Minimizing Nonconvex Nonsmooth Functions via Cutting Planes and Proximity Control

Abstract

We describe an extension of the classical cutting plane algorithm to tackle the unconstrained minimization of a nonconvex, not necessarily differentiable function of several variables. The method is based on the construction of both a lower and an upper polyhedral approximation to the objective function and it is related to the use of the concept of proximal trajectory. Convergence to a stationary point is proved for locally Lipschitz functions.

Piecewise linear approximations in nonconvex nonsmooth optimization

“…We compare the results provided by our code NCNS with those available in the literature for the bundle method NCVX [7] and the variable metric algorithms VN [22] and VMNC [29]. The performance of our algorithm seems comparable with those of the considered methods.…”

“…The bundle approach, primarily devised for dealing with convex minimization, has been specialized in [7,8] to the nonconvex case by splitting the bundle into two subsets related to the points that exhibit, respectively, some kind of "convex" or "nonconvex" behavior.…”

Piecewise-quadratic Approximations in Convex Numerical Optimization

“…Of particular interest in this sense is the fact that aggregation allows restricting the number of quadratic pieces to any fixed value (as low as two), which may ease concerns about dealing with many dense quadratic constraints in the master problem. We also believe that the use of quadratic models could be usefully extended to bundle methods designed to jointly deal with nonconvexity and nonsmoothness [34,12,14,15].…”

A Trust Region Spectral Bundle Method for Nonconvex Eigenvalue Optimization

“…The present trust region method should be compared to the approach of Fuduli, Gaudioso, and Giallombardo [20,21] for general nonsmooth and nonconvex locally Lipschitz functions, where the authors design a trust region with the help of the first order affine approximations a(y) = g (y − y k+1 ) + f (y k+1 ), g ∈ ∂f (y k+1 ) of the objective f at the trial points y k+1 . As these affine models are not support functions to the objective, the authors classify them according to whether a(x) > f (x) or a(x) ≤ f (x), using this information to devise a trust region around the current x.…”