Lyapunov functions are an extremely convenient device for proving that a dynamical system converges.

## Revenue Equivalence

For a number of differing auction settings, we consider the sale of a single item amongst fixed number of auction participants. It is interesting that under a certain game-theoretic construction all these auctions can be seen to be equivalent.

## Cross Entropy Method

In the *Cross Entropy Method*, we wish to estimate the likelihood

Here is a random variable whose distribution is known and belongs to a parametrized family of densities . Further is often a solution to an optimization problem.

## Online Convex Optimization

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.