Convergence of some algorithms for convex minimization
Mathematical Programming: Series A and B - Special issue: Festschrift in Honor of Philip Wolfe part II: studies in nonlinear programming
Solving semidefinite quadratic problems within nonsmooth optimization algorithms
Computers and Operations Research
A bundle-Newton method for nonsmooth unconstrained minimization
Mathematical Programming: Series A and B
Computational Combinatorial Optimization, Optimal or Provably Near-Optimal Solutions [based on a Spring School]
Minimizing Nonconvex Nonsmooth Functions via Cutting Planes and Proximity Control
SIAM Journal on Optimization
A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization
SIAM Journal on Optimization
Globally convergent limited memory bundle method for large-scale nonsmooth optimization
Mathematical Programming: Series A and B
A Method of Centers with Approximate Subgradient Linearizations for Nonsmooth Convex Optimization
SIAM Journal on Optimization
Computing proximal points of nonconvex functions
Mathematical Programming: Series A and B - Nonlinear convex optimization and variational inequalities
Improving an upper bound on the size of k-regular induced subgraphs
Journal of Combinatorial Optimization
Aggregate codifferential method for nonsmooth DC optimization
Journal of Computational and Applied Mathematics
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Proximal bundle methods have been shown to be highly successful optimization methods for unconstrained convex problems with discontinuous first derivatives. This naturally leads to the question of whether proximal variants of bundle methods can be extended to a nonconvex setting. This work proposes an approach based on generating cutting-planes models, not of the objective function as most bundle methods do but of a local convexification of the objective function. The corresponding convexification parameter is calculated “on the fly” in such a way that the algorithm can inform the user as to what proximal parameters are sufficiently large that the objective function is likely to have well-defined proximal points. This novel approach, shown to be sound from both the objective function and subdifferential modelling perspectives, opens the way to create workable nonconvex algorithms based on nonconvex $\mathcal{VU}$ theory. Both theoretical convergence analysis and some encouraging preliminary numerical experience are provided.