Topics in Large-Scale Sparse Estimation and Control

In this thesis, we study two topics related to large-scale sparse/nestimation and control. In the first topic, we describe a method to/neliminate features (variables) in $\ell_{1}$-regularized convex optimization/nproblems. The elimination of features leads to a potentially substantial/nreduction in computational effort needed to solve such problems, especially/nfor large values of the penalty parameter. Our method is not heuristic:/nit only eliminates features that are guaranteed to be absent after/nsolving the optimization problem. The feature elimination step is/neasy to parallelize and can test each feature for elimination independently./nMoreover, the computational effort of our method is negligible compared/nto that of solving the convex problem./nWe study the case of $\ell_{1}$-regularized least-squares problem/n(a.k.a. LASSO) extensively and derive a closed-form sufficient condition/nfor eliminating features. The sufficient condition can be evaluated/nby few vector-matrix multiplications. For comparison purposes, we/npresent a LASSO solver that integrates SAFE with the Coordinate Descent/nmethod. We call our method CD-SAFE, and we report the number of computations/nneeded for solving a LASSO problem using CD-SAFE and using the plain/nCoordinate Descent method. We observe at least a $100$ fold reduction/nin computational complexity for dense and sparse data-sets consisting/nof millions of variables and millions of observations. Some of these/ndata-sets can cause memory problems when loaded, or need specialized/nsolvers. However, with SAFE, we can extend LASSO solvers capabilities/nto treat large-scale problems, previously out of their reach. This/nis possible, because SAFE eliminates variables and thus portions of/nour data at the outset, before loading it into our memory./nWe also show how our method can be extended to general $\ell_{1}$-regularized/nconvex problems. We present preliminary results for the Sparse Support/nVector Machine and Logistic Regression problems./nIn the second topic of the thesis, we derive a method for open-loop/ncontrol of open channel flow, based on the Hayami model, a parabolic/npartial differential equation resulting from a simplification of the/nSaint-Venant equations. The open-loop control is represented as infinite/nseries using differential flatness, for which convergence is assessed./nNumerical simulations show the effectiveness of the approach by applying/nthe open-loop controller to irrigation canals modeled by the full/nSaint-Venant equations./nWe experiment with our controller on the Gignac Canal, located northwest/nof Montpellier, in southern France. The experiments show that it is/npossible to achieve a desired water flow at the downstream of a canal/nusing the Hayami model as an approximation of the real-system. However,/nour observations of the measured water flow at the upstream controlled/ngate made us realize some actuator limitations. For example, deadband/nin the gate opening and unmodeled disturbances such as friction in/nthe gate-opening mechanism, only allow us to deliver piece-wise constant/ncontrol inputs. This fact made us investigate a way to compute a controller/nthat respects the actuator limitations. We use the CD-SAFE algorithm,/nto compute such open-loop control for the upstream water flow. We/ncompare the computational effort needed to obtain an open-loop control/nwith certain dynamics using the CD-SAFE algorithm and the plain Coordinate/nDescent algoirthm. We show that with CD-SAFE we are able to obain/nan open-loop control signal with cheaper computations.

Advisor: El-Ghaoui, Laurent
Bayen, Alexandre