) and penalty branch (quadratic) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.4/39

7 PBM algorithm for convex problems With p i > 0 for i {1,..., m}, we have g i (x) 0 p i ϕ(g i (x)/p i ) 0, i = 1,..., m The corresponding augmented Lagrangian: F(x, u, p) := f(x) + m u i p i ϕ(g i (x)/p i ) i=1 PBM algorithm: = arg min x IR uk, p k ) n u k+1 i = u k i ϕ (g i (x k+1 )/p k i ) i = 1,..., m x k+1 p k+1 i = π p k i i = 1,..., m PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.5/39

8 Properties of the PBM method Theory: {u k } k generated by PBM is the same as for a Proximal Point algorithm applied to the dual problem ( convergence proof) any cluster point of {x k } k is an optimal solution to (CP) f(x k ) f without p k 0 PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.6/39

9 Properties of the PBM method Theory: {u k } k generated by PBM is the same as for a Proximal Point algorithm applied to the dual problem ( convergence proof) any cluster point of {x k } k is an optimal solution to (CP) f(x k ) f without p k 0 Praxis: fast convergence thanks to the barrier branch of ϕ particularly suitable for large sparse problems robust, typically outer iterations and Newton steps PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.6/39

10 PBM in semidefinite programming Problem: { min b T x : A(x) 0 } x R n Question: How can the matrix constraint be treated by PBM approach? A(x) 0 (A : R n S d convex) Idea: Find an augmented Lagrangian as follows: F(x, U, p) = f(x) + U, Φ p (A(x)) Sd PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.7/39

11 PBM in semidefinite programming Problem: { min b T x : A(x) 0 } x R n Question: How can the matrix constraint be treated by PBM approach? A(x) 0 (A : R n S d convex) Idea: Find an augmented Lagrangian as follows: F(x, U, p) = f(x) + U, Φ p (A(x)) Sd Notation: A, B Sd S d+ U S d+ Φ p := tr ( A T B ) inner product on S d = {A S d A positive semidefinite} matrix multiplier (dual variable) penalty function on S d PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.7/39

12 Construction of the penalty function Φ p, first idea Given: scalar valued penalty function ϕ satisfying (ϕ 0 ) (ϕ 5 ) matrix A = S ΛS, where Λ = diag (λ 1, λ 2,..., λ d ) Define A Φ p S T pϕ ( λ1 p ) 0 pϕ ( λ2 p ) pϕ ( λd p ) S any positive eigenvalue of A is penalized by ϕ PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.8/39

13 PBM algorithm for semidefinite problems We have A(x) 0 Φ p (A(x)) 0 and the corresponding augmented Lagrangian: PBM algorithm: F(x, U, p) := f(x) + U, Φ p (A(x)) Sd (i) x k+1 = argmin x R n F(x, Uk, p k ) (ii) U k+1 = D A Φ p (A(x); U k ) (iii) p k+1 < p k PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.9/39

14 PBM algorithm for semidefinite problems The first idea may not be the best one: The matrix function Φ p corresponding to ϕ is convex but may be nonmonotone on H d (r, ) (right branch) may be nonconvex. U, Φ p (A(x)) Sd PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.10/39

15 PBM algorithm for semidefinite problems The first idea may not be the best one: The matrix function Φ p corresponding to ϕ is convex but may be nonmonotone on H d (r, ) (right branch) may be nonconvex. U, Φ p (A(x)) Sd Complexity of Hessian assembling O(d 4 + d 3 n + d 2 n 2 ) Even for a very sparse structure the complexity can be O(d 4 )! n... number of variables d... size of matrix constraint PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.10/39

16 PBM algorithm for semidefinite problems Hessian: [ xx U, Φ p (A(x)) Sd ] d k=1 i,j = ( [ ( sk (x) A i S(x) [ 2 ϕ(λ l (x), λ m (x), λ k (x))] n l,m=1 ] ) [S(x) US(x)] S(x) A j s k (x) ) S : s k : decomposition matrix of A(x) k-th column of S i divided difference of i-th order A : S d R n conjugate operator to A PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.11/39

17 Construction of the penalty function, second idea Find a penalty function ϕ which allows direct computation of Φ, its gradient and Hessian. PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.12/39

18 Construction of the penalty function, second idea Find a penalty function ϕ which allows direct computation of Φ, its gradient and Hessian. Example: (A(x) = x i A i ) ϕ(x) = x 2 Φ(A) = A 2 Then and Φ(A(x)) = A(x)A i + A i A(x) x i 2 x i x j Φ(A(x)) = A j A i + A i A j PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.12/39

19 Construction of the penalty function The reciprocal barrier function in SDP: (A(x) = x i A i ) ϕ := 1 t 1 1 The corresponding matrix function is and we can show that and 2 Φ(A) = (A I) 1 I x i Φ(A(x)) = (A I) 1 A i (A I) 1 x i x j Φ(A(x)) = (A I) 1 A i (A I) 1 A j (A I) 1 PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.13/39

