BFGS-like updates of Constraint Preconditioners for sequences of KKT linear systems

Transcription

1 BFGS-like updates of Constraint Preconditioners for sequences of KKT linear systems Valentina De Simone Dept. of Mathematics and Physics Univ. Campania Luigi Vanvitelli Joint work with Daniela di Serafino (Univ. Campania Luigi Vanvitelli), Luca Bergamaschi and Ángeles Martínez (Univ. Padua) Due giorni di Algebra Lineare Numerica- febbraio 2017

2 Outline 1 Problem, background and motivations 2 BFGS-like updates of Constraint Preconditioners 3 Numerical experiments 4 Conclusions and future work Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

3 Problem, background and motivations Sequences of KKT linear systems Problem Preconditioning sequences of large and sparse KKT (saddle-point) linear systems H k u k = d k, k = 1, 2,... where [ Gk A H k = T A 0 ] G k R n n symmetric positive definite A R m n full rank, with m n [ ] [ ] u1,k d1,k u k =, d k = u 2,k d 2,k Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

4 Problem, background and motivations Sequences of KKT linear systems KKT systems in Interior Point methods for QP KKT linear systems arise at each iteration of Interior Point (IP) methods for convex QP problems 1 minimize 2 x T Qx + c T x, subject to Ax = b, x 0, Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

5 Problem, background and motivations Sequences of KKT linear systems KKT systems in Interior Point methods for QP KKT linear systems arise at each iteration of Interior Point (IP) methods for convex QP problems 1 minimize 2 x T Qx + c T x, subject to Ax = b, x 0, (1,1) block of the KKT matrix H k at kth IP iteration: G k = Q + Θ k, Θ k = X 1 k Z k X k = diag(x k ), Z k = diag(z k ), x k, z k complementary variables As the IP iterate approaches the optimal solution, the entries of Θ k may tend either to zero or to infinity = strongly increasing ill-conditioning of H k [S. Wright, SIAM, 1997; D Apuzzo, DS & di Serafino, COAP, 2010; Gondzio, EJOR, 2012] Large-scale problems = Krylov solvers = effective preconditioners fundamental for the efficiency of the IP method Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

6 Problem, background and motivations Constraint Preconditioners Constraint Preconditioners [ G A T Hu = d, H = A 0 ], G = Q + Θ [ E A T B = A 0 ] Costraint Preconditioner (CP) E simple symmetric approximation to the (1,1) block G such that B is nonsingular able to capture the behaviour of Θ common choice E = diag(g) E also implictly defined by using factorizations of the form B = MCM T, with specially chosen M and C [Dollar, Gould, Schilders & Wathen, 2006; Dollar & Wathen, 2006] Widely investigated and successfully used in Optimization (and PDEs) [surveys: Benzi, Golub & Liesen, 2005; D Apuzzo, DS & di Serafino, 2010] Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

7 Problem, background and motivations Constraint Preconditioners Properties of Constraint Preconditioners [Lukšan & VLček, 1998; Keller, Gould & Wathen, 2000; Bergamaschi, Gondzio & Zilli, 2004] [ ] G A T H = A 0 [ ] E A T (KKT matrix), B = A 0 (CP), E positive definite, Z R n (n m) basis of N (A) (nullspace of A) B 1 H has an eigenvalue at 1 with multiplicity 2m The remaining n m eigenvalues of B 1 H are defined by Z T GZw = λz T EZw λ min (E 1 G) λ((z T EZ 1 )Z T GZ) λ max (E 1 G) If a breakdown does not occur, then CG with preconditioner B computes the solution u of the KKT system Hu = d in at most m n + 2 iterations Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

8 Problem, background and motivations Constraint Preconditioners Properties of Constraint Preconditioners If CG is applied to Hu = d with preconditioner B and starting guess [ ] u (0) u (0) = 1 u (0) such that Au (0) 1 = d 2, 2 then the corresponding iterates u (j) 1 are the same as the ones generated by CG, with preconditioner Z T EZ, applied to (Z T GZ)u 1 = Z T (d 1 Gu (0) ) ( ) Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

9 Problem, background and motivations Constraint Preconditioners Properties of Constraint Preconditioners If CG is applied to Hu = d with preconditioner B and starting guess [ ] u (0) u (0) = 1 u (0) such that Au (0) 1 = d 2, 2 then the corresponding iterates u (j) 1 are the same as the ones generated by CG, with preconditioner Z T EZ, applied to (Z T GZ)u 1 = Z T (d 1 Gu (0) ) ( ) the direction p (j) and the residual r (j) generated by CG take the form [ ] p (j) Z p (j) [ ] = 1 p (j), r (j) (j) r = 1, 2 0 where p (j) 1 and r (j) 1 are the direction and residual generated by CG, with preconditioner Z T EZ, applied to system ( ), and (p (j) ) T Hp (i) = ( p (j) 1 )T Z T GZ p (i) 1 Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

