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tel-00703797, version 2 - 7 Jun 2012 Chapitre 3. Calcul d’équilibre de Nash généralisé x_2 x_2 -2 -1 0 1 2 -10 -5 0 5 10 Contour of ||F(x)|| - Newton Pure (-1,-1.5) 2 1 3 0 4567 -2 -1 0 1 2 x_1
tel-00703797, version 2 - 7 Jun 2012 3.2. Methods to solve nonlinear equations where J denotes the Jacobian ∗ JacF . Let M N k (h) = F (zk) + J(zk)h be the model function. At the kth iterate, the Newton step consists in solving the system M N k (dk) = 0. We get the following equation J(zk)dk = −F (zk). (3.12) Note that there is no guarantee that J(zk) is nonsingular. Another method consists in replacing the Jacobian by an approximate matrix, which will be always invertible. The direction dk solves Hkdk = −F (zk), (3.13) where Hk is updated by a so-called quasi-Newton scheme. This is equivalent to a model function M QN k (h) = F (zk) + Hkh. A quasi-Newton scheme needs an iterative process to update the approximate Jacobian from Hk to Hk+1. The choice of the matrix is large, since there are n 2 terms. We can set first that F (zk) = M QN k+1 (zk+1 − zk) ⇔ Hk+1sk = yk, where sk = zk+1 − zk and yk = F (zk+1) − F (zk). The latter equation is called the secant equation. We could have considered a scheme to approximate directly the inverse of the Jacobian by Wk with the secant equation sk = Wk+1yk. But still, we have an underdetermined system n equations for n × n matrix. Having no other property on the Jacobian of F (e.g. symmetry or positiveness), we generally consider the matrix that makes the smallest possible change to the preceding Jacobian according to the Frobenius norm † , see (Nocedal and Wright, 2006, Chap. 11). For an implementation point of view, the smallest possible change feature reduces the linear algebra computation work from O n 3 to O n 2 cost. If we consider minimizing minH ||H −Hk||F for all matrix H verifying the secant equation, we get the (good) Broyden scheme Hk+1 = Hk + yk − Hksk s T k sk s T k ⇔ Wk+1 = Wk + (sk − Wkyk)y T k Wk sT k Wkyk , using the Sherman-Morrison formula. Otherwise, if we minimize minW ||W − Wk||F for all matrix W verifying the secant equation, then we will obtain the (bad) Broyden scheme Hk+1 = Hk + (sk − Hkyk)y T k Hk y T k Hksk ⇔ Wk+1 = Wk + sk − Wkyk yT k yk y T k . According to Broyden (1965), this method appears often unsatisfactory in practice, so it will be discused no further. ∗. The Jacobian of a function f is defined as usual by ⎛ ∂f1 . . . ∂x1 ⎜ . Jacf(x) = ⎜ . ⎝ . . . . And the ∇f denotes the transpose of the Jacobian. ∂f b ∂x1 ∂f i ∂x j ∂f1 ∂xa . ∂f b ∂xa ⎞ ⎟ ⎠ (x). †. The Frobenius norm (also called the Euclidean norm) for matrix A is defined as ||A||F = i,j |aij|2 . 145
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