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Robust Optimization

Robust Optimization. Multi-dimensional Marquardt Parameters. Multi-dimensional Optimization. The approximating function. A linear fitting function would be. In this work the fitting function is essentially. Expand the first derivatives about the value c 0.

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Robust Optimization

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  1. Robust Optimization Multi-dimensional Marquardt Parameters

  2. Multi-dimensional Optimization

  3. The approximating function • A linear fitting function would be In this work the fitting function is essentially

  4. Expand the first derivatives about the value c0.

  5. The result is a new estimate for the values of c

  6. The usual problem is the inversion of the matrix • If a constant cn is such that its effect is a linear multiple of cm - that is if fA has parts cn Pn and cmPm with Pn and Pm effectively the same for all xi - the matrix has no inverse owing to the fact that there are no unique values for cn and cm

  7. The Marquardt Parameter • The solution is to request small constants. In particular add  c∙c to 2

  8. This results in constant estimates that depend on 

  9. The general result for any single constant is

  10. Relative Smoothers

  11. Define p2 and Fr

  12. Fr versus log 

  13. Fr values • If on a minimization step the actual 2 equals p2, not brave enough • Fr(Fr)3 • If the actual 2 > the last 2 then • Fr(9+Fr)/10 • If the values are slowly changing from one side  Aitkin’s extrapolation.

  14. At the beginning • FR,CHI,CHB,CHL 0.999990000 118785.585 0.100000000E+67 0.500000000E+ • FR,CHI,CHB,CHL 0.999970000 118785.230 118782.022 118785.585 • FR,CHI,CHB,CHL 0.999946001 118784.590 118778.816 118785.230 • FR,CHI,CHB,CHL 0.999902805 118783.438 118773.045 118784.590 • FR,CHI,CHB,CHL 0.999825058 118781.362 118762.657 118783.438 • FR,CHI,CHB,CHL 0.999685131 118777.623 118743.962 118781.362 • FR,CHI,CHB,CHL 0.999433325 118770.875 118710.315 118777.623 • FR,CHI,CHB,CHL 0.998980275 118758.679 118649.761 118770.875 • FR,CHI,CHB,CHL 0.998165430 118736.558 118540.808 118758.679 • FR,CHI,CHB,CHL 0.996700804 118696.154 118344.823 118736.558 • FR,CHI,CHB,CHL 0.994071243 118621.239 117992.433 118696.154 • FR,CHI,CHB,CHL 0.989359873 118476.802 117359.094 118621.239 • FR,CHI,CHB,CHL 0.980949663 118128.813 116219.779 118476.802 • FR,CHI,CHB,CHL 0.966036016 116688.910 114116.688 118128.813 • ierr, iloop 1 0 • FR,CHI,CHB,CHL 0.993990303 116611.223 115987.645 116688.910 • FR,CHI,CHB,CHL 0.989215050 116441.523 115353.576 116611.223 • FR,CHI,CHB,CHL 0.980691773 115680.541 114193.243 116441.523

  15. Aitkin’s extrapolation • FR,CHI,CHB,CHL 0.996558072 115637.868 115282.377 115680.541 • FR,CHI,CHB,CHL 0.993815191 115550.779 114922.670 115637.868 • FR,CHI,CHB,CHL 0.988901771 115232.958 114268.370 115550.779 • FR,CHI,CHB,CHL 0.988901771 114790.155 113954.076 115232.958 • FR,CHI,CHB,CHL 0.988901771 114243.910 113516.188 114790.155 • FR,CHI,CHB,CHL 0.988901771 113659.796 112976.005 114243.910 • FR,CHI,CHB,CHL 0.988901771 113455.616 112398.373 113659.796 • FR,CHI,CHB,CHL 0.988901771 112988.531 112196.459 113455.616 • ierr, iloop 1 0 • AITKIN'S EXTRAPOLATION • FR,CHI,CHB,CHL 0.998890177 112951.863 112863.134 112988.531 • FR,CHI,CHB,CHL 0.998003427 112875.210 112726.346 112951.863 • FR,CHI,CHB,CHL 0.996409757 112716.318 112469.960 112875.210

  16. Non-Linear Irevs • FR,CHI,CHB,CHL 0.999968166 112081.893 112078.975 112082.543 • FR,CHI,CHB,CHL 0.999942699 112080.061 112075.471 112081.893 • FR,CHI,CHB,CHL 0.999896862 112075.828 112068.501 112080.061 • FR,CHI,CHB,CHL 0.999814361 115490.184 112055.022 112075.828 • IREV= 1 IEND= 151 • FR,CHI,CHB,CHL 0.999953590 112070.874 112070.627 112075.828 • FR,CHI,CHB,CHL 0.999860777 113183.400 112055.271 112070.874 • IREV= 1 IEND= 150 • FR,CHI,CHB,CHL 0.999965194 112067.371 112066.973 112070.874 • FR,CHI,CHB,CHL 0.999906028 112064.902 112056.840 112067.371 • ierr, iloop 1 0 • FR,CHI,CHB,CHL 0.999983086 112064.676 112063.007 112064.902 • FR,CHI,CHB,CHL 0.999969555 112063.861 112061.264 112064.676 • FR,CHI,CHB,CHL 0.999945199 112061.774 112057.720 112063.861 • FR,CHI,CHB,CHL 0.999901361 114265.708 112050.720 112061.774 • IREV= 1 IEND= 145 • FR,CHI,CHB,CHL 0.999975340 112059.109 112059.010 112061.774 • FR,CHI,CHB,CHL 0.999926023 113792.644 112050.819 112059.109 • IREV= 1 IEND= 144

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