10 Problem, background and motivations Constraint Preconditioners Application of Constraint Preconditioners E = diag(g) The application of a CP requires its factorization, e.g. [ ] [ ] [ ] [ E A T I 0 E 0 I E B = = 1 A T A 0 AE 1 I 0 LDL T 0 I where AE 1 A T = LDL T (sqrt-free Cholesky factorization) ] Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

11 Problem, background and motivations Constraint Preconditioners Application of Constraint Preconditioners E = diag(g) The application of a CP requires its factorization, e.g. [ ] [ ] [ ] [ E A T I 0 E 0 I E B = = 1 A T A 0 AE 1 I 0 LDL T 0 I where AE 1 A T = LDL T (sqrt-free Cholesky factorization) ] Inverse CP (used in our work) [ P = B 1 I E = 1 A T 0 I ] [ E L T D 1 L 1 ] [ I 0 AE 1 I ] Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

12 Problem, background and motivations Constraint Preconditioners Application of Constraint Preconditioners E = diag(g) The application of a CP requires its factorization, e.g. [ ] [ ] [ ] [ E A T I 0 E 0 I E B = = 1 A T A 0 AE 1 I 0 LDL T 0 I where AE 1 A T = LDL T (sqrt-free Cholesky factorization) ] Inverse CP (used in our work) [ P = B 1 I E = 1 A T 0 I ] [ E L T D 1 L 1 ] [ I 0 AE 1 I ] The factorization of CP may account for a large part of the cost of the IP method (depending on the sparsity of A and the Schur complement AE 1 A T ) Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

13 Problem, background and motivations Preconditioner updates Reducing the cost for building CPs In order to reduce computational cost, inexact CPs have been developed, based on either approximate factorizations of AE 1 A T or sparse approximations of A (CG cannot be used) [Lukšan & Vlček, 1998; Durazzi & Ruggiero, 2003; Perugia & Simoncini, 2000; Benzi & Simoncini, 2006; Bergamaschi, Gondzio, Venturin & Zilli, 2007; Sesana & Simoncini, 2013] Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

14 Problem, background and motivations Preconditioner updates Reducing the cost for building CPs In order to reduce computational cost, inexact CPs have been developed, based on either approximate factorizations of AE 1 A T or sparse approximations of A (CG cannot be used) [Lukšan & Vlček, 1998; Durazzi & Ruggiero, 2003; Perugia & Simoncini, 2000; Benzi & Simoncini, 2006; Bergamaschi, Gondzio, Venturin & Zilli, 2007; Sesana & Simoncini, 2013] Recent development of inexact CPs based on updating techniques General idea: given a (factorized) CP for some KKT matrix of the sequence (seed CP), build preconditioners for subsequent KKT matrices at a moderate cost, by updating the (factorization of) the seed CP with available info Low-rank updates of the Cholesky factorization of AE 1 A T [Bellavia, DS, di Serafino & Morini, SIOPT 2015, COAP 2016] Low-rank quasi-newton updates of inexact CPs with sparse approx of A [Fisher, Gratton, Gürol, Trémolet & Vassuer, CERFACS TR, 2016] Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

15 BFGS-like updates of Constraint Preconditioners The updating technique A limited-memory BFGS-like updating technique [ ] Ĝ A T Notations: Ĥ = A 0 [ ] G A T H = A 0 seed KKT matrix, B = [ Ê A T A 0 ] seed CP, P = B 1 current KKT matrix, B upd updated CP for H, P upd = B 1 upd Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

16 BFGS-like updates of Constraint Preconditioners The updating technique A limited-memory BFGS-like updating technique [ ] Ĝ A T Notations: Ĥ = A 0 [ ] G A T H = A 0 seed KKT matrix, B = [ Ê A T A 0 ] seed CP, P = B 1 current KKT matrix, B upd updated CP for H, P upd = B 1 upd Our aim: extend to KKT systems the limited-memory preconditioners (LMPs) for spd matrices developed in [Gratton, Sartenaer & Tshimanga, SIOPT 2011] [ ] S1 Consider S = R (n+m) q, q m n, rank(s 1 ) = q, AS 1 = 0 S 2 Build a preconditioner for H by performing a rank-q BFGS-like update of B: B upd = B + HS(S T HS) 1 S T H BS(S T BS) 1 S T B or (via the Sherman-Morrison-Woodbury formula): P upd = S(S T HS) 1 S T + ( I S(S T HS) 1 S T H ) P ( I HS(S T HS) 1 S T ) S T HS = S T 1 GS 1 and S T BS = S T 1 ÊS 1 = B upd and P upd are well defined Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

