Bundle methods in the XXIst century: A bird’s-eye view

Bundle methods are often the algorithms of choice for nonsmooth convex optimization, especially if accuracy in the solution and reliability are a concern. We review several algorithms based on the bundle methodology that have been developed recently and that, unlike their forerunner variants, have the ability to provide exact solutions even if most of the … Read more

An approximation scheme for a class of risk-averse stochastic equilibrium problems

We consider two models for stochastic equilibrium: one based on the variational equilibrium of a generalized Nash game, and the other on the mixed complementarity formulation. Each agent in the market solves a one-stage risk-averse optimization problem with both here-and-now (investment) variables and (production) wait-and-see variables. A shared constraint couples almost surely the wait-and-see decisions … Read more

Proximal bundle methods in depth: a unified analysis for inexact oracles

The last few years have seen the advent ofa new generation of bundle methods, capable to handle inexact oracles, polluted by “noise”. Proving convergence of a bundle method is never simple and coping with inexact oracles substantially increases the technicalities. Besides, several variants exist to deal with noise, each one needing an ad hoc proof … Read more

Constrained Bundle Methods for Upper Inexact Oracles with Application to Joint Chance Constrained Energy Problems

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 … Read more

Level Bundle Methods for oracles with on-demand accuracy

For nonsmooth convex optimization, we consider level bundle methods built using an oracle that computes values for the objective function and a subgradient at any given feasible point. For the problems of interest, the exact oracle information is computable, but difficult to obtain. In order to save computational effort the oracle can provide estimations with … Read more

A Class of Dantzig-Wolfe Type Decomposition Methods for Variational Inequality Problems

We consider a class of decomposition methods for variational inequalities, which is related to the classical Dantzig–Wolfe decomposition of linear programs. Our approach is rather general, in that it can be used with set-valued or nonmonotone operators, as well as various kinds of approximations in the subproblems of the functions and derivatives in the single-valued … Read more

Exploiting structure of autoregressive processes in risk-averse multistage stochastic linear programs

We consider a multivariate interstage dependent stochastic process whose components follow a generalized autoregressive model with time varying order. At a given time step, we give some recursive formulae linking future values of the process with past values and noises. We then consider multistage stochastic linear programs with uncertain polyhedral sets depending affinely on such … Read more

Robust management and pricing of LNG contracts with cancellation options

Liquefied Natural Gas contracts offer cancellation options that make their pricing difficult, especially if many gas storages need to be taken into account. We develop a valuation mechanism for such contracts from the buyer’s perspective, a large gas company whose main interest in these contracts is to provide a reliable supply of gas to its … Read more

The value of rolling horizon policies for risk-averse hydro-thermal planning

We consider the optimal management of a hydro-thermal power system in the mid and long terms. From the optimization point of view, this amounts to a large-scale multistage stochastic linear program, often solved by combining sampling with decomposition algorithms, like stochastic dual dynamic programming. Such methodologies, however, may entail prohibitive computational time, especially when applied … Read more

Inexact Bundle Methods for Two-Stage Stochastic Programming

Stochastic programming problems arise in many practical situations. In 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 in the subproblem … Read more