APPROXIMATE NULLSPACE ITERATIONS FOR KKT SYSTEMS IN MODEL BASED OPTIMIZATION. min. s.t. c(x, p) = 0. c(x, p) = 0

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1 Forschungsbericht Nr. 6-5, FB IV Mathematik/Inormatik, Universität Trier, 6 PPROXIMTE NULLSPCE ITERTIONS FOR KKT SYSTEMS IN MODEL BSED OPTIMIZTION KZUFUMI ITO, KRL KUNISCH, ILI GHERMN, ND VOLKER SCHULZ bstract. The aim o the paper is to provide some theoretical basis or approximate reduced SQP methods in contrast to inexact reduced SQP methods, i.e., here the orward and the adjoint problem accuracies are not increased when zooming in to the solution o an optimization problem. Only linear-quadratic problems are treated, where approximate reduced SQP methods can be viewed as null-space iterations or KKT systems. We give theoretical convergence results and show that certain numerical examples possess convergence properties, even i they do not satisy the assumptions or the convergence theorems. Key words. KKT systems, appoximate reduced SQP methods, iterative solvers. MS subject classiications. 65F,65K5,9C, 93C.. Introduction. Karush-Kuhn-Tucker systems are necessary conditions or optimization problems, e.g., o the orm min (x, p) x,p s.t. c(x, p) = where x R nx, p R np are the variable vectors o the optimization problem, : R nx R nx R is the objective o the problem, and c : R nx R nx R nx the equality constraint. The necessary conditions are ormulated as x L(x, p, λ) = p L(x, p, λ) = c(x, p) = where L(x, p, λ) = (x, p)+λ c(x, p) deines the Lagrangian by usage o the adjoint variable vector λ R nx. The distinction o two types o variables, x on the one hand, and p on the other hand, is typical or model based optimization problems, where we assume that the constraint c(x, p) = is a mathematical model or a certain process to be driven into an optimal state deined by the objective. We call x the state vector and assume that the model is always solvable with respect to x, i.e., c/ x is always invertible. In particular or large scale systems, reduced SQP methods [Sch97, Hei96, BNS95, KS9] are used successully as a highly eicient solution approach. These reduced SQP techniques require requent solutions o linear systems with c/ x or ( c/ x) as system matrix. O course, this is usually not perormed exactly, but only approximately. In [HV] inexact reduced SQP techniques are analysed, where the solution accuracy o these systems is increased with the closeness to the optimal solution. In [Sch97] a reormulation o the reduced SQP method is presented or approximate solutions o these systems, which does not require that the solution accuracy is increased but deliveres nevertheless the optimal solution, i convergence is achieved. To the best knowledge o the authors, the convergence itsel is not yet guaranteed theoretically. In the present paper, we simpliy the situation to linear quadratic optimization problems (QP) o the orm min x,p x H x x + p H p p + x x + p p (.) s.t. C x x + C p p + c = (.) Department o Mathematics North Carolina State University, Raleigh, North Carolina, US(kito@math.ncsu.edu) Institut ür Mathematik, Karl-Franzens-Universität Graz, Heinrichstr. 36, -8 Graz, ustria (karl.kunisch@uni-graz.at). Department o Mathematics, University o Trier, Universitätsring 5, 5486 Trier, Germany (ilia.gherman@uni-trier.de). Department o Mathematics, University o Trier, Universitätsring 5, 5486 Trier, Germany (volker.schulz@uni-trier.de).

