optimization_in_scilab.pdf

Ar nap i t e r epsg imp 1 the script produces the

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”ar” , nap , i t e r , epsg , imp = - 1) The script produces the following output. -- > [ fopt , xopt , gradopt ] = optim ( myquadratic , x0 , . . . ”ar” , nap , i t e r , epsg , imp = - 1) | x | =1.562050 e+000, f =7.191736 e+000, | g | =5.255790 e+001 | x | =1.473640 e+000, f =3.415994 e+000, | g | =2.502599 e+001 | x | =1.098367 e+000, f =2.458198 e+000, | g | =2.246752 e+001 | x | =1.013227 e+000, f =1.092124 e+000, | g | =1.082542 e+001 | x | =9.340864 e - 001, f =4.817592e - 001, | g | =5.182592 e+000 [ . . . ] | x | =1.280564 e - 002, f =7.432396e - 021, | g | =5.817126 e - 018 | x | =1.179966 e - 002, f =3.279551e - 021, | g | =2.785663 e - 018 | x | =1.087526 e - 002, f =1.450507e - 021, | g | =1.336802 e - 018 | x | =1.002237 e - 002, f =6.409611e - 022, | g | =6.409898 e - 019 | x | =9.236694 e - 003, f =2.833319e - 022, | g | =3.074485 e - 019 Norm of projected gradient lower than 0.3074485D - 18. 19
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gradopt = 1.0D - 18 * 0.2332982 0.2002412 xopt = 0.0065865 0.0064757 fopt = 2.833D - 22 One can see that the algorithm terminates when the gradient is extremely small g ( x ) 10 - 18 . The cost function is very near zero f ( x ) 10 - 22 , but the solution is not accurate only up to the 3d digit. This is a very difficult test case for optimization solvers. The difficulty is because the function is extremely flat near the optimum. If the termination criteria was based on the gradient, the algorithm would stop in the early iterations. Because this is not the case, the algorithm performs significant iterations which are associated with relatively large steps. 1.8 Quasi-Newton ”qn” with bounds constraints : qnbd The comments state that the reference report is an INRIA report by F. Bonnans [ 4 ]. The solver qnbd is an interface to the zqnbd routine. The zqnbd routine is based on the following routines : calmaj : calls majour, which updates the BFGS matrix, proj : projects the current iterate into the bounds constraints, ajour : probably (there is no comment) an update of the BFGS matrix, rlbd : line search method with bound constraints, simul : computes the cost function The rlbd routine is documented as using an extrapolation method to computed a range for the optimal t parameter. The range is then reduced depending on the situation by : a dichotomy method, a linear interpolation, a cubic interpolation. The stopping criteria is commented as ”an extension of the Wolfe criteria”. The linear algebra does not use the BLAS API. It is in-lined, so that connecting the BLAS may be difficult. The memory requirements for this method are O ( n 2 ), which shows why this method is not recommended for large-scale problems (see [ 21 ], chap.9, introduction). 1.9 L-BFGS ”gc” without constraints : n1qn3 The comments in this solver are clearly written. The authors are Jean Charles Gilbert, Claude Lemarechal, 1988. The BFGS update is based on the article [ 34 ]. The solver n1qn3 is an interface to the n1qn3a routine. The architecture is clear and the source code is well commented. The n1qn3a routine is based on the following routines : 20
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prosca : performs a vector x vector scalar product, simul : computes the cost function, nlis0 : line search based on Wolfe criteria, extrapolation and cubic interpolation, ctonb : copies array u into v, ddd2 : computed the descent direction by performing the product hxg.
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