ENSIEEHT-IRIT, 2, rue Camichel, Toulouse (France) LMS SAMTECH, A Siemens Business,15-16, Lower Park Row, BS1 5BN Bristol (UK)

Transcription

1 Quasi-Newton updates with weighted secant equations by. Gratton, V. Malmedy and Ph. L. oint Report NAXY October ENIEEH-IRI, 2, rue Camichel, 3000 oulouse France LM AMECH, A iemens Business,5-6, Lower Park Row, B 5BN Bristol UK University of Namur, 6, rue de Bruxelles, B5000 Namur Belgium

2 Quasi-Newton updates with weighted secant equations. Gratton, V. Malmedy and Ph. L. oint 6 October 203 Abstract We provide a formula for variational quasi-newton updates with multiple weighted secant equations. he derivation of the formula leads to a ylvester equation in the correction matrix. Examples are given. Introduction Quasi-Newton methods have long been at the core of nonlinear optimization, both in the constrained and unconstrained case. uch methods are characterized by their use of an approximation for the Hessian of the underlying nonlinear functions which is typically recurred by low rank modifications of an initial estimate. he corresponding updates are derived variationally to enforce minimal correction in a suitable norm while guaranteeing the preservation of available secant equations which provide information about local curvature. By far the most common case is when a single such secant equation is considered for updating the Hessian approximation, and several famous formulae have been derived in this context, such as the DFP Davidon, 959, Fletcher and Powell, 963, BFG Broyden, 970, Fletcher, 970, Goldfarb, 970, hanno, 970 or PB Powell, 970 updates. More unusual is the proposal by chnabel 983 see also Byrd, Nocedal and chnabel, 994 to consider the simultaneous enforcement of several secant equations. he purpose of this small note, whose content follows up from Malmedy 200, is to pursue this line of thought, but in a slightly relaxed context where the deviation from the multiple secant equations is penalized rather than strictly forced to zero. Our development is primarily motivated by an attempt to explain the influence of the order in which secant pairs are considered in limited-memory BFG, as described in Malmedy 200, where a weighed combination of secant updates is used to conduct this investigation. his motivation was recently reinforced by a presentation of a stochastic context in which secant equations are enforced on average Nocedal, 203. his short paper is organized as follows. he new penalized multiple secant update is derived in ection 2, while a few examples are considered in ection 3. Conclusions are drawn in ection 4. 2 Penalized multisecant quasi-newton updates We consider deriving a new quasi-newton approximation B + for the Hessian of a nonlinear function from IR n into IR from an existing one B, while at the same time attempting to satisfy m secant equations. More specifically, we assume that m secant pairs of the form s i,y i i =,...,m are at our disposal. hese pairs are supposed to contain useful curvature information and are typically derived by computing differences y i in the gradient of the underlying nonlinear function corresponding to steps s i in the variable space IR n. he matrix E holds the correction B + B. he vectors s i and y i form the columns of the n m matrices and Y, respectively. ENEEIH-IRI, 2, rue Camichel, 3000 oulouse, France. serge.gratton@enseeiht.fr LM AMECH, A iemens Business, 5-6, Lower Park Row, B 5BN Bristol UK. vincent.malmedy@gmail.com Namur Center for Complex ystems naxys and Department of Mathematics, University of Namur, 6, rue de Bruxelles, B-5000 Namur, Belgium. philippe.toint@unamur.be

3 Gratton, Malmedy, oint: Quasi-Newton updates with weighted secant equations 2 As is well-known see Dennis and chnabel, 983, for instance, most secant updates may be derived by a variational approach. o define multisecant versions of the classical formulae, it would therefore make sense to solve the constrained variational problem min E 2 W EW 2 F 2.a s.t. B + Es i = y i for i =,...,m, and E = E, 2.b using some particular nonsingular weighting matrix W, whose choice determines the formula that is obtained. For instance, choosing W = I yields the multisecant PB update, while any matrix W such that W W = Y yields the multisecant DFP update. he multisecant BFG may then be obtained by duality from its DFP counterpart. his is the approach followed by chnabel 983 see also Byrd et al., 994 for a multisecant analog of the BFG formula. We propose here to consider a slightly different framework : instead of solving 2., we aim to solve the penalized problem min E 2 W EW 2 F + 2 ω i B + Es i y i 2 Ŵ 2.2a s.t. E = E, 2.2b with Ŵ = W W, in effect relaxing the secant equations by penalizing their violation with independent nonnegative penalty parameters ω i for i =,...,m. 2. olving the penalized variational problem Zeroing the derivative of the Lagrangian function of problem 2.2 with respect to E in the matrix direction K, we obtain that trw EW W KW + trk M M + ω i s i K Ŵ B + Es i y i = 0, 2.3 where M is the Lagrange multiplier corresponding to the symmetry constraint. Using the vectorization operator vec, this equation may be rewritten as vecŵ EŴ + M M,vecK + ω i s i Ŵ Es i s i Ŵ r i,veck = 0, 2.4 where r i = y i Bs i is the residual on the i-th secant equation at the current Hessian. As the equation 2.4 must hold for every matrix K, we thus have that which is equivalent to vecŵ EŴ + M M + Ŵ EŴ + M M + ω i s i Ŵ Es i s i Ŵ r i = 0, ω i Ŵ Es i s i Ŵ r i s i = 0. By summing this equation with its transpose and then, pre- and post-multiplying by Ŵ, we eliminate the Lagrangian multiplier M, and obtain that E I + ω i s i s i Ŵ + I + ω i Ŵs i s i E = ω i Ŵs iri + r i s i Ŵ.

