# Category Archives: Optimization

## ORF523: Mirror Prox

Today I’m going to talk about a variant of Mirror Descent invented by Nemirovski in 2004 (see this paper). This new algorithm is called Mirror Prox and it is designed to minimize a smooth convex function over a compact convex … Continue reading

## ORF523: Mirror Descent, part II/II

We start with some of the standard setups for the Mirror Descent algorithm we described in the previous post. Standard setups for Mirror Descent ‘Ball’ setup. The simplest version of Mirror Descent is obtained by taking on The function is … Continue reading

## ORF523: Mirror Descent, part I/II

In this post we are interested in minimizing a convex function over a compact convex set under the assumption that is -Lipschitz on with respect to some arbitrary norm (precisely we assume that ). As we have seen in this … Continue reading

## ORF523: ISTA and FISTA

As we said in the previous lecture it seems stupid to consider that we are in a black-box situation when in fact we know entirely the function to be optimized. Consider for instance the LASSO objective where one wants to minimize over . By … Continue reading

## ORF523: Conditional Gradient Descent and Structured Sparsity

In the following table we summarize our findings of previous lectures in terms of oracle convergence rates (we denote ). Note that in the last two lines the upper bounds and lower bounds are not matching. In both cases one can … Continue reading

## Guest post by Amir Ali Ahmadi: Sum of Squares (SOS) Techniques: An Introduction, Part II/II

The reader is encouraged to read Part I of this series before reading this post. For ease of reference, we recall the polynomial optimization problem from our previous post: (1) This is the task of minimizing a multivariate polynomial … Continue reading

## ORF523: Strong convexity

Today we will talk about another property of convex functions that can significantly speed-up the convergence of first-order methods: strong convexity. We say that is -strongly convex if it satisfies (1) Of course this definition does not require differentiability … Continue reading

## Guest post by Amir Ali Ahmadi: Sum of Squares (SOS) Techniques: An Introduction, Part I/II

Sum of squares optimization is an active area of research at the interface of algorithmic algebra and convex optimization. Over the last decade, it has made significant impact on both discrete and continuous optimization, as well as several other disciplines, … Continue reading

## ORF523: Nesterov’s Accelerated Gradient Descent

In this lecture we consider the same setting than in the previous post (that is we want to minimize a smooth convex function over ). Previously we saw that the plain Gradient Descent algorithm has a rate of convergence of … Continue reading

## ORF523: Oracle complexity of smooth convex functions

In this lecture we consider the unconstrained minimization of a function that satisfy the following requirements: (i) admits a minimizer on . (ii) is continuously differentiable and convex on . (iii) is smooth in the sense that the gradient mapping … Continue reading