IOR Preprint 4/2011
M. Diehl, B. Houska, O. Stein, P. Steuermann:
A lifting method for generalized semi-infinite programs based on lower level Wolfe duality
Abstract: This paper introduces novel numerical solution strategies for generalized semi-infinite optimization problems (GSIP), a class of mathematical optimization problems which occur naturally in the context of design centering problems, robust optimization problems, and many fields of engineering science. GSIPs can be regarded as bilevel optimization problems, where a parametric lower-level maximization problem has to be solved in order to check feasibility of the upper level minimization problem. The current paper discusses several strategies to reformulate this class of problems into equivalent standard minimization problems by exploiting the concept of lower level Wolfe duality. Here, the main contribution is the discussion of the non-degeneracy of the corresponding formulations under various assumptions. Finally, these non-degenerate re-formulations of the original GSIP allow us to apply standard nonlinear optimization algorithms.
Keywords: Semi-infinite optimization, lower level duality, lifting approach, adaptive convexification, mathematical program with complementarity constraints.
AMS Subject Classification: 90C34, 90C25, 90C46, 65K05.
IOR Preprint 3/2011
O. Stein, P. Steuermann:
On smooth relaxations of obstacle sets
Abstract: We present and discuss a method to relax sets described by finitely many smooth convex inequality constraints by the level set of a single smooth convex inequality constraint. Based on error bounds and Lipschitz continuity, special attention is paid to the maximal approximation error and a guaranteed safety margin. Our results allow to safely avoid the obstacle by obeying a single smooth constraint. Numerical results indicate that our technique gives rise to a smoothing method which performs well even for smoothing parameters very close to zero.
Keywords: Relaxation, error bound, Lipschitz continuity, hyperbolic smoothing, entropic smoothing, obstacle problem.
AMS Subject Classification: 90C25, 90C30, 90C31.
IOR Preprint 2/2011
O. Stein:
Twice differentiable characterizations of convexity notions for functions on full dimensional convex sets
Abstract: We derive C²-characterizations for convex, strictly convex, as well as uniformly convex functions on full dimensional convex sets. In the cases of convex and uniformly convex functions this weakens the well-known openness assumption on the convex sets. We also show that, in a certain sense, the full dimensionality assumption cannot be weakened further. In the case of strictly convex functions we weaken the well-known sufficient C²-condition for strict convexity to a characterization. Several examples illustrate the results.
Keywords: Convexity, strict convexity, true convexity, differentiable characterization, full dimensional convex set.
IOR Preprint 1/2011
V. Shikhman,O. Stein:
On jet-convex functions and their tensor products
Abstract: In this paper we introduce necessary and sufficient conditions for the tensor product of two convex functions to be convex. For our analysis we introduce the notions of true convexity, jet-convexity, true jet-convexity as well as true log-convexity. The links between jet-convex and log-convex functions are elaborated. As an algebraic tool we introduce the jet product of two symmetric matrices and study some of its properties. We illustrate our results by an application from global optimization, where a convex underestimator for the tensor product of two functions is constructed as the tensor product of convex underestimators of the single functions.
Keywords: 2-jet matrix, jet-convexity, true convexity, log-convexity, tensor product of functions, jet product of matrices.