Y.F. HU and C. STOREY. A family of optimally conditioned quasiNewton updates for unconstrained optimization. Technical report, Department of Mathematical Sciences, Loughborough University of Technology, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... the BFGS and DFP updates; see e.g. 11] the measure (A) trace(A) Gamma log(det(A) which is used in the convergence analysis in [4] and also results in the BFGS and DFP updates, see [13] the standard condition number measure (A) 1 (A) n (A) which results in a curve of sized updates, see [2, 25, 18]; the uniform condition number (A) trace(A) n) det(A) 1 n ) which results in the sized DFP and inverse sized BFGS updates, also called the Oren Luenberger self scaling updates, see [10] the optimal volume measure oe(A) 1 (A) det(A) 1 n ) and the resulting optimally conditioned, ....

....t ff and 0 OE = ac ac Gammab 2 t Gamma b c t if ac Gamma b 2 0 OE arbitrary if ac Gamma b 2 = 0 1 A 9 = 3.5) and call Phi C the Self Scaling Efficient Curve. Note that this curve contains optimally conditioned updates, i.e. updates optimal for the measure, see [2, 25, 18]. See figures 1 and 2 for illustrations of the various efficient sets. We now state and prove our main results. Theorem 3.1 The efficient updates for problem (3.1) are the self scaling efficient region updates B (t; OE) with (t; OE) 2 Phi R . Before we prove the above theorem, we present ....

Y.F. HU and C. STOREY. A family of optimally conditioned quasiNewton updates for unconstrained optimization. Technical report, Department of Mathematical Sciences, Loughborough University of Technology, 1991.

....this idea to the Broyden class. From their numerical tests, we can see that the results are encouraging. Also recently factorized selective column scaling was introduced by Siegel [23] which seems very encouraging. There are also some interesting numerical results in Al Baali [24] Hu and Storey [25] and Luksan [26] on optimal conditioning and self scaling. Khalfan [27] gives some numerical tests on the SR1. The above various test results are all inconclusive and many are contradictary. This motivates us to use various measures to explain these results. Chapter 3 Volume of Ellipsoids 3.1 ....

....if 2ac (a c)b: The proof of Theorem 5.1 can be found in [3] Wolkowicz in [5] gave a general form of optimal updates by considering all the s:p:d: updates satisfying the secant equation. Byrd independently showed that optimal update is a sized Broyden class update. Also, Hu and Storey in [25] gave a family of optimal updates. We summarize the discussion in section 5 of [5] in the following theorem which generalizes Theorem 5.1. Theorem 5.2 Consider the problem min (B ) subject to B s = y; B s:p:d: 5.1.27) 35 (i) The optimal updates of I for (5.1.27) are of the form B = BQ B; ....

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Y. F. HU and C. STOREY. A family of optimally conditioned quasi-newton updates for unconstrained optimization. Technical Report Mathematics Re- 97 port Number A151, Department of Mathematical Sciences, Loughborough University of Technology, 1991.

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