The augmented Lagrangian method (ALM) is a benchmark for solving the convex minimization problem with linear constraints. We consider the special case where the objective is in form of the sum of m functions without coupled variables. For solving this separable convex programming model, it is usually required to decompose the ALM subproblem at each … Read more
We propose restricted memory level bundle methods for minimizing constrained convex nonsmooth optimization problems whose objective and constraint functions are known through oracles (black-boxes) that might provide inexact information. Our approach is general and covers many instances of inexact oracles, such as upper, lower and on-demand oracles. We show that the proposed level bundle methods … Read more
We consider convex nonsmooth optimization problems whose objective function is known through a (fine) oracle together with some additional (cheap but poor) information – formalized as a second coarse oracle with uncontrolled inexactness. It is the case when the objective function is itself the output of an optimization solver, using a branch-and-bound procedure, or decomposing … Read more
We consider the problem of identifying the densest k-node subgraph in a given graph. We write this problem as an instance of rank-constrained cardinality minimization and then relax using the nuclear and l1 norms. Although the original combinatorial problem is NP-hard, we show that the densest k-subgraph can be recovered from the solution of our … Read more
This paper presents a proximal point approach for computing the Riemannian or intrinsic Karcher mean of symmetric positive definite matrices. Our method derives from proximal point algorithm with Schur decomposition developed to compute minimum points of convex functions on symmetric positive definite matrices set when it is seen as a Hadamard manifold. The main idea … Read more
In this paper we analyze the randomized block-coordinate descent (RBCD) methods for minimizing the sum of a smooth convex function and a block-separable convex function. In particular, we extend Nesterov’s technique (SIOPT 2012) for analyzing the RBCD method for minimizing a smooth convex function over a block-separable closed convex set to the aforementioned more general … Read more
Motivated by mixed-integer, nonlinear optimization problems, we derive linear inequality characterizations for sets of the form conv{(x, q ) \in R^d × R : q \in Q(x), x \in R^d – int(P )} where Q is convex and differentiable and P \subset R^d . We show that in several cases our characterization leads to polynomial-time … Read more
In this paper we provide a generalization of two well-known positivstellensätze, namely the positivstellensatz from Pólya [Georg Pólya. Über positive darstellung von polynomen vierteljschr. In Naturforsch. Ges. Zürich, 73: 141-145, 1928] and the positivestellensatz from Putinar and Vasilescu [Mihai Putinar and Florian-Horia Vasilescu. Positive polynomials on semialgebraic sets. Comptes Rendus de l’Académie des Sciences – … Read more
The rediscovery of the affine scaling method in the late 80s was one of the turning points which led to a new chapter in Modern Optimization – the interior point methods (IPMs). Simultaneously and independently the theory of exterior point methods for convex optimization arose. The two seemingly unconnected fields turned out to be intrinsically … Read more
While there is no lack of efficient Krylov subspace solvers for Hermitian systems, there are few for complex symmetric, skew symmetric, or skew Hermitian systems, which are increasingly important in modern applications including quantum dynamics, electromagnetics, and power systems. For a large consistent complex symmetric system, one may apply a non-Hermitian Krylov subspace method disregarding … Read more