Abstract
We propose a new algorithm for the optimization of convex functions over a polyhedral set in {mathbb {R}}^n. The algorithm extends the spectral projected-gradient method with limited-memory BFGS iterates restricted to the present face whenever possible. We prove convergence of the algorithm under suitable conditions and apply the algorithm to solve the Lasso problem, and consequently, the basis-pursuit denoise problem through the root-finding framework proposed by van den Berg and Friedlander (SIAM J Sci Comput 31(2):890–912, 2008). The algorithm is especially well suited to simple domains and could also be used to solve bound-constrained problems as well as problems restricted to the simplex.
Highlights
In this paper we propose an algorithm for optimization problems of the form minimize f (x) subject to x ∈ C, (1)x where f : Rn → R is a convex, twice continuously differentiable function, and C is a polyhedral set in Rn
As this work was motivated by improving the Lasso problem, we focus on the weighted one-norm ball: Cw,1 = {x ∈ Rn | x w,1 ≤ τ }, where x w,1 := i wi |xi | positive wi
The first thing to notice is that the runtime of qpOASES is insensitive to the value of τ, whereas the runtime for spg and the hybrid method increases with τ
Summary
In this paper we propose an algorithm for optimization problems of the form minimize f (x) subject to x ∈ C,. Ax − b 2 subject to x 1 ≤ τ, and that solving (BPσ ) can be reduced to finding the smallest τ for which the Lasso solution xτ∗ satisfies Axτ∗ − b ≤ σ Denoting by τσ this critical value of τ and assuming that b lies in the range space of A it was shown in [2] that the Pareto curve is convex and differentiable at all τ ∈ [0, τ0) with gradient AT r ∞/ r 2 where r denotes the misfit Axτ∗ − b. In this paper we propose a hybrid algorithm that switches between the two methods in a seamless and lightweight fashion
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
