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Data-driven Piecewise Affine Decision Rules for Stochastic Programming with Covariate Information
Focusing on stochastic programming (SP) with covariate information, this
paper proposes an empirical risk minimization (ERM) method embedded within a
nonconvex piecewise affine decision rule (PADR), which aims to learn the direct
mapping from features to optimal decisions. We establish the nonasymptotic
consistency result of our PADR-based ERM model for unconstrained problems and
asymptotic consistency result for constrained ones. To solve the nonconvex and
nondifferentiable ERM problem, we develop an enhanced stochastic
majorization-minimization algorithm and establish the asymptotic convergence to
(composite strong) directional stationarity along with complexity analysis. We
show that the proposed PADR-based ERM method applies to a broad class of
nonconvex SP problems with theoretical consistency guarantees and computational
tractability. Our numerical study demonstrates the superior performance of
PADR-based ERM methods compared to state-of-the-art approaches under various
settings, with significantly lower costs, less computation time, and robustness
to feature dimensions and nonlinearity of the underlying dependency
Price decomposition in large-scale stochastic optimal control
We are interested in optimally driving a dynamical system that can be
influenced by exogenous noises. This is generally called a Stochastic Optimal
Control (SOC) problem and the Dynamic Programming (DP) principle is the natural
way of solving it. Unfortunately, DP faces the so-called curse of
dimensionality: the complexity of solving DP equations grows exponentially with
the dimension of the information variable that is sufficient to take optimal
decisions (the state variable). For a large class of SOC problems, which
includes important practical problems, we propose an original way of obtaining
strategies to drive the system. The algorithm we introduce is based on
Lagrangian relaxation, of which the application to decomposition is well-known
in the deterministic framework. However, its application to such closed-loop
problems is not straightforward and an additional statistical approximation
concerning the dual process is needed. We give a convergence proof, that
derives directly from classical results concerning duality in optimization, and
enlghten the error made by our approximation. Numerical results are also
provided, on a large-scale SOC problem. This idea extends the original DADP
algorithm that was presented by Barty, Carpentier and Girardeau (2010)
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