RegularizedOptimization

How to cite

If you use RegularizedOptimization.jl in your work, please cite using the format given in CITATION.bib.

Synopsis

This package contains solvers to solve regularized optimization problems of the form

minₓ f(x) + h(x)

where f: ℝⁿ → ℝ has Lipschitz-continuous gradient and h: ℝⁿ → ℝ is lower semi-continuous and proper. The smooth term f describes the objective to minimize while the role of the regularizer h is to select a solution with desirable properties: minimum norm, sparsity below a certain level, maximum sparsity, etc. Both f and h can be nonconvex.

Installation

To install the package, hit ] from the Julia command line to enter the package manager and type julia pkg> add https://github.com/JuliaSmoothOptimizers/RegularizedOptimization.jl

What is Implemented?

Please refer to the documentation.

Related Software

• RegularizedProblems.jl
• ShiftedProximalOperators.jl

References

1. A. Y. Aravkin, R. Baraldi and D. Orban, A Proximal Quasi-Newton Trust-Region Method for Nonsmooth Regularized Optimization, SIAM Journal on Optimization, 32(2), pp.900–929, 2022. Technical report: https://arxiv.org/abs/2103.15993
2. R. Baraldi, R. Kumar, and A. Aravkin (2019), Basis Pursuit De-noise with Non-smooth Constraints, IEEE Transactions on Signal Processing, vol. 67, no. 22, pp. 5811-5823.

bibtex @article{aravkin-baraldi-orban-2022, author = {Aravkin, Aleksandr Y. and Baraldi, Robert and Orban, Dominique}, title = {A Proximal Quasi-{N}ewton Trust-Region Method for Nonsmooth Regularized Optimization}, journal = {SIAM Journal on Optimization}, volume = {32}, number = {2}, pages = {900--929}, year = {2022}, doi = {10.1137/21M1409536}, abstract = { We develop a trust-region method for minimizing the sum of a smooth term (f) and a nonsmooth term (h), both of which can be nonconvex. Each iteration of our method minimizes a possibly nonconvex model of (f + h) in a trust region. The model coincides with (f + h) in value and subdifferential at the center. We establish global convergence to a first-order stationary point when (f) satisfies a smoothness condition that holds, in particular, when it has a Lipschitz-continuous gradient, and (h) is proper and lower semicontinuous. The model of (h) is required to be proper, lower semi-continuous and prox-bounded. Under these weak assumptions, we establish a worst-case (O(1/\epsilon^2)) iteration complexity bound that matches the best known complexity bound of standard trust-region methods for smooth optimization. We detail a special instance, named TR-PG, in which we use a limited-memory quasi-Newton model of (f) and compute a step with the proximal gradient method, resulting in a practical proximal quasi-Newton method. We establish similar convergence properties and complexity bound for a quadratic regularization variant, named R2, and provide an interpretation as a proximal gradient method with adaptive step size for nonconvex problems. R2 may also be used to compute steps inside the trust-region method, resulting in an implementation named TR-R2. We describe our Julia implementations and report numerical results on inverse problems from sparse optimization and signal processing. Both TR-PG and TR-R2 exhibit promising performance and compare favorably with two linesearch proximal quasi-Newton methods based on convex models. } }