In this thesis we investigate the use of first-order convex optimization methods applied to problems in signal and image processing. First we make a general introduction to convex optimization, first-order methods and their iteration com-plexity. Then we look at different techniques, which can be used with first-order methods such as smoothing, Lagrange multipliers and proximal gradient meth-ods. We continue by presenting different applications of convex optimization and notable convex formulations with an emphasis on inverse problems and sparse signal processing. We also describe the multiple-description problem. We finally present the contributions of the thesis. The remaining parts of the thesis consist of five research papers. The first paper addresses non-smooth first-order convex optimization and the trade-off between accuracy and smoothness of the approximating smooth function. The second and third papers concern discrete linear inverse problems and reliable numerical reconstruction software. The last two papers present a convex opti-mization formulation of the multiple-description problem and a method to solve it in the case of large-scale instances. i i
