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    Robust production management
    (2016) Guigues, Vincent Gérard Yannick
    The problem of production management can often be cast in the form of a linear program with uncertain parameters and risk constraints. Typically, such problems are treated in the framework of multi-stage Stochastic Programming. Recently, a Robust Counterpart (RC) approach has been proposed, in which the decisions are optimized for the worst realizations of problem parameters. However, an application of the RC technique often results in very conservative approximations of uncertain problems. To tackle this drawback, an Adjustable Robust Counterpart (ARC) approach has been proposed in (Ben-Tal et al. 2003). In ARC, some decision variables are allowed to depend on past values of uncertain parameters. A restricted version of ARC, introduced in (Ben-Tal etal. 2003), which can be efficiently solved, is referred to as Affinely Adjustable Robust Counterpart (AARC). In this paper, we consider an application of the ARC and AARC methodologies to the problem of yearly electricity production management in France. We provide tractable formulations for the AARC of quadratic and of some conic quadratic optimization problems, as well as for the ARC and AARC of the electricity production problem. We then give the quality of robust solutions obtained by using different uncertainty sets estimated using simulated and historical data. Our methodology is finally compared with other management methods.
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    Non-asymptotic confidence bounds for the optimal value of a stochastic program
    (EMAp - Escola de Matemática Aplicada, 2016) Guigues, Vincent Gérard Yannick; Juditsky, Anatoli; Nemirovski, Arkadi Semenovich
    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 time, such bounds are more reliable than 'standard' confidence bounds obtained through the asymptotic approach. We also discuss bounding the optimal value of MinMax Stochastic Optimization and stochastically constrained problems. We conclude with a small simulation study illustrating the numerical behavior of the proposed bounds.
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    Multistep stochastic mirror descent for risk-averse convex stochastic programs based on extended polyhedral risk measures
    (EMAp - Escola de Matemática Aplicada, 2016) Guigues, Vincent Gérard Yannick
    We consider risk-averse convex stochastic programs expressed in terms of extended polyhedral risk measures. We derive computable con dence 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 con dence intervals on both the optimal values and optimal solutions. Numerical simulations show that our con dence intervals are much less conservative and are quicker to compute than previously obtained con dence intervals for SMD and that the multistep Stochastic Mirror Descent algorithm can obtain a good approximate solution much quicker than its nonmultistep counterpart. Our con dence intervals are also more reliable than asymptotic con dence intervals when the sample size is not much larger than the problem size.
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    Joint dynamic probabilistic constraints with projected linear decision rules
    (EMAp - Escola de Matemática Aplicada, 2016) Guigues, Vincent Gérard Yannick; Henrion, René
    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 (in nite dimensional) problem and approximating problems working with projections from di erent subclasses of decision policies. Considering the subclass of linear decision rules and a generalized linear model for the underlying stochastic process with noises that are Gaussian or truncated Gaussian, we show that the value and gradient of the objective and constraint functions of the approximating problems can be computed analytically.
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    Convergence analysis of sampling-based decomposition methods for risk-averse multistage stochastic convex programs
    (EMAp - Escola de Matemática Aplicada, 2016) Guigues, Vincent Gérard Yannick
    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.