17 BFGS-like updates of Constraint Preconditioners Theoretical properties Structure of B upd Structure of B upd B upd = B + HS(S T HS) 1 S T H BS(S T BS) 1 S T B [ ] HS(S T HS) 1 S T Γ 0 H =, 0 0 Γ = (GS 1 + A T S 2 )(S1 T GS 1 ) 1 (S1 T G + S2 T A) [ ] BS(S T BS) 1 S T Φ 0 B =, 0 0 Φ = (ÊS 1 + A T S 2 )(S1 T ÊS 1) 1 (S1 T Ê + S 2 T A) [Ê ] + Γ Φ A T B upd = A 0 exact CP Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

18 BFGS-like updates of Constraint Preconditioners Theoretical properties Spectral properties of B 1 upd H Theorem: B 1 updh has an eigenvalue at 1 with multiplicity 2m + q q more eigs equal to 1 Theorem: Any eigenvalue λ of (Z T GZ)w = λz T (Ê + Γ Φ)Zw satisfies: { } { } min λ min (Ê 1 G), 1 λ max λ max (Ê 1 G), 1 B upd is not worse than B Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

19 BFGS-like updates of Constraint Preconditioners Implementation issues Choice of S [ ] All the directions p (j) p (j) = 1 p (j) generated by applying CG with a CP to any 2 KKT system of the sequence are such that Ap (j) 1 = 0 Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

20 BFGS-like updates of Constraint Preconditioners Implementation issues Choice of S [ ] All the directions p (j) p (j) = 1 p (j) generated by applying CG with a CP to any 2 KKT system of the sequence are such that Ap (j) 1 = 0 Set S = [ p (1),..., p (q)] with p (j) generated by preconditioned CG applied to the previous KKT system in the sequence Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

21 BFGS-like updates of Constraint Preconditioners Implementation issues Choice of S [ ] All the directions p (j) p (j) = 1 p (j) generated by applying CG with a CP to any 2 KKT system of the sequence are such that Ap (j) 1 = 0 Set S = [ p (1),..., p (q)] with p (j) generated by preconditioned CG applied to the previous KKT system in the sequence What about applying q iterations of CG with B to the current system Hu = d, building S with the resulting directions, and restarting CG with B upd from the last computed iterate? (Two-phase CG) Exact arithmetic: two-phase CG with Bupd built from the current directions is equivalent to CG with B (follows from the equivalence between BFGS and CG on the KKT systems, see also [Nazareth, 1979]) Floating-point arithmetic:?? Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

22 BFGS-like updates of Constraint Preconditioners Implementation issues H-orthogonality of preconditioned CG directions CVXQP3 problem (CUTEst), n = 20000, m = 15000, KKT systems from PRQP IP solver [Cafieri, D Apuzzo, DS, di Serafino, Toraldo, ] PCG applied to KKT sys at IP iter 24, B from IP iter 19, q = 50 Normalized scalar Products (p (j) ) T Hp (l ) l = 0, 25, 50 j > l p (j) Hp (l ) 1e+00 1e+00 1e+00 1e+00 1e+00 1e+00 1e-04 1e-04 1e-04 1e-04 1e-04 1e-04 Normalized scalar product 1e-08 1e-08 FIXED (no ortho) BFGS-C FIXED (ortho) Normalized scalar product 1e-08 1e-08 FIXED (no ortho) BFGS-C FIXED (ortho) Normalized scalar product 1e-08 1e-08 FIXED (no ortho) BFGS-C FIXED (ortho) 1e-12 1e-12 1e-12 1e-12 1e-12 1e-12 1e-16 1e PCG iteration j l = 0 1e-16 1e PCG iteration j l = 25 1e-16 1e PCG iteration j l = 50 FIXED (with and without ortho): CG with B, BFGS-C: two-phase CG, with B upd built from the current CG directions ortho: H-orthogonalization of the first q (normalized) CG directions if (p (i) ) T Hp (j) > Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

