A bundle-Newton method for nonsmooth unconstrained minimization

An Approximate Quasi-Newton Bundle-Type Method for Nonsmooth Optimization

and Applied Analysis 3 rate of convergence under some additional assumptions, and it should be noted that we only use the approximate values of the objective function and its subgradients which makes the algorithm easier to implement. Some notations are listed below for presenting the algorithm. (i) ∂f(x) = {ξ ∈ Rn | f(z) ≥ f(x) + ξT(z − x), ∀z ∈ R}, the subdifferential of f at x, and each such...

A Bundle Method for a Class of Bilevel Nonsmooth Convex Minimization Problems

We consider the bilevel problem of minimizing a nonsmooth convex function over the set of minimizers of another nonsmooth convex function. Standard convex constrained optimization is a particular case in this framework, corresponding to taking the lower level function as a penalty of the feasible set. We develop an explicit bundle-type algorithm for solving the bilevel problem, where each itera...

A modified secant method for unconstrained minimization

Truncated Nonsmooth Newton Multigrid Methods for Simplex-Constrained Minimization Problems

We present a multigrid method for the minimization of strongly convex functionals defined on a finite product of simplices. Such problems result, for example, from the discretization of multi-component phase-field problems. Our algorithm is globally convergent, requires no regularization parameters, and achieves multigrid convergence rates. We present numerical results for the vector-valued All...

Double Bundle Method for Nonsmooth DC Optimization

The aim of this paper is to introduce a new proximal double bundle method for unconstrained nonsmooth DC optimization, where the objective function is presented as a difference of two convex (DC) functions. The novelty in our method is a new stopping procedure guaranteeing Clarke stationarity for solutions by utilizing only DC components of the objective function. This optimality condition is s...