Some test results by Baker Kearfott on some functions from the Mor'e
et al. test set, namely on 
Beale, Brown--Dennis, Wood, Kowalik--Osborne.

In these problems, the minimizers are easily found by local 
optimization, but to verify that these are global minimizers 
(which must be done by an exhaustive branch and bound procedure)
appears to be much more difficult. 


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The times are clock times on a 486 DX4-100 laptop with Windows 95 and 
Microsoft Fortran. The code was compiled in debug mode, with a factor 
of 5 penalty over the highest level of optimization. The raw speed of 
a Sparc 20 is probably another factor of 5 or so faster.


------------------------------------------------------------------------

Beale

 Initial box:
    -.2000D+01    .5000D+01    
    -.2000D+01    .5000D+01
 Optimizer:
     .3000D+01    .5012D+00
 Minimum value:
     .6094D-04 

Without subst.
 Number of bisections:         33
 Number of Gauss--Seidel steps on the dense system:        212
 Total number of dense slope matrix evaluations:        331
 Total number second-order interval evaluations of the
 original function:         79
 wall clock time:	 5s

With subst.
 Number of bisections:         11
 Number of Gauss--Seidel steps on the dense system:         48
 Total number of dense slope matrix evaluations:         97
 Total number second-order interval evaluations of the
 original function:         33
 wall clock time:	 3s

------------------------------------------------------------------------

Brown--Dennis

 Initial box:
    -.1000D+02    .1000D+02
    -.1000D+02    .1000D+02
    -.1000D+02    .1000D+02
    -.1000D+02    .1000D+02
 Optimizer:
    .0000D+00    .0000D+00    .0000D+00    .0000D+00
 Minimum value:
    .1615D+08    

Without subst.
 Number of bisections:        150
 Number of Gauss--Seidel steps on the dense system:       1645
 Total number of dense slope matrix evaluations:       2014
 Total number second-order interval evaluations of the
 original function:        314
 wall clock time:	 263s

With subst.
 Number of bisections:         72
 Number of Gauss--Seidel steps on the dense system:        634
 Total number of dense slope matrix evaluations:        818
 Total number second-order interval evaluations of the
 original function:        158
 wall clock time:	 179s


------------------------------------------------------------------------

Kowalik

 Initial box:
     .0000D+00    .1000D+01
     .0000D+00    .1000D+01
     .0000D+00    .1000D+01
     .0000D+00    .1000D+01
 Optimizer:
     .1928D+00    .1913D+00    .1231D+00    .1361D+00
 Minimum value:
     .3075D-03    

Without subst.
 Number of bisections:      14268
 Number of Gauss--Seidel steps on the dense system:     185794
 Total number of dense slope matrix evaluations:     226720
 Total number second-order interval evaluations of the
 original function:      28560
 wall clock time:	 17140s

With subst.
 Number of bisections:      10644
 Number of Gauss--Seidel steps on the dense system:     151622
 Total number of dense slope matrix evaluations:     183411
 Total number second-order interval evaluations of the
 original function:      21331
 wall clock time:	 15350s


------------------------------------------------------------------------

Wood

 Initial box:
    -.1000D+02    .1000D+02
    -.1000D+02    .1000D+02
    -.1000D+02    .1000D+02
    -.1000D+02    .1000D+02
 Optimizer:
     .1000D+01    .1000D+01    .1000D+01    .1000D+01
 Minimum value:
     .0000D+00

Without subst.
 Number of bisections:         11
 Number of Gauss--Seidel steps on the dense system:        140
 Total number of dense slope matrix evaluations:        193
 Total number second-order interval evaluations of the
 original function:         45
 wall clock time:	 4s

With subst.
 Number of bisections:         10
 Number of Gauss--Seidel steps on the dense system:        110
 Total number of dense slope matrix evaluations:        165
 Total number second-order interval evaluations of the
 original function:         45
 wall clock time:	 4s

------------------------------------------------------------------------