2 K. ITO, K. KUNISCH, V. SCHULZ, ND I. GHERMN The QP might stand alone as such or might be a QP-subproblem within an SQP method or anonlinear optimization problem as studied above, so that equation (.) can be thought o as the linearization o a process model which is to be optimized in the sense o (.). The matrices H x R nx nx and H p R np np are supposed to be symmetric and the (stiness) matrix C x R nx nx is supposed to be invertible. The matrix C p R nx np determines the inluence o the parameter vector p on the system. The dimensions o the respective vectors are x R nx, p R np and c R nx. In order to have a well posed problem, we assume that the reduced Hessian, S = H p + Cp C x H x Cx C p (.3) is positive deinite (or coercice, i we think o it as an operator in a unction space). The QP (.,.) is equivalent to the system o linear equations H x C x H p C p C x C p x p λ = x p c (.4) Model based optimization problems usually start rom an already established solution technique o the model equation C x x + C p p =. That means that there is a-priori knowledge, e.g., in the orm o an approximation or C x, which is easily invertible and can be used or an iterative solution o the state equation (.). For that, we assume ρ(i C x ) < where ρ denotes the spectral radius. nd we assume to have some approximation B o the reduced Hessian S, as well. It is illustrative or the subsequent discussion to have a look at an SQP step or problem (.,.), which gives the exact solution o it. We need the ollowing deinitions: T := [ C x C p I ] [ ] Hx, H :=, C := [ ] C H x C p p Thus, we obtain another representation o the reduced Hessian S = T HT which can be considered the null-space Schur complement ( ) o the system ( matrix ) (KKT matrix) in (.4). We collect x, p in the vector y := and also := x x p p The solution o the QP, which is exactly one SQP step ormulated as in [KS93], can be written as ( ) ( ) y = T S T + T S T Hx Cx c C x c λ = C x (H x x + x ) In contrast to that, a reduced SQP step omits terms containing H x. Thereore a reduced SQP method does not give the solution ater one step. The iterations are o the orm y k+ = y k + y λ k+ = λ k + λ with C x S C p C x C p x p = λ H x C x H p C p C x C p x k p k λ k x p c

3 PPROXIMTE NULLSPCE ITERTIONS FOR KKT SYSTEMS 3 In the present paper, we substitute C x in T by some approximation and the exact reduced Hessian S also by some approximation B and investigate convergence conditions or the resulting iteration. The resulting method is reormulated as the linear iteration: x k+ p k+ λ k+ = x k p k λ k a B C p C p H x C x H p C p C x C p x k p k λ k + x p c (.5) where we use a slightly more general ormulation so that is some approximation to C x and a is some approximation to C x. One can think about circumstances, where it might prove useul to choose a. In practical examples [GS5] the ollowing act has been observed which seems surprising at the irst glance: The method works better, i B is an approximation or a wrong reduced Hessian S rather than the exact reduced Hessian S, where S = H p + Cp a H x C p (.6) We will also give a theoretical explanation or this observation. One should note that this observation is in line with similar studies or variational saddle point problems as in [BWY9]. However, the convergence theory there cannot be caried over to the iteration (.5). Because the iteration concept is based on an approximate nullspace representation in T (C x is substituted by ) we call it an approximate nullspace iterative technique. The resulting method resembles preconditioning approaches in [BS], where theoretical results are derived or the case that C x is not substituted by something else. One should note that the iterations considered in this paper are related to the so-called piggy-back iterations in [GF, Gri5], where also the exact reduced Hessian is ound to lead not to optimal convergence results. In the next section, we will give the main theoretical results o this paper. Section 3 is devoted to the application to a generic optimal control problem oten ound in literature. Results o numerical experiments supporting the theory o section are given in section 4.. Convergence results. In this section, we will show that the above iteration (.5) is convergent and we will also give criteria or the convergence. The whole convergence theory o this section is based on a perturbation analysis. First, we show a inite step convergence or a reduced-type exact solver. Lemma.. The matrices, a are assumed to be invertible and also the Schur complement S rom equation (.6). Then, the iteration xk+ p k+ λ k+ = xk p k λ k a S C p C p H x a H p C p C p xk p k λ k converges ater three steps to the exact solution o the (perturbed) problem H x a H p Cp x p + x p = C p λ c + x p c Proo. One could proo this result by proceeding in parallel to reduced SQP methods, i.e. to perorm actually three iterations o a reduced SQP method applied to the QP starting rom zero and observe that the outcome is actually the exact solution. However, this will not lead towards the right proo strategy or the remainder o this section. So, we just study the iteration matrix o the iteration above and show that it is nilpotent. First we give the exact inverse o the inverse matrix above in a block orm a S C p C p = C p a S C p a S a

