Life is full of best. The cheapest route, the strongest bridge for the least steel, the portfolio with the most reward for the least risk, the neural network that fits the data best. Optimization is the mathematics of best — of finding, among all the choices allowed, the one that makes some quantity as large or as small as it can possibly be.

It is quietly one of the most powerful tools humans have. Airlines schedule crews with it, factories plan production with it, and nearly every piece of machine learning is, at its heart, an optimization problem in disguise — a giant search for the parameters that make the model least wrong. This course teaches you both the geometry of what "best" looks like and the algorithms that actually go and find it.

The big idea: shape decides everything

One theme runs through the whole subject. The shape of the problem — the landscape of the objective and the fence of the constraints — decides whether "best" is easy or essentially hopeless. When the landscape is convex (a single bowl with no false valleys), any downhill step leads to the one true minimum, and we can find it fast and be sure. When it is not, we hunt with cleverness and no guarantees. Understanding that divide — convex versus not — is the key that unlocks everything else.

The shape of the journey

This course moves in three stages, from what to how.

• Stage A — The landscape. What optimization is, and the beautiful special case — convexity — that makes it tractable.
• Stage B — Structured problems. Linear programs, the duality that shadows every problem, and the conditions that pin down an optimum.
• Stage C — Algorithms that search. How machines actually descend a landscape, from a single gradient step to curvature-aware leaps, and out into the harder wild.

Stage A — The landscape

1. What Is Optimization?
2. Convex Sets and Functions

Stage B — Structured problems and optimality

1. Linear Programming and the Simplex Method
2. Duality
3. The KKT Conditions

Stage C — Algorithms that search

1. Gradient Descent
2. Newton and Quasi-Newton Methods
3. Constrained and Integer Programming

Let's get started

We begin with the question itself: what does it even mean to optimize? Once you can name the objective, the variables and the constraints, half of any problem is already solved.

Let's get started → What Is Optimization?