R.B. SCHNABEL. Analysing and improving quasi-Newton methods for unconstrained optimization. PhD thesis, Department of Computer Science, Cornell University, Ithaca, NY, 1977. Also available as TR-77-320.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 2 c and B 1 2 c H B 1 2 c , respectively, see [6] The measure (A) 1 (A) n (A) 1. 3) the 2 condition number) where 1 and n are the largest and smallest eigenvalues, respectively, has been used to choose an optimally conditioned update in the Broyden class, 7] See also [8]) In [9] Theorem 5.1, we showed that the measure (A) trace(A) n det(A) 1 n (1.4) yields the inverse sized BFGS update (the BFGS update of a b B c ) and sized DFP update (the DFP update of b c B c ) The sized updates are often referred to as the Oren Luenberger self scaling updates, ....

....problem (i) given in a) Suppose that b a . For simplicity, we again use the matrix B in (3.3) with corresponding secant equation (3.4) Note that the SR1 update is s.p.d. when Gammac b Gammac ac ac Gammab 2 or equivalently Gammaa b Gammaa ac ac Gammab 2 , see e.g. [8], 9] Therefore b a implies that the SR1 update is s.p.d. and in addition, that b c since b 2 ac . Moreover the SR1 update has n Gamma 1 unit eigenvalues and the other eigenvalue is smaller than 1 if v t s = b Gamma c 0 , or equivalently if (s Gamma H c y) t y = b Gamma a 0 . ....

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R.B. SCHNABEL. Analysing and improving quasi-Newton methods for unconstrained optimization. PhD thesis, Department of Computer Science, Cornell University, Ithaca, NY, 1977. Also available as TR-77-320.

.... if OE ac= ac Gamma b 2 ) Choosing OE = 0; 1 yields the well known DFP and BFGS updates, respectively; the set of updates with OE 2 [0; 1] is called the convex class; while the symmetric rank one update, denoted SR1, corresponds to the OE value OE SR1 = Gammac= b Gamma c) See e.g. [11, 21] for details. The current update is sized by the positive scalar t means it is changed to tB c ; see e.g. 10] We work in the space of symmetric matrices equipped with the trace inner product hA; Bi = traceAB: For a symmetric matrix A, we let 1 (A) 2 (A) Delta Delta Delta n (A) denote ....

....h 1 ( h n ( Gamma h n (OE 0 ) h 1 (OE 0 ) Gamma h 1 (OE 1 ) h n (OE 1 ) Gamma h n (OE 0 ) OE 0 ) Gamma (OE 1 ) Gamma (OE 0 ) Gamma Gamma (OE 1 ) 0 ( OE) 0 Gamma ( OE) 2.23) for some OE 0 OE OE 1 with 0 Gamma ( OE) 6= 0. From chapter 7 in [21] or Lemma 2.1, we have 0 Sigma (OE) Gamma(ac Gamma b 2 ) 2b 2 (1 Sigma g(OE) where g(OE) f 1 (OE) Gamma b c ) f 1 (OE) Gamma b c ) 2 ac Gammab 2 c 2 ] 1 2 1: Let h(OE) 0 (OE) 0 Gamma (OE) 1 g(OE) 1 Gamma g(OE) 2.24) Then h 0 (OE) ....

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R.B. SCHNABEL. Analysing and improving quasi-Newton methods for unconstrained optimization. PhD thesis, Department of Computer Science, Cornell University, Ithaca, NY, 1977. Also available as TR-77-320.

....define the distance in some metric of an s:p:d: matrix to the identity or a multiple of the identity. For example the Frobenius norm (and weighted Frobenius norm) gives rise to various rank two updates of B c . These include the popular BFGS and DFP updates from the Broyden class, see e.g. 2] and [3]. In [4] it is shown that the measure (A) trace A) n det(A) 1=n ; 1.1.3) where det denotes determinant, gives rise to sized updates. Sized update means that B is the update of ffB c , ff 0, instead of B c . In [5] a study of the volumes of the ellipsoids corresponding to B c and B ....

....semi definite. If B satisfies the secant equation, then the columns of B Gamma B c are in the span of fB c s; yg if and only if B = B OE , given in (2.1.7) 11 We include the following explicit representation of the eigenvalues of a scaled Broyden class update. This result is in Schnabel [3]. Lemma 2.1 The matrix B Gamma 1 2 c B OE B Gamma 1 2 c has n Gamma 2 unit eigenvalues and the two remaining eigenvalues are Sigma (OE) f 1 (OE) Sigma (f 1 (OE) 2 Gamma f 2 (OE) 1 2 ; 2.1.14) where 8 : f 1 (OE) a(b c) GammaOE(ac Gammab 2 ) 2b 2 f 2 (OE) a b ....

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R.B. SCHNABEL. Analysing and improving quasi-Newton methods for unconstrained optimization. PhD thesis, Department of Computer Science, Cornell University, Ithaca, NY, 1977. Also available as TR-77-320.

....t s 0, and w = 1 y t s y Gamma 1 s t B c s B c s: If OE = 1 we get the BFGS update and OE = 0 yields the DFP update. The updates for OE 2 [0; 1] are called the convex class. This is not the most common parameterization, but it is well known and it allows us to use results directly from [15] without reproving them here and uselessly lengthening the paper. If we form the Fletcher dual updates, i.e. we exchange the roles of y and s and let H c = B Gamma1 c , then we get the inverse updates H OE = H c Gamma 1 y t H c y H c yy t H c 1 y t s ss t (1 Gamma OE)y ....

....B OE s = y : Furthermore, let a = y t H c y; b = y t s; c = s t B c s: Note that b 2 ac with equality if and only if B Gamma 1 2 c y and B 1 2 c s are collinear, which is true if and only if y and B c s are collinear, which is true if and only if H c y and s are collinear. From [15], B OE is s.p.d. if b 0 and OE ac ac Gammab 2 . A optimal rank two update is found in [2] by minimizing the measure (H c B OE ) over the Broyden family of rank two updates. Note that the spectrum of a matrix product C 1 C 2 is equal to the spectrum of C 2 C 1 and (C) C Gamma1 ) ....

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R.B. SCHNABEL. Analysing and improving quasi-Newton methods for unconstrained optimization. PhD thesis, Department of Computer Science, Cornell University, Ithaca, NY, 1977. Also available as TR-77320.

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