Non-asymptotic confidence bounds for the optimal value of a stochastic program

We discuss a general approach to building non-asymptotic confidence bounds for stochastic optimization problems. Our principal contribution is the observation that a Sample Average Approximation of a problem supplies upper and lower bounds for the optimal value of the problem which are essentially better than the quality of the corresponding optimal solutions. At the same … Read more

Joint dynamic probabilistic constraints with projected linear decision rules

We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto hard constraints of wait-and-see type. We establish the relation between the original (infinite dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of … Read more

Multistep stochastic mirror descent for risk-averse convex stochastic programs based on extended polyhedral risk measures

We consider risk-averse convex stochastic programs expressed in terms of extended polyhedral risk measures. We derive computable confidence intervals on the optimal value of such stochastic programs using the Robust Stochastic Approximation and the Stochastic Mirror Descent (SMD) algorithms. When the objective functions are uniformly convex, we also propose a multistep extension of the Stochastic … Read more

Convergence analysis of sampling-based decomposition methods for risk-averse multistage stochastic convex programs

We consider a class of sampling-based decomposition methods to solve risk-averse multistage stochastic convex programs. We prove a formula for the computation of the cuts necessary to build the outer linearizations of the recourse functions. This formula can be used to obtain an efficient implementation of Stochastic Dual Dynamic Programming applied to convex nonlinear problems. … Read more

Hypotheses testing on the optimal values of several risk-neutral or risk-averse convex stochastic programs and application to hypotheses testing on several risk measure values

Given an arbitrary number of risk-averse or risk-neutral convex stochastic programs, we study hypotheses testing problems aiming at comparing the optimal values of these stochastic programs on the basis of samples of the underlying random vectors. We propose non-asymptotic tests based on confidence intervals on the optimal values of the stochastic programs obtained using the … Read more

SDDP for multistage stochastic linear programs based on spectral risk measures

We consider risk-averse formulations of multistage stochastic linear programs. For these formulations, based on convex combinations of spectral risk measures, risk-averse dynamic programming equations can be written. As a result, the Stochastic Dual Dynamic Programming (SDDP) algorithm can be used to obtain approximations of the corresponding risk-averse recourse functions. This allows us to define a … 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

SDDP for some interstage dependent risk averse problems and application to hydro-thermal planning

We consider interstage dependent stochastic linear programs where both the random right-hand side and the model of the underlying stochastic process have a special structure. Namely, for stage $t$, the right-hand side of the equality constraints (resp. the inequality constraints) is an affine function (resp. a given function $b_t$) of the process value for this … Read more

A stabilized model and an efficient solution method for the yearly optimal power management

We propose a stabilized model for the electricity generation management problem on a yearly scale. We also introduce an original and efficient solution method in a particular case. Our model is compared to other management methods and offers the best average cost while preserving a reasonable standard deviation of the cost over a set of … Read more