We consider the setting of sequentially optimizing the average of a sequence of functions, so called *online convex optimization*.

## Gradient Descent

We consider one of the simplest iterative procedures for solving the (unconstrainted) optimization

## A Network Decomposition

We consider a decomposition of the following network utility optimization problem

SYS:

## Congestion Control

We argue, in a slightly informal manner, that queueing networks implicitly optimize a utility function subject to constraints on network capacity. We start with the simple example of a closed queueing network and, as we shall discuss, a motivating example is the Transmission Control Protocol which controls the number of packets in transfer on an Internet connection.

## Gale-Eisenberg Market

The Gale-Eisenberg is a nice example were the distributed decisions of buyers and sellers have an equilibrium which solves an optimization problem.

## Sanov’s Theorem

Sanov’s asks how *likely* is it that the empirical distribution some IIDRV’s is *far* from the distribution. And shows that the relative entropy determines the likelihood of being far.

## Entropy and Boltzmann’s Distribution

Entropy and Relative Entropy occur sufficiently often in these notes to justify a (somewhat) self-contained section. We cover the discrete case which is the most intuitive.