23 BFGS-like updates of Constraint Preconditioners Implementation issues Algorithmic/implementation details The inverse preconditioner P upd is used in practice P upd = S(S T HS) 1 S T + ( I S(S T HS) 1 S T H ) P ( I HS(S T HS) 1 S T ) Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

24 BFGS-like updates of Constraint Preconditioners Implementation issues Algorithmic/implementation details The inverse preconditioner P upd is used in practice P upd = S(S T HS) 1 S T + ( I S(S T HS) 1 S T H ) P ( I HS(S T HS) 1 S T ) Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

25 BFGS-like updates of Constraint Preconditioners Implementation issues Algorithmic/implementation details The inverse preconditioner P upd is used in practice P upd = S(S T HS) 1 S T + ( I S(S T HS) 1 S T H ) P ( I HS(S T HS) 1 S T ) 1. S = Hk+1 S = (H k + H k+1 H k ) S = S H + (Θ k+1 Θ k ) S k O(nq) flops 2. Computation of S T S O( q2 2 n) flops 3. Cholesky factorization: L T S L S := S T S O(q 3 ) flops Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

26 BFGS-like updates of Constraint Preconditioners Implementation issues Algorithmic/implementation details The inverse preconditioner P upd is used in practice P upd = S(S T HS) 1 S T + ( I S(S T HS) 1 S T H ) P ( I HS(S T HS) 1 S T ) 1. S = Hk+1 S = (H k + H k+1 H k ) S = S H + (Θ k+1 Θ k ) S k O(nq) flops 2. Computation of S T S O( q2 2 n) flops 3. Cholesky factorization: L T S L S := S T S O(q 3 ) flops Computation of w = P upd r (at each PCG iteration) 1. v 1 = S T 4 triangular solves: O(q r 2 ) 2. v 2 = r SL T S L 1 S v 4 mat-vet: O((n + m)q) (BLAS2 routines) 1 3. v 2 = Pv 3 saxpy: O(n + m), O(q) 2 4. v 1 = v 1 S T v 2 1 application of P (factors available) 5. w = v 2 + S L T S L 1 S v 1 further cost reduction by exploiting AS 1 = 0 The cost of the reorthogonalization in the worst case: O(q 2 n/2) flops Choice of q: tradeoff between cost and effectiveness (q n m) Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

27 Numerical experiments Numerical experiments: testing details Comparison of 5 preconditioners (Fortran 95 implementation): RECOM: CP recomputed from scratch FIXED: seed preconditioner P BFGS-P: Pupd built with CG directions computed at previous IP iteration BFGS-C: Pupd built with current CG directions (two-phase CG) DOUBLE: BFGS-P in the first q CG iterations, then Pupd built by updating BFGS-P with the directions computed in the first q CG iters (two-phase CG) convex QP problems from the CUTEst collection and modified versions of them (to reduce sparsity of the Schur complement AE 1 A T ) problem n m nz(a) nz(ae 1 A T ) nz(l) CVXQP CVXQP3N STCQP STCQP2N Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

28 Numerical experiments Numerical experiments: testing details (cont d) Sequences of KKT systems, with right-hand sides and adaptive tolerances, extracted from PRQP, a Fortran 95 primal-dual Potential Reduction solver for convex Quadratic Programming [Cafieri, D Apuzzo, DS, di Serafino & Toraldo, ] CP recomputed every s IP iterations with 2 s 9 actual value of q: q = min{q max, nit prev }, q max = 5, 10, 20, 50, 100, with nit prev = # of CG iterations for solving the previous KKT system in the sequence Failure declared if a KKT system in the sequence in not solved to required accuracy within 600 iterations Computational environment: Intel Core i7 CPU (quad-core, 2.67 GHz) with 6 GB RAM and 8 MB cache, Linux O.S., gfortran compiler (-O4 option), MA57 from HSL Mathematical Software Library Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

29 Numerical experiments Some numerical results CVXQP3N n = 20000, m = 15000, nnz(ae 1 A T ) = total # IP iterations = 36 Prec s q max GC iters Tf-Schur Ta-seed Tupd Ttot RECOM FIXED BFGS-P BFGS-C DOUBLE Tf-Schur: time for factorization of AE 1 A T Ta-seed: time for application of seed Tupd: time for other updating operations, including H-orthogonalization Ttot: total time for preconditioned CG (all times are in seconds) Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

30 Numerical experiments Some numerical results CVXQP3N n = 20000, m = 15000, nnz(ae 1 A T ) = total # IP iterations = 36 Prec s q max GC iters Tf-Schur Ta-seed Tupd Ttot RECOM FIXED BFGS-P failure on system 34 BFGS-C DOUBLE failure on system 34 Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

