ADMM for l1 Regularized Optimization Problems and Applications Oriented Input Design for MPC

Sammanfattning: This licentiate thesis is divided into two main parts. The first part considers alternating direction method of multipliers (ADMM) for ℓ1 regularized optimization problems and the second part considers applications oriented input design for model predictive control (MPC).Many important problems in science and engineering can be formulated as convex optimization problems. As such, they have a unique solution and there exist very efficient algorithms for finding the solution. We are interested in methods that can handle big, in terms of the number of variables, optimization problems in an efficient way. Large optimization problems are common in many fields of research, for example, the problem of feature selection from huge medical data sets. ADMM is a method capable of handling such problems. We derive a scalable and efficient algorithm based on ADMM for two ℓ1 regularized optimization problems: ℓ1 mean and covariance filtering that occur in signal processing, and ℓ1 regularized MPC that is a specific type of model based control.System identification provides tools for estimating models of dynamical systems from experimental data. The application of such models can be divided into three main categories: prediction, simulation and control. We focus on identifying models used for control, with special attention to MPC. The objective is to minimize a cost related to the identification experiment while guaranteeing, with high probability, that the obtained model gives an acceptable control performance. We use applications oriented input design to find such a model. We present a general procedure of implementing applications oriented input design to unknown, and possibly nonlinear, systems controlled using MPC. In addition, we show that the input design problem obtained for output-error systems has the same simple structure as for finite impulse response systems.