Vincent Guigues

Sampling-Based Decomposition Methods for Multistage Stochastic Programs Based on Extended Polyhedral Risk Measures

We define a risk-averse nonanticipative feasible policy for multistage stochastic programs and propose a methodology to implement it. The approach is based on dynamic programming equations written for a risk-averse formulation of the problem. This formulation relies on a new class of multiperiod risk functionals called extended polyhedral risk measures. Dual representations of such risk functionals are given and used to derive conditions of coherence. In the one-period case, conditions for convexity and consistency with second order stochastic dominance are also provided. The risk-averse dynamic programming equations are specialized considering convex combinations of one-period extended polyhedral risk measures such as spectral risk measures. To implement the proposed policy, the approximation of the risk-averse recourse functions for stochastic linear programs is discussed. In this context, we detail a stochastic dual dynamic programming algorithm which converges to the optimal value of the risk-averse p...

Risk-averse feasible policies for large-scale multistage stochastic linear programs

We consider risk-averse formulations of stochastic linear programs having a structure that is common in real-life applications. Specifically, the optimization problem corresponds to controlling over a certain horizon a system whose dynamics is given by a transition equation depending affinely on an interstage dependent stochastic process. We put in place a rolling-horizon time consistent policy. For each time step, a risk-averse problem with constraints that are deterministic for the current time step and uncertain for future times is solved. To each uncertain constraint corresponds both a chance and a Conditional Value-at-Risk constraint. We show that the resulting risk-averse problems are numerically tractable, being at worst conic quadratic programs. For the particular case in which uncertainty appears only on the right-hand side of the constraints, such risk-averse problems are linear programs. We show how to write dynamic programming equations for these problems and define robust recourse functions that can be approximated recursively by cutting planes. The methodology is assessed and favourably compared with Stochastic Dual Dynamic Programming on a real size water-resource planning problem.

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 risk-averse nonanticipative feasible policy for the stochastic linear program. Formulas for the cuts that approximate the recourse functions are given. In particular, we show that some cut coefficients have analytic formulas.

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. We prove the almost sure convergence of these decomposition methods when the relatively complete recourse assumption holds. We also prove the almost sure convergence of these algorithms when applied to risk-averse multistage stochastic linear programs that do not satisfy the relatively complete recourse assumption. The analysis is first done assuming the underlying stochastic process is interstage independent and discrete, with a finite set of possible realizations at each stage. We then indicate two ways of extending the methods and convergence analysis to the case when the process is interstage dependent.

Robust Management and Pricing of Liquefied Natural Gas Contracts with Cancelation Options

Liquefied Natural Gas contracts offer cancelation options that make their pricing difficult, especially if many gas storages need to be taken into account. We develop a valuation mechanism from the buyer’s perspective, a large gas company whose main interest in these contracts is to provide to clients a reliable supply of gas. The approach combines valuation with hedging, taking into account that price-risk is driven by international markets, while volume-risk depends on local weather and is stage-wise dependent. The methodology is based on setting risk-averse stochastic mixed 0-1 programs, for different contract configurations. These difficult problems are solved with light computational effort, thanks to a robust rolling-horizon approach. The resulting pricing mechanism not only shows how a specific set of contracts will impact the company business, but also provides the manager with alternative contract configurations to counter-propose to the contract seller.

Exploiting the structure of autoregressive processes in chance-constrained multistage stochastic linear programs

Abstract We consider an interstage dependent stochastic process whose components follow an autoregressive model with time varying order. At a given time, we give some recursive formulaexa0linking future values of the process with past values and noises. We then consider multistage stochastic linear programs with uncertain sets depending affinely on such processes. At each stage, dealing with uncertainty using probabilistic constraints, the recursive relations can be used to obtain explicit expressions for the feasible set.

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 Mirror Descent algorithm and obtain confidence intervals on both the optimal values and optimal solutions. Numerical simulations show that our confidence intervals are much less conservative and are quicker to compute than previously obtained confidence intervals for SMD and that the multistep Stochastic Mirror Descent algorithm can obtain a good approximate solution much quicker than its nonmultistep counterpart.

A Central Limit Theorem and Hypotheses Testing for Risk-averse Stochastic Programs

We study statistical properties of the optimal value and optimal solutions of the sample average approximation of risk-averse stochastic problems. Central limit theorem-type results are derived for the optimal value when the stochastic program is expressed in terms of a law invariant coherent risk measure having a discrete Kusuoka representation. The obtained results are applied to hypotheses testing problems aiming at comparing the optimal values of several risk-averse convex stochastic programs on the basis of samples of the underlying random vectors. We also consider nonasymptotic tests based on confidence intervals on the optimal values of the stochastic programs obtained using the stochastic mirror descent algorithm. Numerical simulations show how to use our developments to choose among different distributions and on the considered class of risk-averse stochastic programs the asymptotic tests show better results.

Nonparametric multivariate breakpoint detection for the means, variances, and covariances of a discrete time stochastic process

We introduce a nonparametric breakpoint detection method for the means and covariances of a multivariate discrete time stochastic process. Breakpoints are defined as left or right endpoints of maximal intervals of local time homogeneity for the means and covariances. The breakpoint detection method is an adaptive algorithm that estimates the last maximal interval of homogeneity. Applied recursively, it allows us to find an arbitrary number of breakpoints. We then study a second breakpoint detection algorithm that makes use of a sliding window. The quality of both methods is analysed. For the adaptive algorithm, we provide the quality of the estimation of the one-step-ahead means and covariance matrix as well as upper bounds on the type I and type II errors when applying the procedure to a change-point model. Regarding the second method, the probability of correctly detecting the breakpoint of a change-point model is bounded from below. Numerical simulations assess the performance of both methods using simulated data.