31 Numerical experiments Some numerical results CVXQP3N: details on CG iterations s = 8, q max = 20, 50 IPit RECOM FIXED BFGS-C BFGS-P DOUBLE Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

32 Numerical experiments Some numerical results CVXQP3N: convergence histories q max = 50, s = 8 1e+08 1e+06 FIXED (8,0) FIXED(8,50) BFGS-C(8,50) DOUBLE(8,50) BFGS-P(8,50) 1e+08 1e+06 1e+04 FIXED (8,0) FIXED(8,50) BFGS-C(8,50) DOUBLE(8,50) BFGS-P(8,50) residual norm 1e+04 residual norm 1e+02 1e+00 1e+02 1e-02 1e-04 1e+00 1e PCG iteration number IP iter= 24, seed from it = PCG iteration number IP iter= 32, seed from it =25 Rank: 1 DOUBLE 2 BFGS-P 3 BFGS-C 4 FIXED Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

33 Numerical experiments Some numerical results STCQP2N n = 16385, m = 8190, nnz(ae 1 A T ) = total # IP iterations = 12 Prec s q max GC iters Tf-Schur Ta-seed Tupd Ttot ORIG SEED BFGS-P BFGS-C DOUBLE Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

34 Conclusions and future work Some conclusions The proposed limited-memory BFGS-like update of a CP is still an exact CP The preconditioned matrix P upd H has q more unit eigenvalues than in the case of the usual CP computed from scratch, and enjoys a nonexpansion property of the spectrum w.r.t. PH The preconditioner updating technique yields a reduction of the overall execution time when the factorization of the Schur complement is expensive Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

35 Conclusions and future work Some conclusions The proposed limited-memory BFGS-like update of a CP is still an exact CP The preconditioned matrix P upd H has q more unit eigenvalues than in the case of the usual CP computed from scratch, and enjoys a nonexpansion property of the spectrum w.r.t. PH The preconditioner updating technique yields a reduction of the overall execution time when the factorization of the Schur complement is expensive... and future work Investigate some issues (analysis of failures; adaptive/automatic choice of q and s) Extend the updating technique to CPs for sequences of KKT systems with nonzero (2,2) blocks Extend the updating technique to CPs (?) for more general sequences of KKT systems (variable off-diagonal blocks, A A T,...) Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

36 Conclusions and future work Thank you for your attention! L. Bergamaschi, D. di Serafino, V. De Simone, A. Martínez BFGS-like updates of constraint preconditioners for sequences of KKT linear systems January 2017, available from Optimization Online. Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

37 Conclusions and future work CVXQP3 n = 20000, m = 15000, nnz(ae 1 A T ) = total # IP iterations = 35 Prec s q max GC iters Tf-Schur Ta-seed Tupd Ttot RECOM FIXED BFGS-P BFGS-C DOUBLE Tf-Schur: time for factorization of AE 1 A T Ta-seed: time for application of seed Tupd: time for other updating operations, including H-orthogonalization Valentina Ttot: Detotal Simone time (Second for preconditioned Univ. Naples) BFGS-like CG updates of Constraint Precs (all times2gal are2017 in seconds) 24 / 27

38 Conclusions and future work CVXQP3 n = 20000, m = 15000, nnz(ae 1 A T ) = Prec s q max GC iters Tf-Schur Ta-seed Tupd Ttot RECOM FIXED BFGS-P BFGS-C DOUBLE Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27

39 Conclusions and future work The residual norm decreases up to about at the 38-th PCG iteration, and then it keeps increasing, without being able to reach the tolerance τ = 10 6 (nevertheless, a reduction of the residual norm of about 12 orders of magnitude is obtained). Note that BFGS-P(8,20) is able to satify the stopping criterion, thus showing that using a large number of directions coming from the previous KKT system may not be beneficial in the last IP iterations. Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27 Failure of BFGS-P(8,50) 1e+06 1e+04 residual norm 1e+02 1e+00 1e-02 1e-04 1e PCG iteration number IP iter= 34, seed from it =33

40 Conclusions and future work STCQP2 n = 16385, m = 8190, nnz(ae 1 A T ) = total # IP iterations = 12 Prec s q max GC iters Tf-Schur Ts-Schur Tupd Ttot ORIG SEED BFGS-P BFGS-C DOUBLE Valentina De Simone (Second Univ. Naples) BFGS-like updates of Constraint Precs 2gAL / 27