joint work with Martina Cerulli, Elisabeth Gaar, Angelika Wiegele
Augmented Lagrangian methods are among the most popular first-order approaches to handle large scale semidefinite programming problems. We focus on doubly nonnegative problems, namely semidefinite programming problems where elements of the matrix variable are constrained to be nonnegative. Starting from two methods already proposed in the literature on conic programming, we introduce Augmented Lagrangian methods with the possibility of employing a factorization of the dual variable. We present numerical results for instances of the DNN relaxation of the stable set problem.
Bundle methods form a cutting model from collected subgradient information and determine the next candidate as the minimizer of the model augmented with a proximal term. The choice of cutting model and proximal term offers a lot of flexibility and may even allow to move towards second order behavior. After a short introduction to bundle methods in general, we discuss the current approach for integrating such choices in the forthcoming ConicBundle 1.0 for the general and the semidefinite case and report on experience gathered on several test instances.
In this talk, I consider the problem of shape-constrained regression, where response variables are assumed to be a function f of some features (corrupted by noise) and f is constrained to be monotonous or convex with respect to these features. Problems of these type occur in many areas, such as, e.g., pricing. Using semidefinite programming, I construct a regressor for the data which has the same monotonicity and convexity properties as f. I further show that this regressor is a consistent estimator of f.