Search

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]

Stochastic programming problems arise in many practical situations, hi general, the deterministic equivalents of these problems can be very large and may not be solvable directly by general-purpose optimization approaches. For the particular case of two-stage stochastic programs, we consider decomposition approaches akin to a regularized L-shaped method that can handle inexactness ill the subproblem solution. From a nonsmooth optimization perspective, these variants amount to applying a proximal bundle method to an oracle that gives inaccurate values for the objective function and a subgradient. Rather than forcing early termination of the subproblem optimization to define inexact oracles, we select a small subset of scenarios for which the subproblem solution is exact, and we replace the information for the remaining scenarios by a fast procedure that does not involve solving an optimization problem. The inaccurate oracle information creates inexact cuts in the master program, which are well handled by the recently introduced inexact bundle methods. The proposed approaches are validated by encouraging numerical results on several two-stage stochastic linear programs found in the literature. [ABSTRACT FROM AUTHOR]