4 Gratton, Malmedy, oint: Quasi-Newton updates with weighted secant equations 3 If we define the n m matrices R = Y B, R = R Ω /2, = Ω /2 and Z = Ŵ, where Ω = diagω i, we thus have to solve the Lyapunov equation AE + EA = C, 2.5 where A = I + Z and C = R Z + Z R. We know from that equation 2.5 has a unique solution whenever A does not have opposite eigenvalues see p. 44 of Lancaster and ismenetsky, 985. But A = I + Z = I + Ŵ, and from the relation spa {} = spi + Ŵ {}, we may deduce that A has only positive eigenvalues. hus 2.5 has a unique solution. ransposing the left and right hand-side of 2.5, and using the symmetry of C, we furthermore see that E also satifies 2.5. herefore, since the solution of this equation is unique, E is symmetric. We now show that E can be written in the form E = R,Z X X 2 R X2 X 3 Z 2.6 where X and X 3 are symmetric. ubstituting this expression into 2.5 and using Z R Z + Z R = R, 0 Im R I m 0 Z yields [ R,Z X X 2 2 X2 X RX + ZX 2 RX2 + ZX3 0 X R + X2 Z + 0 X2 R + X3 Z 0 Im I m 0 A solution of 2.5, and thus the unique one, can thus be obtained by setting and choosing X 3 to be the solution of the Lyapunov equation But the matrix ] R Z = 0. X = 0, X 2 = 2I m + Z 2.7 I m + ZX3 + X 3 I m + Z = RX2 X 2 R. 2.8 I m + Z = Im + Ŵ 2.9 is symmetric and positive definite. Hence equation 2.8 also has a unique solution, which can directly be computed using the eigen-decomposition of the m m symmetric matrix I m + Z. herefore, using 2.6, 2.7 and the fact that 2.9 implies the symmetry of X 2, we obtain that the solution of 2.3 can be written as E = R 2I m + Z Z + Z 2I m + Z R + ZX3 Z, 2.0 where X 3 solves 2.8. Note that the core of the computation to obtain E the determination of X 2 and X 3 only involves forming m m matrices from m n factors and then working in this smaller space, resulting in an Om 2 n + m 3 computational complexity.

5 Gratton, Malmedy, oint: Quasi-Newton updates with weighted secant equations 4 An important particular case is the case where we have only one secant equation, i.e. m =. In this case, the Lyapunov equation 2.8 is a scalar equation, and, defining r = y Bs, r = ωr and s = ωs, we obtain that E = rz + zr 2 + z s s r 2 + z s + z s zz, = rs Ŵ + Ŵsr 2ω + s Ŵs s r 2ω + s Ŵs ω + s Ŵs Ŵss Ŵ 2. As the scalar ω goes to infinity, we get the update E = rs Ŵ + Ŵsr s Ŵs s r s Ŵs 2 Ŵss Ŵ. 3 Examples of penalized updates 3. he penalized PB update If we choose Ŵ = I, we obtain from 2.0 and 2.8 that where X 3 solves B + = B + R 2Ω + + 2Ω + R + Ω /2 X 3 Ω /2, 3.2 I m + Ω /2 Ω /2 X 3 + X 3 I m + Ω /2 Ω /2 = Ω /2 R Ω /2 X 2 X 2 Ω /2 R Ω /2 3.3 with X 2 = 2I + Ω /2 Ω /2. If we consider the single-secant case m =, we derive from 2. that B + = B + rs + sr 2ω + s 2 + ω + s 2 2 s,r 2ω + s 2 s 2 ss, he penalized DFP update If we now assume that the matrices B and Y are symmetric, and that Y is positive definite, we deduce that B + = B + R 2Ω + Y Y + Y 2Ω + Y R + Y Ω /2 X 3 Ω /2 Y, 3.5 where X 2 = 2I + ΩY Ω and X3 solves X 2 IX 3 + X 3 X 2 I = Ω /2 R Ω /2 X 2 X 2 Ω /2 R Ω / with X 2 = 2I + Ω /2 Y Ω /2. Note also that a direct computation shows that X 2 I + IX 2 Ω /2 R Ω /2 X 2 + X 2 Ω /2 R Ω /2 X 2 X 2 I + I = Ω /2 R Ω /2 X 2 + X 2 Ω /2 R Ω / o simplify and better understand the relation between the penalized and a standard DFP formula in block form, we set X 4 = X 3 + X 2 Ω /2 R Ω /2 X 2, 3.8 sum up equations 3.7 and 3.6, and obtain that I + Ω /2 Y Ω /2 X 4 + X 4 I + Ω /2 Y Ω /2 = 2X 2 Ω /2 R Ω /2 X

