The task of recovering a low-rank matrix from its noisy linear measurements plays a central role in computational science. Smooth formulations of the problem often exhibit an undesirable phenomenon: the condition number, classically defined, scales poorly with the dimension of the ambient space. In contrast, we here show that in a variety of concrete circumstances, … Read more
This paper derives new inexact variants of the Douglas-Rachford splitting method for maximal monotone operators and the alternating direction method of multipliers (ADMM) for convex optimization. The analysis is based on a new inexact version of the proximal point algorithm that includes both an inertial step and overrelaxation. We apply our new inexact ADMM method … Read more
One of the most frequently used approaches to solve linear bilevel optimization problems consists in replacing the lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions and by reformulating the KKT complementarity conditions using techniques from mixed-integer linear optimization. The latter step requires to determine some big-M constant in order to bound the lower level’s dual feasible … Read more
In this paper we study the minimization of a nonsmooth black-box type function, without assuming any access to derivatives or generalized derivatives and without any knowledge about the analytical origin of the function nonsmoothness. Directional methods have been derived for such problems but to our knowledge no model-based method like a trust-region one has yet … Read more
Stochastic variance reduced methods have recently surged into prominence for solving large scale optimization problems in the context of machine learning. Tan, Ma and Dai et al. first proposed the new stochastic variance reduced gradient (SVRG) method with the Barzilai-Borwein (BB) method to compute step sizes automatically, which performs well in practice. On this basis, … Read more
This paper researches combinatorial algorithms for the multi-commodity flow problem. We relax the capacity constraints and introduce a \emph{penalty function} \(h\) for each arc. If the flow exceeds the capacity on arc \(a\), arc \(a\) would have a penalty cost. Based on the \emph{penalty function} \(h\), a new conception , \emph{equilibrium pseudo-flow}, is introduced. Then … Read more
Learning directed acyclic graphs (DAGs) from data is a challenging task both in theory and in practice, because the number of possible DAGs scales superexponentially with the number of nodes. In this paper, we study the problem of learning an optimal DAG from continuous observational data. We cast this problem in the form of a … Read more
We study the convergence rate of a hierarchy of upper bounds for polynomial minimization prob-lems, proposed by Lasserre [SIAM J. Optim.21(3) (2011), pp.864-885], for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r of the hierarchy is defined as the minimal expected value of the polynomial over … Read more
This paper researches combinatorial algorithms for the multi-commodity flow problem. We relax the capacity constraints and introduce a \emph{penalty function} \(h\) for each arc. If the flow exceeds the capacity on arc \(a\), arc \(a\) would have a penalty cost. Based on the \emph{penalty function} \(h\), a new conception , \emph{equilibrium pseudo-flow}, is introduced. Then … Read more
Let $V$ be an Euclidean Jordan algebra and $\Omega$ be a cone of invertible squares in $V$. Suppose that $g:\mathbb{R}_{+} \to \mathbb{R}$ is a matrix monotone function on the positive semiaxis which naturally induces a function $\tilde{g}: \Omega \to V$. We show that $-\tilde{g}$ is compatible (in the sense of Nesterov-Nemirovski) with the standard self-concordant … Read more