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 Pg: min{f(x)/maxω∊Ω min ᵧ∊T ᶠj(x,w,t) < 0, x>0}, x,ω,τ∊Rm, f: Rn→R1, ᶠRn × Rn× Rn→R1·, continuously differentiable, Ω and T compact subsets of Rn, J= { 1,…,p}. A simplified form of Pg’ P: min { f(x)Ψ(x)maxω∊Ωminð „∊T ᶠ(x,w,t)< 0, x>0}, is discussed. It is shown that Ψ (.) 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 subalgorithm for nondifferentiable problems of the form Pi: min{f(x) maxω∊Ω, minð„∊T ᶠj(x,w,t) < 0}, whereΩiis a discrete set, is presented. The subalgorithm, an extension of Polak's method of feasible directions to nondiffferentiable 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. © 1979 IEEE