Getting started, the first goal is to formalize the mathematics of the problem. By this I mean writing down the the three approaches formally. I hope to manage at least the first one by today.

This is now my official research blog. I will write once or twice a day regarding my progress. So, lets kick of the first post.

I'm currently trying to figure out a mathematical framework for including estimate uncertainty in my resource allocation scheme. I aim to write a paper about how to handle this and to get started, I want to set up and compare a few different alternatives. First some preliminaries.

Assume I have a system of N components C_i which consume resources u_i and produce results y_i according to the linear model y_i = k_i * u_i. The constraints on u_i are so that sum(u) <= 1, u_i > 0. Assume I want to allocate resources so that the cost-function J = sum(f_i(y_i)) is maximized, where f_i is some convex function. I now want to expand the model to include the case where the parameters of the model are not completely known and include a cost in J so that it reflects the benefits of probing by allocating resources.

I now desire to try three approaches to this.

1. The values of k is in some interval k_min - k_max. the cost J should be based on the tightness of that intervall.

2. J is augmented with some cost based on the variance of the estimate of k

3. The estimate of k is increased over time unless there is information that refutes that.

More about this tomorrow.