20 Construction of the penalty function Complexity of Hessian assembling: O(d 3 n + d 2 n 2 ) for dense matrices O(n 2 K 2 ) for sparse matrices (K... max. number of nonzeros in A i, i = 1,..., n) Compare to O(d 4 + d 3 n + d 2 n 2 ) in the general case { min b T x : A(x) 0 } x R n A : R n S d PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.14/39

21 Handling sparsity... essential for code efficiency { min b T x : A(x) 0 } x R n A = x i A i Three basic sparsity types: many (small) blocks sparse Hessian (multi-load truss/material) (A I) 1 A i (A I) 1 A j (A I) 1 PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.15/39

22 Handling sparsity... essential for code efficiency { min b T x : A(x) 0 } x R n A = x i A i Three basic sparsity types: many (small) blocks sparse Hessian (multi-load truss/material) few (large) blocks (A I) 1 A i (A I) 1 A j (A I) 1 PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.15/39

23 Handling sparsity... essential for code efficiency { min b T x : A(x) 0 } x R n A = x i A i Three basic sparsity types: many (small) blocks sparse Hessian (multi-load truss/material) few (large) blocks A dense, A i sparse (most of SDPLIB examples) (A I) 1 A i (A I) 1 A j (A I) 1 PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.15/39

24 Handling sparsity x + 1 x + 2 x full version as inefficient as general sparse version Recently, 3 matrix-matrix multiplication routines: full full full sparse sparse sparse PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.16/39

25 Handling sparsity essential for code efficiency { min b T x : A(x) 0 } x R n A = x i A i Three basic sparsity types: many (small) blocks sparse Hessian (multi-load truss/material) few (large) blocks A dense, A i sparse (most of SDPLIB examples) A sparse (truss design with buckling/vibration, maxg,... ) (A I) 1 A i (A I) 1 A j (A I) 1 PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.17/39

26 Handling sparsity Fast inverse computation of sparse matrices Z = M 1 N Explicite inverse of M: O(n 3 ) Assume M is sparse and Cholesky factor L of M is sparse Z i = (L 1 ) T L 1 N i, i = 1,..., n Complexity: n times nk O(n 2 K) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.18/39

27 Numerical results New code called PENNON Comparison with DSDP by Benson and Ye SDPT3 by Toh, Todd and Tütüncü SeDuMi by Jos Sturm SDPLIB problems: sdplib/ PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.19/39

28 Numerical results problem variables matrix DSDP SDPT3 PENNON arch control control gpp mcp qap ss theta equalg equalg maxg maxg PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.20/39

29 Examples from Mechanics Multiple-load Free Material Optimization After reformulation, discretization, further reformulation: min α R,x (R n ) L { α } L (c l ) T x l A i (α, x) 0, i = 1,..., m l=1 Many ( 5000) small (11 19) matrices. Large dimension (nl ) In a standard form: { min x (R n ) L a T x nl i=1 x i B i 0 } x 1 + x PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.21/39

30 Examples from Mechanics no. of size of problem var. matrix DSDP SDPT3 SeDuMi PENNON mater mater mater m mater m m PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.22/39

31 Truss design with free vibration control Lowest eigenfrequency of the optimal structure should be bigger than a prescribed value min t,u s.t. ti A(t)u = f σ σ l (g(u) c) t T min. eigenfrequency a given value PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.23/39

32 Truss design with free vibration control Formulated as SDP problem: ti min t subject to A(t) ( λm(t) ) 0 c f T 0 f A(t) t i 0, i = 1,..., n where A(t) = t i A i A i = E i l 2 i γ i γ T i M(t) = t i M i M i = c diag(γ i γ T i ) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.24/39

33 truss test problems trto: problems from single-load truss topology design. Normally formulated as LP, here reformulated as SDP for testing purposes. vibra: single load truss topology problems with a vibration constraint. The constraint guarantees that the minimal self-vibration frequency of the optimal structure is bigger than a given value. buck: single load truss topology problems with linearized global buckling constraint. Originally a nonlinear matrix inequality, the constraint should guarantee that the optimal structure is mechanically stable (does not buckle). All problems characterized by sparsity of the matrix operator A. PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.25/39

34 truss test problems problem n m DSDP SDPT3 PENNON trto trto trto buck buck buck vibra vibra vibra PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.26/39

35 Benchmark tests by Hans Mittelmann: Implemented on the NEOS server: Homepage: kocvara/pennon/ Available with MATLAB interface through TOMLAB PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.27/39

36 When PCG helps (SDP)? Linear SDP, dense Hessian Complexity of Hessian evaluation A = n A i, A i R m m i=1 O(m 3 A n + m2 A n2 ) for dense matrices O(m 2 A n + K2 n 2 ) for sparse matrices (K... max. number of nonzeros in A i, i = 1,..., n) Complexity of Cholesky algorithm - linear SDP O(n 3 ) (... from PCG we expect O(n 2 )) Problems with large n and small m: CG better than Cholesky (expected) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.28/39

37 Iterative algorithms Conjugate Gradient method for Hd = g, H S n y = Hx complexity O(n 2 ) Exact arithmetics: convergence in n steps overall complexity O(n 3 ) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.29/39