4 4 K. ITO, K. KUNISCH, V. SCHULZ, ND I. GHERMN Now, we compute explicitly the iteration matrix a H x a M = I S Cp H p Cp C p C p = = = a S C p C p C p a S C p a a S C p C p H x a H p C p C p S H x S H p a C p a H x (S H p ) S C p a H x S (S H p ) a H x When studying M, we have to keep in mind the deinition (.6) or S We investigate each block o the 3 3-block matrix M, which is not obviously zero, separately. (M ) (,) = C p a H x C p a H x = ( C p a H x C p S + H p }{{} (M ) (,) = C p a H x (S H p ) = ( C p a H x C p S + H p }{{} ) S C p a H x = (S H p )S C p a H x (S H p )S (S H p ) ) S (S H p ) = (M ) (,) = S C p a H x C p a H x + S (S H p )S C p a H x = S ( C p a H x C ) p S + H p S }{{} C p a H x = (M ) (,) = S C p a H x (S H p ) + S (S H p )S (S H p ) = S ( C p a H x C ) p S + H p S }{{} (S H p ) = (M ) (3,) = a (M ) (3,) = a Thereore, M is o the orm H x H x M = C p a H x (in general) (S H p ) (in general) and obviously M 3 = M M =. Now, we use the ollowing abbreviations in order to obtain some insight rom a more abstract point o view R := K := a B C p C p H x a Hp C p C p K := H x C x H p C p C x C p Hp := H p S + B

5 PPROXIMTE NULLSPCE ITERTIONS FOR KKT SYSTEMS 5 Note that the matrix B is the exact Schur complement o the matrix K, because H p + C p a H x Thereore, we can conclude with lemma. C p = H p S + B + C p a (I R K) 3 = The iteration matrix o the iteration (.5) is now the matrix I R K = I R K + R ( K K) =: M + N H x C p = B where M is a nilpotent matrix o nilpotency degree 3 and N can be considered as a perturbation o N. For any norm., this yields (M + N) 3 [ M + MN + NM + N + M + NM + M ] N The ollowing lemma is based on a perturbation argument and estimates the inluence o the perturbation induced by K. Lemma.. Deine θ := M + MN + NM + N + M + NM + M, r := N = R ( K K) I θ r <, then the iteration (.5) converges with the upper bound or the convergence rate κ := 3 θ r <. Proo. By Geland s theorem, we obtain ρ(i R K) = 3 ρ((i R K) 3 ) = 3 lim (I n R K) 3n /n = 3 lim (M + n N)3n /n 3 θ r < Remark. O course, we might obtain a simpler expression or θ by observing that θ 3( M + N ) M + N But using the right hand side instead o the θ as deined above is a too coarse estimation, as we observed numerically. In order to proceed, we have to choose a speciic norm. We choose the l norm. Theorem.3. We deine the numerical spectral norms r := I C x, r a := I a C x, r S := I B S I max{r, ra, r S} < / θ, with θ := θ ϕ, and ϕ := C pb Cp C p I B Cp I I the iteration (.5) converges with the upper bound or the convergence rate κ = 3 θ r <

6 6 K. ITO, K. KUNISCH, V. SCHULZ, ND I. GHERMN Proo. We observe that R ( K K) = C pb Cp a C pb B Cp a B a C pb Cp C p I B Cp I I where pplication o lemma. gives I a C x I B S I C x a Cx B S C x I a Cx I B S I C x = max{r, ra, r S} = ρ(i R K) 3 (I R K) 3 3 θ R ( K K) 3 }{{} κ 3 θ ϕ max{r, ra, r S} < The theorem shows that the convergence behaviour o the approximate nullspace iteration is limited both by the approximation quality in the orward and in the adjoint system and by the approximation quality o the wrong Schur complement in a worst case ashion. Oten, one might choose a = =:. In this situation, we can give a reined version o the theorem above, i additionally Cx = C x and one chooses the spectral norm x p := λ R / x / B / pb / / λ / Corollary.4. We choose in theorem.3 a = = and the norm. :=. R. We deine I max{ρ, ρ S} < / θ, with θ := θ ϕ, ρ := ρ(i C x ), ρ S := ρ(i B S ) and ϕ := ρ / C p B Cp / / C p B / I B / Cp / I I the iteration (.5) converges with the upper bound or the convergence rate κ = 3 θ r < Proo. We give more reined representations o the actors C p B Cp C p I B Cp I = I R / C p B Cp / / C p B / I B / Cp / I = I / C p B Cp / / C p B / I ρ B / Cp / I I

