Lagrangian formulation for nonlinear stochastic programming

 1 2 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 19 Apr '16, 17:48 ocramz 41●6 accept rate: 0%
Be the first one to answer this question!
 toggle preview community wiki

By Email:

Once you sign in you will be able to subscribe for any updates here

Markdown Basics

• *italic* or _italic_
• **bold** or __bold__
• image?![alt text](/path/img.jpg "Title")
• numbered list: 1. Foo 2. Bar
• to add a line break simply add two spaces to where you would like the new line to be.
• basic HTML tags are also supported

Tags:

Asked: 19 Apr '16, 17:48

Seen: 339 times

Last updated: 20 Apr '16, 05:12

Related questions

OR-Exchange! Your site for questions, answers, and announcements about operations research.