38 Iterative algorithms Conjugate Gradient method for Hd = g, H S n y = Hx complexity O(n 2 ) Exact arithmetics: convergence in n steps Praxis: may be much worse (ill-conditioned problems) overall complexity O(n 3 ) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.29/39

39 Iterative algorithms Conjugate Gradient method for Hd = g, H S n y = Hx complexity O(n 2 ) Exact arithmetics: convergence in n steps Praxis: may be much worse (ill-conditioned problems) Praxis: may be much better preconditioning overall complexity O(n 3 ) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.29/39

40 Iterative algorithms Conjugate Gradient method for Hd = g, H S n y = Hx complexity O(n 2 ) Exact arithmetics: convergence in n steps Praxis: may be much worse (ill-conditioned problems) Praxis: may be much better preconditioning Convergence theory: number of iterations depends on condition number distribution of eigenvalues overall complexity O(n 3 ) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.29/39

41 Iterative algorithms Conjugate Gradient method for Hd = g, H S n y = Hx complexity O(n 2 ) Exact arithmetics: convergence in n steps Praxis: may be much worse (ill-conditioned problems) Praxis: may be much better preconditioning Convergence theory: number of iterations depends on condition number distribution of eigenvalues overall complexity O(n 3 ) Preconditioning: solve M 1 Hd = M 1 g with M H 1 PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.29/39

42 Conditioning of Hessian Solve Hd = g, H Hessian of F(x, u, U, p, P) = f(x) + U, Φ P (A(x)) SmA Condition number depends on P Example: problem Theta2 from SDPLIB (n = 498) κ 0 = 394 κ opt = PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.30/39

43 Theta2 from SDPLIB (n = 498) Behaviour of CG: testing Hd + g / g PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.31/39

44 Theta2 from SDPLIB (n = 498) Behaviour of QMR: testing Hd + g / g PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.31/39

45 Theta2 from SDPLIB (n = 498) QMR: effect of preconditioning (for small P ) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.31/39

46 Control3 from SDPLIB (n = 136) κ 0 = κ opt = PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.32/39

47 Control3 from SDPLIB (n = 136) Behaviour of CG: testing Hd + g / g PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.32/39

48 Control3 from SDPLIB (n = 136) Behaviour of QMR: testing Hd + g / g PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.32/39

49 Preconditioners Should be: efficient (obvious but often difficult to reach) simple (low complexity) only use Hessian-vector product (NOT Hessian elements) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.33/39

50 Preconditioners Should be: efficient (obvious but often difficult to reach) simple (low complexity) only use Hessian-vector product (NOT Hessian elements) Diagonal Symmetric Gauss-Seidel L-BFGS (Morales-Nocedal, SIOPT 2000) A-inv (approximate inverse) (Benzi-Collum-Tuma, SISC 2000) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.33/39

51 Preconditioners Monteiro, O Neil, Nemirovski (2004): Adaptive Ellipsoid Preconditioner Improves the CG performance on extremely ill-conditioned systems. preconditioner: M = C k C T k, C k+1 αc k + βc k p k p T k, C 1 = γi α, β, p k... by matrix-vector products VERY preliminary results (MATLAB implementation) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.34/39

52 Preconditioners example Example: problem Theta2 from SDPLIB (n = 498) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.35/39

53 Preconditioners example Example: problem Theta2 from SDPLIB (n = 498) CG CG-diag CG-SGS CG-MON. PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.36/39

54 Hessian free methods Use finite difference formula for Hessian-vector products: with h = (1 + x k 2 ε) 2 F(x k )v F(x k + hv) F(x k ) h Complexity: Hessian-vector product = gradient evaluation Complexity: need for Hessian-vector-product type preconditioner Limited accuracy (4 5 digits) PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.37/39

55 Test results: linear SDP, dense Hessian Stopping criterium for PENNON Exact Hessian: 10 7 (7 8 digits in objective function) Approximate Hessian: 10 4 (4 5 digits in objective function) Stopping criterium for CG/QMR??? Hd = g, stop when Hd + g / g ǫ PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.38/39

56 Test results: linear SDP, dense Hessian Stopping criterium for PENNON Exact Hessian: 10 7 (7 8 digits in objective function) Approximate Hessian: 10 4 (4 5 digits in objective function) Stopping criterium for CG/QMR??? Hd = g, stop when Hd + g / g ǫ Experiments: ǫ = 10 2 sufficient. often very low (average) number of CG iterations Complexity: n 3 kn 2, k 4 8 Practice: effect not that strong, due to other complexity issues PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.38/39

57 Problems with large n and small m Library of examples with large n and small m (courtesy of Kim Toh) CG-exact much better than Cholesky CG-approx much better than CG-exact PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.39/39

58 problem n m PENSDP PENSDP (APCG) SDPLR CPU CPU CG CPU iter ham_7_5_ ham_9_ ham_8_3_ ham_9_5_ theta theta theta theta theta m theta m theta theta m theta m theta m theta t keller sanr problem theta83 theta103 theta123 n m PENSDP (APCG) RENDL