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We present a framework for analyzing convergence and local rates of convergence of a class of descent algorithms, assuming the objective function is weakly convex. The framework is general, in the sense that it combines the possibility of explicit iterations (based on the gradient or a subgradient at the current iterate), implicit iterations (using a subgradient at the next iteration, like in the proximal schemes), as well as iterations when the associated subgradient is specially constructed and does not correspond either to the current or the next point (this is the case of descent steps in bundle methods). Under the subdifferential-based error bound on the distance to critical points, linear rates of convergence are established. Our analysis applies, among other techniques, to prox-descent for decomposable functions, the proximal-gradient method for a sum of functions, redistributed bundle methods, and a class of algorithms that can be cast in the feasible descent framework for constrained optimization. [ABSTRACT FROM AUTHOR]

Joint chance constrained problems give rise to many algorithmic challenges. Even in the convex case, i.e., when an appropriate transformation of the probabilistic constraint is a convex function, its cutting-plane linearization is just an approximation, produced by an oracle providing subgradient and function values that can only be evaluated inexactly. As a result, the cutting-plane model may lie above the true constraint. For dealing with such upper inexact oracles, and still solving the problem up to certain precision, a special numerical algorithm must be put in place. We introduce a family of constrained bundle methods, based on the so-called improvement functions, that is shown to be convergent and encompasses many previous approaches as well as new algorithms. Depending on the oracle accuracy, we analyze to what extent the considered methods solve the joint chance constrained program. The approach is assessed on real-life energy problems, arising when dealing with stochastic hydroreservoir management. [ABSTRACT FROM AUTHOR]

We consider the problem of minimizing nonsmooth convex functions, defined piecewise by a finite number of functions each of which is either convex quadratic or twice continuously differentiable with positive definite Hessian on the set of interest. This is a particular case of functions with primal-dual gradient structure, a notion closely related to the so-called VU space decomposition: At a given point, nonsmoothness is locally restricted to the directions of the subspace V, while along the subspace U the behavior of the function is twice differentiable. Constructive identification of the two subspaces is important, because it opens the way to devising fast algorithms for nonsmooth optimization (by following iteratively the manifold of smoothness on which superlinear U-Newton steps can be computed). In this work we show that, for the class of functions in consideration, the information needed for this identification can be obtained from the output of a standard bundle method for computing proximal points, provided a minimizer satisfies the nondegeneracy and strong transversality conditions. [ABSTRACT FROM AUTHOR]