مرکز منطقه ای اطلاع رساني علوم و فناوري - Theoretical and computational aspects of the optimal design centering, tolerancing, and tuning problem

Abstract :

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 $P_g$ , $P: min \\{ f(x) \\mid \\Psi (x) \\buildrel\\triangle\\over{=} max_{\\omega \\in \\Omega} min_{\\tau \\in T} \\zeta (x,\\omega , \\tau ) \\leq 0 \\}$ 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.