Robust Optimization: Design in MEMS - University of California ...

19where x k−1 was the current minimum at the last iteration. The single variable optimizationproblem ismintN(t)D(t)s.t. g i (t) ≤ 0 (3.6)We will refer to this problem as a line search in IR n . This rational polynomial minimizationin one variable, subject to polynomial constraints is easy to solve in closedform.3.3 Exact Line Search MinimizationThe solution to the problem posed in equation (3.6) is presented here. We beginby defining the set of feasible points, Ω i , for each constraint, g i (t), asΩ i := {t : g i (t) ≤ 0} (3.7)Computing Ω i is done by finding the real roots of g i (t), and then checking the intervalsbetween the roots to identify where g i (t) is negative.Assuming that the roots of g i (t) can be computed numerically, the real roots,{λ r1 , λ r2 , . . . , λ rp }, satisfy g i (λ rk ) = 0 for k = 1, ..., p. Once the real roots are identified,the set Ω i can be found by simply checking the sign of g i (t) on either side of λ rkfor k = 1, ..., p. Double roots are easily handled because the algorithm computes thesgn(g i (t)) on both sides of λ rk . If no real roots exist, then the function g i (t) is eitherstrictly positive, in which case Ω i is the empty set, or strictly negative which impliesΩ i ∈ IR. In this case, evaluating g i (0) is sufficient to determine the sign of g i (t) for allt. Storing Ω i is can be done by maintaining a list of numbered pairs that representthe intervals of feasible regions.Once all of the feasible sets have been found, the next step is to find their intersection.We will define the intersection to beΩ := Ω 1 ∩ Ω 2 ∩ ... ∩ Ω p (3.8)Ω is the set of feasible points for which all the constraints are satisfied. If Ω is theempty set, then the search has failed because no where along the line x = x k−1 + tv