Theoretical and computational aspects of the optimal design centering, tolerancing, and tuning problem

The optimal design centering, tolerancing, and tuning problem is transcribed into a mathematical programming problem of the form
P_g: \min\{f(x)|\max_{\omega\in\Omega}\min_{\tau\in\Gamma} \zeta^{j}(x,\omega, \tau) \leq 0\} , x \geq 0, x, \omega, \tau \in R^{n}
,
f: R^n \rightarrow R^1
,
\zeta: R^n \times R^n \times R^n \rightarrow R^1
, continuously differentiable,
\Omega
and
T
compact subsets of
R^n
,
J=\{1, \cdots , p\}
. A simplified form of
P_g
,
P: \min \{f(x) \Psi (x) \underset{=}{\triangle} \max_{omega\in \Omega \min_{\tau \in T} \zeta(x,\omega, \tau ) \leq 0 \}
is discussed. It is shown that
$\Psi(\cdot )
is locally Lipschitz continuous but not continuously differentiable. Optimality conditions for
P
based on the concept of generalized gradients are derived. An algorithm, consisting of a master outer approximations algorithm proposed by Gonzaga and Polak and of a new subalgorlthm for nondifferentiable problems of the form
P_{i}: \min\{f(x)| \max_{\omega\in\Omega_i\} \min_{\tau \in T} \zeta (x, \omega, \tau ) \leq 0 \}
, where
\Omega_i
is a discrete set, is presented. The subalgorlthm, an extension of Polak's method of feasible directions to nondifferentlable problems, is shown to converge under suitable assumptions. Moreover, the optimality function used in the subalgorithm is proven to satisfy a condition which guarantees that the overall algorithm converges.