7 PPROXIMTE NULLSPCE ITERTIONS FOR KKT SYSTEMS 7 and I C x I B S = I C x R I / C x / = I B / S B / I / C x / I / C x / = ρ I B / S B / = max{ρ I / C x /, ρ S} We conclude analogously to the proo o lemma.3 ρ(i R K) 3 (I R K) 3 R 3 θ R ( K K) 3 R }{{} κ 3 θ ϕ max{ρ, ρ S } < In many cases, one might have a pretty good idea about the proper choice o approximating C x. However, the only part o S which is easily accessible is H p. Thereore, a natural question arises about the useuleness o just using H p as an approximation to S. For the analsis o this eect, one has to take into account more reined problem characteristics. This will be perormed in the next section. 3. pplication to optimal control. The iterative methods discussed above are o particular importance or the solution o optimal control problems. generic version o them is the problem s.t. min x,p (x ˆx, x ˆx) + µ (p, p) L y + Πp where L : H (Ω) H Ω) L Ω) is a linear mapping or unctions deined on an open region Ω and (.,.) is the scalar product in L. The operator Π is assumed to be bounded. Discretization, e.g., by inite dierences with meshsize h gives H x = h d M, H p = h d M, C x = L h = h M, C p = Π h where d is the space dimension o Ω and M m <, M m <, Π h π < For iteration purposes, we may assume that the approximation to L h is o the orm = h W, with w I W w I or some w R, w. Then, the wrong Schur complement takes the orm S = µ h d M + Π h h W h d M h W Π h = µ h d M + h d+4 Π h W M W Π h The H p -part o the Schur complement is easily available. Thereore we consider the choice o B = H p in the Schur complement part o the iterations. The resulting iteration matrix takes the orm I B S = I H p S = h4 µ Π h W M W Π h

8 8 K. ITO, K. KUNISCH, V. SCHULZ, ND I. GHERMN Now, we can make the ollowing observations. Corollary 3.. For discretized optimal control problems with the characteristics described above, we obtain a good convergence rate in the Schur complement or B = H p, provided that µ is large enough or the discretization h is ine enough. Proo. For the norm. = (.,.) / we see that ρ S = ρ(i Hp S ) h4 µ Π h W M W Π h h4 µ π w m Thereore, ρ S <, i µ large enough or h small enough. Now, we can easily achieve ρ S <. The orward and adjoint system can also be assumed to be solvable with ρ <. I we take a close look at theorems.3 and.4, we see that these properties or ρ S, ρ are not enough to guarantee overall convergence. t least in corollary.4, we observe, that ϕ is close to, i h is small enough or µ large enough. However, the parameter θ increases noticably or decreasing h, when perorming numerical experiments as below. Thereore, or small h the conditions or ρ S, ρ become very restrictive in order to be able to apply the convergence theorems.3 and.4. But, the numerical results below show, that convergence is also achieved in cases, where theorems.3 and.4 are not applicable. 4. Numerical experiments. Here, we illustrate the theoretical results rom the previous section by application to a common model problem, which serves as a standard test problem in PDE constrained optimization. For a given open computational region Ω, we investigate the problem min x,p Ω (x(ξ) ˆx(ξ)) dξ + µ s.t. y(ξ) = p(ξ), ξ Ω x(ξ) = ξ Ω Ω p(ξ) dξ The variables x and p are thought o as unctions deined on the domain Ω and denotes the Laplacian operator. The aim o the problem is to track a given unction ˆx with the solution o a dierential equation. Since we only want to illustrate certain numerical eects and do not pretend to attack any real lie problem in this paper (this has been done and will be done again by the authors in dierent publications), we even downsize the problem to D, i.e., Ω = [, ]. Then, this model problem simpliies to min x,p (x(ξ) ˆx(ξ)) dξ + µ s.t. x (ξ) = p(ξ), ξ [, ] x() = = x() p(ξ) dξ This problem is discretized by inite dierences on an equidistant mesh with meshsize h = /N, N N, i.e. x l := x(l h), p l := p(l h), l =,..., N l =,..., N x (l h) h ( x l + x l x l+ ), l =,..., N N (x(ξ) ˆx(ξ)) dξ h (x l ˆx(l h)) l= N p(ξ) dξ h p(ξ) l=

