Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

Table 3.3: Algorithm performance: anti-optimization.
J ∗ iter fevals time x ∗ [m] x ∗ [kg]
Alg. Form [µm] # # [min] d1 d2 m1 m2
SA Eq 3.4 307.17 59 1800 11.52 0.0481 0.0580 0.1234 0.0164
SQP Eq 3.4 306.86 25 86 1.48 0.0486 0.0581 0.0 0.0
SQP Eq 3.5 306.86 17 35 2.00 0.0486 0.0581 0.0 0.0
MC SQP Eq 3.5 306.86 698 1431 53.52 0.0486 0.0581 0.0 0.0
mulations using the SA design as an initial guess. In both cases the same design is
found, but the simple minimization problem (Equation 3.5) is more efficient. The
min-max problem (Equation 3.4) takes 2.0 minutes and requires 25 iterations to con-
verge, while the simple optimization converges in only 1.5 minutes and 17 iterations.
The MC SQP algorithm finds the same optimal design as the SA-SQP combination
indicating that this design is likely to be a global optimum. The combination of SA
and SQP proves to be a much quicker way to heuristically search the space than MC
SQP, requiring a combined 13.52 minutes to find the optimal design in contrast to
53.32 minutes needed for the ten MC SQP runs.
The SA design is slightly sub-optimal, but is close to the SQP designs and pro-
vides a good starting point for the gradient search. The min-max formulation is used
for SA since gradients are not required, and the algorithm may have trouble finding
feasible iterates due to the additional constraints in the alternate formulation (Equa-
tion 3.5). The SA algorithm requires far more function evaluations then the SQP
optimizers because the search is not guided by gradient information. Although the
SA optimization does not perform quite as well as SQP, the worst-case performance
of the resulting design, J ∗ , is significantly lower than that of the worst-case PT de-
sign indicating that a more robust design has indeed been found. Note that in the
SQP-optimized design the design masses are all zero, while those in the SA design are
small, but non-zero. Due to the random nature of the SA search, it is highly unlikely
that the resulting design is an optimal solution, especially if that solution is along a
constraint boundary, as is the case with the design masses.
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