Abstract
The following problem is studied: Given a compact setS inR n and a Minkowski functionalp(x), find the largest positive numberr for which there existsx ∈ S such that the set of ally ∈ R n satisfyingp(y−x) ≤ r is contained inS. It is shown that whenS is the intersection of a closed convex set and several complementary convex sets (sets whose complements are open convex) this “design centering problem” can be reformulated as the minimization of some d.c. function (difference of two convex functions) overR n. In the case where, moreover,p(x) = (x T Ax)1/2, withA being a symmetric positive definite matrix, a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.K. Brayton, S.W. Director, G.D. Hachtel and L.M. Vidigal, “A new algorithm for statistical circuit deign based on quasi-Newton methods and function splitting,”IEEE Transactions on Circuits Systems 26 (1979) 784–794.
R.B. Holmes,Geometric Functional Analysis and Its Applications (Springer-Verlag, New York, Heidelberg, Berlin, 1975).
O.L. Mangasarian and T.-H. Shiau, “A variable complexity norm maximization problem,” Technical Summary Report No. 2780, University of Wisconsin-Madison, Mathematics Research Center (Madison, WI, 1985).
E. Polak, “An implementable algorithm for the optimal design centering, tolerancing, and tuning problem,”Journal of Optimization Theory and Applications 37 (1982) 45–67.
T.-H. Shiau, “Finding the largestl p-ball in a polyhedral set,” Technical Summary Report No. 2778, University of Wisconsin-Madison, Mathematics Research Center (Madison, WI, 1984).
J.J. Strodiot, V.H. Nguyen and Ng.V. Thoai, “On an optimum shape design problem,” Preprint, Facultés Universitaires de Namur, 1985.
P.T. Thach and H. Tuy, “Global optimization under Lipschitzian constraints,”Japan Journal of Applied Mathematics 4 (1987) 205–217.
T.V. Thieu, B.T. Tam and V.T. Ban, “An outer approximation method for globally minimizing a concave function over a compact convex set,” IFIP Working Conference on Recent Advances on System Modeling and Optimization (Hanoi, 1983).
H. Tuy and Ng. V. Thuong, “Minimizing a convex function over the complement of a convex set,”Methods of Operations Research 49 (1984) 85–99.
H. Tuy, “On outer approximation methods for solving concave minimization problems,”Acta Mathematica Vietnamica 8 (1983), No. 2, 3–34.
H. Tuy, “Convex programs with an additional reverse convex constraint,”Journal of Optimization Theory and Applications 52 (1987) 463–486.
H. Tuy, “A general deterministic approach to global optimization via d.c. programming,” presented at the ConferenceMathematics for Optimization (Toulouse, 1985).
L.M. Vidigal and S.W. Director, “A design centering algorithm for non-convex regions of acceptability,”IEEE Transactions on Computer Aided Design of Integrated Circuits and Systems I (1982) 11–24.
L.M. Vidigal and S.W. Director, “Design centering: The quasiconvex, quasiconcave performance function case,”Proceedings IEEE International Symposium on Circuits and System (1980).
Z.Q. Xia, J.J. Strodiot and V.H. Nguyen, “Some optimality conditions for theC M -embedded problem with Euclidean norm,”IIASA Workshop on Nondifferentiable Optimization: Motivations and Applications (Sopron, Hungary, 1984).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Thach, P.T. The design centering problem as a D.C. programming problem. Mathematical Programming 41, 229–248 (1988). https://doi.org/10.1007/BF01580765
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01580765