The well-known basis-pursuit denoise problem can be solved using the root-finding approach that was introduced in the SPGL1 solver. In this talk we present several techniques that can improve the performance of the solver. For the Lasso subproblems we develop a hybrid quasi-Newton method that enables light-weight switching between the regular spectral projected-gradient steps and quasi-Newton steps restricted to the current face of the one-norm ball. We also introduce additional improvements to the root-finding procedure and the generation of dual-feasible points.
joint work with Andre Milzarek, Zaiwen Wen, Tong Zhang
We present a stochastic quasi-Newton extragradient method for solving composite-type optimization problems. We assume that gradient information of the smooth part of the objective function can only be accessed via stochastic oracles. The proposed algorithm combines semismooth quasi-Newton steps for a fixed-point equation with stochastic proximal extragradient steps and converges to stationary points in expectation. Local convergence results are discussed for a variant of the approach using variance reduction and numerical comparisons are provided illustrating the efficiency of the method.
joint work with Armin Rund
We present an algorithm for the efficient solution of structured nonsmooth operator equations in Banach spaces. The term structured indicates that we consider equations which are composed of a smooth and a semismooth mapping. The novel algorithm combines a semismooth Newton method with a quasi-Newton method. This hybrid approach retains the local superlinear convergence of both these methods under standard assumptions and can be embedded in a matrix-free limited-memory truncated trust-region framework. Numerical results on nonconvex and nonsmooth large-scale real-world problems are provided.