joint work with Michael Ulbrich
We propose a globalized semismooth Newton framework for solving convex-composite optimization problems involving smooth nonconvex and nonsmooth convex terms in the objective function. A prox-type fixed point equation representing the stationarity conditions forms the basis of the approach. The proposed algorithm utilizes semismooth Newton steps for the fixed point equation to accelerate an underlying globally convergent descent method. We present both global and local convergence results and provide numerical experiments illustrating the efficiency of the semismooth Newton method.
joint work with Abraham Engle
We discuss the local convergence theory of Newton and quasi-Newton methods for convex-composite optimization: minimize $f (x)=h(c(x))$, where h is an infinite-valued closed proper convex function and $c$ is smooth. The focus is on the case where $h$ is infinite-valued convex piecewise linear-quadratic. The optimality conditions are embedded into a generalized equation, and we show the strong metric subregularity and strong metric regularity of the corresponding set-valued mappings. The classical convergence of Newton-type methods is extended to this class of problems
We study conjugacy results of compositions of a convex function $g$ and a nonlinear mapping $F$ (possibly extended-valued in a generalized sense) such that the composition $g\circ F$ is still convex. This composite structure which preserves convexity models a myriad of applications comprising e.g. conic programming, convex additive composite minimization, variational Gram functions or vector optimization. Our methodology is largely based on infimal convolution combined with cone-induced convexity.