Related papers: Around a Farkas type Lemma
We study randomized algorithms for constrained optimization, in abstract frameworks that include, in strictly increasing generality: convex programming; LP-type problems; violator spaces; and a setting we introduce, consistent spaces. Such…
We consider the problem of decision-making with side information and unbounded loss functions. Inspired by probably approximately correct learning model, we use a slightly different model that incorporates the notion of side information in…
This paper considers the inversion of ill-posed linear operators. To regularise the problem the solution is enforced to lie in a non-convex subset. Theoretical properties for the stable inversion are derived and an iterative algorithm akin…
We present a convex formulation of dictionary learning for sparse signal decomposition. Convexity is obtained by replacing the usual explicit upper bound on the dictionary size by a convex rank-reducing term similar to the trace norm. In…
For a collection of convex bodies $P_1,\dots,P_n \subset \mathbb{R}^d$ containing the origin, a Minkowski complex is given by those subsets whose Minkowski sum does not contain a fixed basepoint. Every simplicial complex can be realized as…
We prove the log-Brunn-Minkowski conjecture for convex bodies with symmetries to $n$ independent hyperplanes, and discuss the equality case and the uniqueness of the solution of the related case of the logarithmic Minkowski problem. We also…
Motivated by recent increased interest in optimization algorithms for non-convex optimization in application to training deep neural networks and other optimization problems in data analysis, we give an overview of recent theoretical…
We introduce the notion of consistent error bound functions which provides a unifying framework for error bounds for multiple convex sets. This framework goes beyond the classical Lipschitzian and H\"olderian error bounds and includes…
In exact sparse optimization problems on Rd (also known as sparsity constrained problems), one looks for solution that have few nonzero components. In this paper, we consider problems where sparsity is exactly measured either by the…
We construct and parametrize solutions to the constraint equations of general relativity in a neighborhood of Minkowski spacetime with arbitrary prescribed decay properties at infinity. We thus provide a large class of initial data for the…
We consider the problem of minimizing a function, which is the sum of a linear function and a composition of a strongly convex function with a linear transformation, over a compact polyhedral set. Jaggi and Lacoste-Julien [14] showed that…
Predictive models are being increasingly used to support consequential decision making at the individual level in contexts such as pretrial bail and loan approval. As a result, there is increasing social and legal pressure to provide…
Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these…
This paper provides characterizations of the weak solutions of optimization problems where a given vector function $F,$ from a decision space $X$ to an objective space $Y$, is "minimized" on the set of elements $x\in C$ (where $C\subset X$…
This paper considers online convex optimization with long term constraints, where constraints can be violated in intermediate rounds, but need to be satisfied in the long run. The cumulative constraint violation is used as the metric to…
Non-convex optimization plays a key role in a growing number of machine learning applications. This motivates the identification of specialized structure that enables sharper theoretical analysis. One such identified structure is…
Convex sample approximations of chance-constrained optimization problems are considered, in which chance constraints are replaced by sets of sampled constraints. We propose a randomized sample selection strategy that allows tight bounds to…
We examine the duality theory for a class of non-convex functions obtained by composing a convex function with a continuous one. Using Fenchel duality, we derive a dual problem that satisfies weak duality under general assumptions. To…
Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the…
It is by now well-known that one can recover a potential in the wave equation from the knowledge of the initial waves, the boundary data and the flux on a part of the boundary satisfying the Gamma-conditions of J.-L. Lions. We are…