Related papers: Upper-bounding $\ell_1$-optimization weak threshol…
This note studies a method for the efficient estimation of a finite number of unknown parameters from linear equations, which are perturbed by Gaussian noise. In case the unknown parameters have only few nonzero entries, the proposed…
We consider a model of an electric circuit, where differential algebraic equations for a circuit part are coupled to partial differential equations for an electromagnetic field part. An uncertainty quantification is performed by changing…
We study sparsity in the max-plus algebraic setting. We seek both exact and approximate solutions of the max-plus linear equation with minimum cardinality of support. In the former case, the sparsest solution problem is shown to be…
The efficacy of robust optimization spans a variety of settings with uncertainties bounded in predetermined sets. In many applications, uncertainties are affected by decisions and cannot be modeled with current frameworks. This paper takes…
We consider optimization problems containing nonconvex quadratic functions for which semidefinite programming (SDP) relaxations often yield strong bounds. We investigate linear inequalities that outer approximate the positive semidefinite…
We introduce LiPopt, a polynomial optimization framework for computing increasingly tighter upper bounds on the Lipschitz constant of neural networks. The underlying optimization problems boil down to either linear (LP) or semidefinite…
We introduce a general framework to handle structured models (sparse and block-sparse with possibly overlapping blocks). We discuss new methods for their recovery from incomplete observation, corrupted with deterministic and stochastic…
This investigation is motivated by PDE-constrained optimization problems arising in connection with electrocardiograms (ECGs) and electroencephalography (EEG). Standard sparsity regularization does not necessarily produce adequate results…
Error bounds have been studied for more than seventy years, beginning with the seminal result of Hoffman (1952) [{\it J. Res. Natl. Bur. Standards}, 49 (1952), 263--265], which establishes an upper bound for the distance from an arbitrary…
The sparse optimization problems arise in many areas of science and engineering, such as compressed sensing, image processing, statistical and machine learning. The $\ell_{0}$-minimization problem is one of such optimization problems, which…
Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel…
Uncertainty relations are a fundamental feature of quantum mechanics. How can these relations be found systematically? Here we develop a semidefinite programming hierarchy for additive uncertainty relations in the variances of non-commuting…
This paper investigates a general class of problems in which a lower bounded smooth convex function incorporating $\ell_{0}$ and $\ell_{2,0}$ regularization is minimized over a box constraint. Although such problems arise frequently in…
Solving sparse linear systems from discretized PDEs is challenging. Direct solvers have in many cases quadratic complexity (depending on geometry), while iterative solvers require problem dependent preconditioners to be robust and…
This paper considers sparsity in linear regression under the restriction that the regression weights sum to one. We propose an approach that combines $\ell_0$- and $\ell_1$-regularization. We compute its solution by adapting a recent…
We consider the general nonlinear optimization problem where the objective function has an additional term defined by the $ \ell_0 $-quasi-norm in order to promote sparsity of a solution. This problem is highly difficult due to its…
We study a regularization framework that combines a convex fidelity term with multiple $\ell_1$-based regularizers, each linked to a distinct linear transform. This multi-penalty model enhances flexibility in promoting structured sparsity.…
Most inverse problems from physical sciences are formulated as PDE-constrained optimization problems. This involves identifying unknown parameters in equations by optimizing the model to generate PDE solutions that closely match measured…
We develop both first and second order numerical optimization methods to solve non-smooth optimization problems featuring a shared sparsity penalty, constrained by differential equations with uncertainty. To alleviate the curse of…
One way to understand time-series data is to identify the underlying dynamical system which generates it. This task can be done by selecting an appropriate model and a set of parameters which best fits the dynamics while providing the…