9 PPROXIMTE NULLSPCE ITERTIONS FOR KKT SYSTEMS 9 For the sake o simplicity, we omit values at and so that our vectors o unknowns are x = (x,..., x N ), p = (p,..., p N ) The discretized problem is now o the orm (.,.) with H x = h I, H p = µ h I, C p = I, x = ˆx, p =, c = where I is the identity in R N and C x = h For ease o computations, we choose x = ˆx =. Then, we know the exact solution, which is zero or all variables. In order to perorm numerical convergence tests with varying approximations to C x, we construct these approximations by Jacobi steps, i.e., or D := diag(c x ) we deine and thereore := D := D (I + (I C x D )) := D (I + (I + (I C x D ))(I C x D )).. Thus i = (( C x ) i + I). > ρ(i C x) > ρ(i C x) = ρ(i C x) > ρ(i 3 C x) = ρ(i C x) 3 >... In the case o, we can give the Schur complement analytically S = H p + C p H x C p = (µ h + h5 4 ) I (4.) In all other cases, the ormulas become more complicated. We treat B analogously: we choose B := H p and B i as the approximation to S ater i Richardson iterations with H p. That means B := H p = µ hi := B (I C p H x C p B B j i i dditionally, we investigate the cases = C x and B = S resp. B = S rom (.3). The norms to be used later are chosen as ( ( ( and x := h N l= x l) /, p := h N l= j ) p l) /, λ := x p λ := ( x + p + λ ) / h N l= λ l) /, First, we investigate or N = (h = ) and µ =. the convergence rates o corollary.4. Table 4. summarizes the results. In column i, we denote, how many Jacobi iterations

10 K. ITO, K. KUNISCH, V. SCHULZ, ND I. GHERMN i j ρ ρ B ρ It θ r θ κ S S E S S E S S E S S E S S E C x C x C x C x S C x S E Table 4. Convergence results or N = and µ =. are perormed in i (C x means that we choose = C x, i.e., the orward annd the adjoint problem is solved exactly). The column j gives the analogous inormation or B j. Furthermore, ρ denotes the spectral readius o the matrix I i C x. nalogously, ρ B denotes the spectral readius o the iteration matrix o the design equation and ρ It denotesb the spectral readius o the overall iteration matrix(i R K). For the deinitions o θ, r, θ and κ see above and use the. R -norm. We observe that the estimation or κ in corollary.4 is very cautious, because we have in all cases κ > ρ It. The convergence condition max{ρ, ρ B } < / θ ormulated in corollary.4 is only satisied rom the row (i = 5, j = ) on downward. The igure 4. gives the convergence history and and the eigenvalues o the iteration matrix o the KKT system or two characteristic cases. We can state the act that the approximate KKT iteration converges more oten than the thery ormulated in the above sections guarantees. To observe also numerically more drastically, that the choice B = S is not optimal in most cases, we perorm urther test or the case N = h =. and µ =.. From table 4. we conclude that the choice B = S is inerior to other choices, since the spectral radius o the KKT iteration matrix is larger than in cases, where B is chosen closely to S (c., the cases i = 5,, 5). In this setting, we can also check that it is not enough to achieve convergence in the orward, the adjoint and the design system in order to obtain convergence or the overall KKT iteration. In the tests above (N = ), we have achieved convergence in all cases.

