Abstract We study the Compressed Sensing (CS) problem, which is the problem of finding the most sparse vector that satisfies a set of linear measurements up to some numerical tolerance. CS is a central problem in Statistics, Operations Research and Machine Learning which arises in applications such as signal processing, data compression, image reconstruction, and multi-label learning. We introduce an $$\ell _2$$ ℓ 2 regularized formulation of CS which we reformulate as a mixed integer second order cone program. We derive a second order cone relaxation of this problem and show that under mild conditions on the regularization parameter, the resulting relaxation is equivalent to the well studied basis pursuit denoising problem. We present a semidefinite relaxation that strengthens the second order cone relaxation and develop a custom branch-and-bound algorithm that leverages our second order cone relaxation to solve small-scale instances of CS to certifiable optimality. When compared against solutions produced by three state of the art benchmark methods on synthetic data, our numerical results show that our approach produces solutions that are on average $$6.22\%$$ 6.22 % more sparse. When compared only against the experiment-wise best performing benchmark method on synthetic data, our approach produces solutions that are on average $$3.10\%$$ 3.10 % more sparse. On real world ECG data, for a given $$\ell _2$$ ℓ 2 reconstruction error our approach produces solutions that are on average $$9.95\%$$ 9.95 % more sparse than benchmark methods ( $$3.88\%$$ 3.88 % more sparse if only compared against the best performing benchmark), while for a given sparsity level our approach produces solutions that have on average $$10.77\%$$ 10.77 % lower reconstruction error than benchmark methods ( $$1.42\%$$ 1.42 % lower error if only compared against the best performing benchmark). When used as a component of a multi-label classification algorithm, our approach achieves greater classification accuracy than benchmark compressed sensing methods. This improved accuracy comes at the cost of an increase in computation time by several orders of magnitude. Thus, for applications where runtime is not of critical importance, leveraging integer optimization can yield sparser and lower error solutions to CS than existing benchmarks.

MOTIVATION: Various computational biology calculations require a probabilistic optimization protocol to determine the parameters that capture the system at a desired state in the configurational space. Many existing methods excel at certain scenarios, but fail in others due, in part, to an inefficient exploration of the parameter space and easy trapping into local minima. Here, we developed a general-purpose optimization engine in R that can be plugged to any, simple or complex, modelling initiative through a few lucid interfacing functions, to perform a seamless optimization with rigorous parameter sampling. RESULTS: ROptimus features simulated annealing and replica exchange implementations equipped with adaptive thermoregulation to drive Monte Carlo optimization process in a flexible manner, through constrained acceptance frequency but unconstrained adaptive pseudo temperature regimens. We exemplify the applicability of our R optimizer to a diverse set of problems spanning data analyses and computational biology tasks. AVAILABILITY AND IMPLEMENTATION: ROptimus is written and implemented in R, and is freely available from CRAN (http://cran.r-project.org/web/packages/ROptimus/index.html) and GitHub (http://github.com/SahakyanLab/ROptimus).

We study the Sparse Plus Low-Rank decomposition problem (SLR), which is the problem of decomposing a corrupted data matrix into a sparse matrix of perturbations plus a low-rank matrix containing the ground truth. SLR is a fundamental problem in Operations Research and Machine Learning which arises in various applications, including data compression, latent semantic indexing, collaborative filtering, and medical imaging. We introduce a novel formulation for SLR that directly models its underlying discreteness. For this formulation, we develop an alternating minimization heuristic that computes high-quality solutions and a novel semidefinite relaxation that provides meaningful bounds for the solutions returned by our heuristic. We also develop a custom branch-and-bound algorithm that leverages our heuristic and convex relaxations to solve small instances of SLR to certifiable (near) optimality. Given an input n-by-n matrix, our heuristic scales to solve instances where n = 10000 in minutes, our relaxation scales to instances where n = 200 in hours, and our branch-and-bound algorithm scales to instances where n = 25 in minutes. Our numerical results demonstrate that our approach outperforms existing state-of-the-art approaches in terms of rank, sparsity, and mean-square error while maintaining a comparable runtime.