Are greedy methods such as orthogonal matching pursuit considered obsolete for finding sparse solutions? When researchers first began seeking sparse solutions to $Ax = b$, they used greedy methods such as orthogonal matching pursuit (OMP).  In OMP, we activate components of $x$ one by one, and at each stage we select the component $i$ such that the $i$th column of $A$ is most correlated with the residual $Ax - b$.
Researchers then developed methods such as Basis Pursuit and Lasso, which are based on solving optimization problems with sparsity-inducing regularizers. The Basis Pursuit problem is
\begin{align}
\underset{x}{\text{minimize}} & \quad \| x \|_1 \\
\text{subject to} & \quad Ax = b.
\end{align}
The Lasso problem is
$$
\underset{x}{\text{minimize}} \quad \frac12 \| Ax - b \|_2^2 + \lambda \| x \|_1.
$$
This new strategy was made possible by new optimization algorithms (interior point methods) which were able to solve these large scale optimization problems efficiently.
Question: Are greedy methods such as orthogonal matching pursuit and its variants now considered to be obsolete?  Is there a consensus that they do not work as well as the approaches based on optimization with sparsity-inducing regularizers? Has OMP been abandoned?
Here is a 1994 paper by Chen and Donoho that gives a brief overview of early attempts to find sparse solutions to $Ax = b$, leading up to Basis Pursuit and Lasso:
Atomic Decomposition by Basis Pursuit
 A: If by ''obsolete'' you mean ''researchers are not working on them anymore'', the answer is no - or, at least, I shall hope not! Here are few papers that have been published among the iterative / greedy approaches, that are used to analyze various methods: 


*

*for some new results on orthogonal matching pursuit 2015

*Some results on the efficacy of Graded Hard Thresholding Pursuit (yes, this is shameless advertisement) 2016

*IHT for subexponential measurements 2017

*Some results on hard thresholding 2016


This list is clearly not exhaustive, but represents 4 main papers (in my opinion) that treat of iterative and greedy approaches for sparse approximation. Should be included: all resources related to co-sparse model, dictionary-sparse approaches, one bit approaches & co. 
If by ''obsolete'' you mean ''has not practical applications'', well allow me to retort! I have personally used such greedy approaches in the following contexts: 


*

*Approximation of numerical solutions to high dimensional PDEs via sparse polynomial chaos expansions

*Analysis of metagenomics data survey by greedy methods and information theory (this is quite old, needs to be refreshed, and is still pretty much work in progress)


Bottom line, yes! These methods are alive, and are being researched from both a theoretical and a practical point of view! 