11 PPROXIMTE NULLSPCE ITERTIONS FOR KKT SYSTEMS Norm o residuum orward design adjoint total (KKT) Im 4 x Norm o residuum Iteration 4 6 case i =, j = orward design adjoint total (KKT) Im Re Iteration. case i = 5, j =.5.5 Re Fig. 4.. Convergence hostory and eigenvalues or h = and µ =. Table 4. gives the results or N = h =. and µ =.. The columns i, j, ρ, ρ B, ρ It are deined as above. In addition, we give the numbers or iterations in the column [# It] until xk x p k p λ k λ = xk p k λ k 4 The start o the iterations is x = p = λ = (,..., ). We observe that the KKT iteration does not converge or the choice = D (i = ), although the iterations in the components (orward, adjoint, design) are convergent. This is in line with theorems.3 and.4, which state that each spectral readius o the components has to below a certain limit, which may be signiicantly below, in order to guarantee the convergence o the overall KKT iteration. Furthermore, we observe the eect o choosing B = S: The convergence deteriorates signiicantly, i.e., the choice B = S (resp. B S ) is signiicantly better than B S E. Finally, we give the history o the resuduals and the respective eigenvalue distribution or the cases i = 3 and B = S resp. B = S E in igure 4..

12 K. ITO, K. KUNISCH, V. SCHULZ, ND I. GHERMN i j ρ ρ B ρ It # It S S E S S E S S E S S E Table 4. Convergence results or n = and µ =. 5. Conclusions. The aim o this paper is to investigate deect correcting iterations o the type (.5) or the solution o linear-quadratic optimization problems. The intention o the paper is to provide some theoretical oundation or practically well-established iterations. In particular, we try to give answers to the ollowing questions: Is iteration (.5) convergent and under which conditions? We shown that the iteration is convergent, i is chosen close enough to C x, a close enough to C x and B close enough to S. What matrix should be chosen as the matrix B, i.e., as a preconditioner in the Schur complement system? We have ound theoretical justiication or the statement that B should be chosen close o S, rather than S which would the the canonical irst guess. re there examples which satisy the restrictive assumptions o the convergence theorems? In the numerical results section, examples are given which show that the convergence theorems do not talk about the empty set. lthough presenting theoretical investigations into iterations o type (.5) and giving some positive theoretical results, we observe, that there is still some theoretical gap, insoar, as there are many more cases, where the iteration converges numerically than where the assummptions o the convergence theorems are satiaied. knowledgements. The third autors wishes to thank the university o Graz or providing support or his stay at the university o Graz, during which most o the ideas in this paper have been developed. REFERENCES [BNS95] [BS] [BWY9] L.T. Biegler, J. Nocedal, and C. Schmid, reduced Hessian method or large-scale constrained optimization, SIM Journal on Optimization 5 (995), no., Battermann and E. Sachs, Block preconditioners or KKT systems in PDE-governed optimal control problems, Fast solution o discretized optimization problems (K.-H. Homann, R.H.W. Hoppe, and V. Schulz, eds.), ISNM, no. 38, Birkhaeuser,, pp. 8. R. Bank, B Welert, and H Yserentant, class o iterative methods or solving saddle point problems, Numer. Math. 56 (99),

13 PPROXIMTE NULLSPCE ITERTIONS FOR KKT SYSTEMS 3 5 orward design adjoint total (KKT) 8 x Norm o residuum 5 Im Iteration Re Norm o residuum 5 5 case i =, B = S orward design adjoint total (KKT) Im x Iteration Re case i = 3, B = S E Fig. 4.. Convergence history and eigenvalues or N = [GF] [Gri5] [GS5] [Hei96] [HV] [KS9] [KS93] [Sch97]. Griewank and C. Faure, Reduced unctions, gradients and Hessians rom ixed point iterations or state equations, Numer. lgor. 3 (), Griewank, Projected Hessians or preconditioning in one-step one-shot design optimization, Large Scale Nonlinear Optimization, Kluwer, 5, to appear. I. Gherman and V. Schulz, Preconditioning o one-shot pseudo-timestepping methods or shape optimization, PMM 5 (5), no., M. Heinkenschloss, Projected sequential quadratic programming methods, SIM Journal on Optimization 6 (996), M. Heinkenschloss and L. Vicente, nalysis o inexact trust-region sqp algorithms, SIM Journal on Optimization (), K. Kunisch and E. Sachs, Reduced SQP methods or parameter identiication problems, SIM Journal on Numerical nalysis 9 (99), F.-S. Kuper and E.W. Sachs, Reduced SQP methods or nonlinear heat conduction control problems, International Series o Numerical Mathematics (993), V.H. Schulz, Solving discretized optimization problems by partially reduced SQP methods, Comput. Vis. Sci. (997), no.,