What are the conditions for formulating a Lagrangian, if the system's state is stochastic? Some notation: \( x \in X \subset \mathbb{R}^d\) is the control variable, \(u \in \mathbb{R}^n\) is the state vector, \( ( \Omega, S, P ) \) with \( \omega \in \Omega , 0 \leq P(A \in S) \leq 1\) is a probability space. We know that the state u obeys an equilibrium equation, \( G(u(x, \omega)) = 0\), and we wish to optimize a differentiable function of the state, \( F(u(x, \omega))\). Given any realization \( \omega \), \( x \mapsto u(x, \cdot)\) is differentiable, whereas \(F(u)\) is a simple function e.g. \( \|u\|^2 \). If the equilibrium equation is linear, we can write it as \(G(x,\omega) u = b\), where \(b\) is a known forcing vector. In general, if we formulate a Lagrangian functional \(L = F(x(u, \omega)) + \langle \lambda, G(u(x, \omega)) \rangle\), the multiplier vector \(\lambda\) will itself be a random variable (since the state function \(x \mapsto G(x, \omega)\) is random, there is a distinct multiplier for each realization, i.e. scenario, let's say we have \(N\) of them), and consequently, we will need to solve \(N\) adjoint problems, in addition to \(N\) "forward" solves, for a sample-average estimate of the subgradient \(\partial_x L\). I wonder if this reasoning is fruitful and whether it ties in with the more established linear stochastic programming theory. Some doubts: in this case, what is the definition of the inner product \( \langle \cdot, \cdot \rangle\)? How is the notion of adjoint operator related to that of non-anticipating constraints? Also, I would be happy if anyone could point me to related references. Thank you in advance
asked
ocramz |