6 Gratton, Malmedy, oint: Quasi-Newton updates with weighted secant equations 5 We also deduce from 3.8 that Ω /2 X 3 Ω /2 = 2Ω + Y R 2Ω + Y + Ω /2 X 4 Ω /2, and inserting this expression into the penalized DFP update 3.5, we obtain that B + = I Y 2Ω 2 + Y B I Y 2Ω 2 + Y +Y 22Ω 2 + Y 2Ω 2 + Y Y 2Ω 2 + Y + X 5 Y 3.20 where, in view of 3.9, X 5 = Ω /2 X 4 Ω /2 solves I + ΩY X 5 + X 5 I + Y Ω = 2 2Ω + Y R 2Ω + Y. 3.2 But, using Y = Y, 22Ω + Y 2Ω + Y Y 2Ω + Y and hence 3.20 finally becomes B + = = 2Ω + Y [ 2 2Ω + Y 2Ω + Y Y 2Ω + Y ] = 2Ω + Y 4Ω + Y 2Ω + Y, I Y 2Ω + Y B I Y 2Ω + Y 2Ω +Y + Y 4Ω + Y 2Ω + Y X5 Y where X 5 solves 3.2. If we set Ω = ωi and assume that Y is non singular, we obtain from 3.2 that X 5 = Oω, and we see from 3.22 that the penalized update converges to the DFP update in block form when ω goes to infinity. If m =, 2. gives that B + = B + ry + yr 2ω + s,y he penalized BFG update ω + s,y 2 2ω + s,y s,r s,y yy Let us assume again that Y is symmetric positive definite and set P = HY, with H being an approximation of the inverse Hessian. Exchanging the role of Y and in the penalized DFP formula, we derive the BFG penalized update defined by where X 6 solves H + = I 2Ω + Y Y H I 2Ω + Y Y 2Ω + + Y 4Ω 2 + Y 2Ω + Y + X6, 3.24 I + ΩY X 6 + X 6 I + Y Ω = 2 2Ω + Y P 2Ω + Y If we set Ω = ωi, and consider large values for ω we see that the penalized update again converges to the BFG formula. We also conclude that for ω sufficiently large, the penalized update will be positive definite. If Y is not symmetric and positive definite, an approximate update could be computed by 2Ω replacing the matrix + Y 4Ω + Y 2Ω + Y + X6 between and in expression 3.24 by any symmetric and positive definite approximation.

7 Gratton, Malmedy, oint: Quasi-Newton updates with weighted secant equations 6 In the case of a single secant pair m =, we obtain that 2ω + s,y + H + = H + ps + sp = I ρsy + 2ρω with ρ = s,y, p = s Hy and θ = + ρ ω 4 Conclusions H ω + s,y 2 2ω + s,y I ρys + 2ρω p,y s,y ss θρss, 3.27 [ ρ y,hy + 2ρ 2 ω + + ρ ω 2 + ρ ] ω. We have used variational techniques to derive a new algorithm for updating a quasi-newton Hessian approximation using multiple secant equations. he cost of the associated updates has been discussed and remains reasonable in the sense that it does not involve terms in the cube of the problem dimension. It is hoped that the new algorithm may prove useful beyond its initial application in Malmedy 200. References C. G. Broyden. he convergence of a class of double-rank minimization algorithms. Journal of the Institute of Mathematics and its Applications, 6, 76 90, 970. R. H. Byrd, J. Nocedal, and R. B. chnabel. Representation of quasi-newton matrices and their use in limited memory methods. Mathematical Programming, 63, 29 56, 994. W. C. Davidon. Variable metric method for minimization. Report ANL-5990Rev., Argonne National Laboratory, Research and Development, 959. Republished in the IAM Journal on Optimization, vol., pp. 7, 99. J. E. Dennis and R. B. chnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, NJ, UA, 983. Reprinted as Classics in Applied Mathematics 6, IAM, Philadelphia, UA, 996. R. Fletcher. A new approach to variable metric algorithms. Computer Journal, 3, , 970. R. Fletcher and M. J. D. Powell. A rapidly convergent descent method for minimization. Computer Journal, 6, 63 68, 963. D. Goldfarb. A family of variable metric methods derived by variational means. Mathematics of Computation, 24, 23 26, 970. P. Lancaster and M. ismenetsky. he heory of Matrices. Academic Press, London, second edn, 985. V. Malmedy. Hessian approximation in multilevel nonlinear optimization. PhD thesis, Department of Mathematics, University of Namur, Namur, Belgium, 200. J. Nocedal. private communication, oulouse, 203. M. J. D. Powell. A new algorithm for unconstrained optimization. in J. B. Rosen, O. L. Mangasarian and K. Ritter, eds, Nonlinear Programming, pp. 3 65, London, 970. Academic Press. R. B. chnabel. Quasi-newton methods using multiple secant equations. echnical Report CU-C , Departement of Computer cience, University of Colorado at Boulder, Boulder, UA, 983. D. F. hanno. Conditioning of quasi-newton methods for function minimization. Mathematics of Computation, 24, , 970.