Weak convexity and approximate subdifferentials

We explore and construct an enlarged subdifferential for weakly convex functions. The resulting object turns out to be continuous with respect to both the function argument and the enlargement parameter. We carefully analyze connections with other constructs in the literature and particularize to the weakly convex setting well-known variational principles. By resorting to the new … Read more

Minimizing the difference of convex and weakly convex functions via bundle method

We consider optimization problems with objective and constraint being the difference of convex and weakly convex functions. This framework covers a vast family of nonsmooth and nonconvex optimization problems, particularly those involving certain classes of composite and nonconvex value functions. We investigate several stationary conditions and extend the proximal bundle algorithm of [van Ackooij et … Read more

A bundle method for nonsmooth DC programming with application to chance-constrained problems

This work considers nonsmooth and nonconvex optimization problems whose objective and constraint functions are defined by difference-of-convex (DC) functions. We consider an infeasible bundle method based on the so-called improvement functions to compute critical points for problems of this class. Our algorithm neither employs penalization techniques nor solves subproblems with linearized constraints. The approach, which … Read more

Quantifying the value of flexibility: demand response versus storage

Intermittent sources of energy represent a challenge for electrical networks, particularly regarding demand satisfaction at peak times. Energy management tools such as load shaving or storage systems can be used to mitigate abrupt variations in the network.The value of different mechanisms to move energy through time is determined by a multi-objective programming approach, that aims … Read more

Distributionally robust optimization with multiple time scales: valuation of a thermal power plant

The valuation of a real option is preferably done with the inclusion of uncertainties in the model, since the value depends on future costs and revenues, which are not perfectly known today. The usual value of the option is defined as the maximal expected (discounted) profit one may achieve under optimal management of the operation. … Read more

Optimizing power generation in the presence of micro-grids

In this paper we consider energy management optimization problems in a future wherein an interaction with micro-grids has to be accounted for. We will model this interaction through a set of contracts between the generation companies owning centralized assets and the micro-grids. We will formulate a general stylized model that can, in principle, account for … Read more

On level regularization with normal solutions in decomposition methods for multistage stochastic programming problems

We consider well-known decomposition techniques for multistage stochastic programming and a new scheme based on normal solutions for stabilizing iterates during the solution process. The given algorithms combine ideas from finite perturbation of convex programs and level bundle methods to regularize the so-called forward step of these decomposition methods. Numerical experiments on a hydrothermal scheduling … Read more

An Adaptive Partition-based Level Decomposition for Solving Two-stage Stochastic Programs with Fixed Recourse

We present a computational study of several strategies to solve two-stage stochastic linear programs by integrating the adaptive partition-based approach with level decomposition. A partition-based formulation is a relaxation of the original stochastic program, obtained by aggregating variables and constraints according to a scenario partition. Partition refinements are guided by the optimal second-stage dual vectors … Read more

Probabilistic optimization via approximate p-efficient points and bundle methods

For problems when decisions are taken prior to observing the realization of underlying random events, probabilistic constraints are an important modelling tool if reliability is a concern. A key concept to numerically dealing with probabilistic constraints is that of p-efficient points. By adopting a dual point of view, we develop a solution framework that includes … Read more

Decomposition algorithm for large-scale two-stage unit-commitment

Everyday, electricity generation companies submit a generation schedule to the grid operator for the coming day; computing an optimal schedule is called the unit-commitment problem. Generation companies can also occasionally submit changes to the schedule, that can be seen as intra-daily incomplete recourse actions. In this paper, we propose a two-stage formulation of unit-commitment, wherein … Read more