convex-optimization
5 free lessons tagged convex-optimization across Math. Each one is a short sequence of focused steps with narration and a five-question quiz at the end — take them in any order, no signup required.
Newton's method and the interior point revolution
Second derivatives buy something gradients cannot: a step shaped by curvature, immune to conditioning, converging quadratically. This lesson builds Newton's method, then layers it on a log barrier to get interior point methods, the machinery that made large constrained problems solvable with a certificate rather than a hope.
Gradient descent: choosing the step and knowing the rate
Gradient descent is three lines of code and a hundred years of theory. This lesson derives why a safe step size is one over the smoothness constant, why the condition number governs everything, and why acceleration reaching order one over k squared is provably the best any first-order method can do.
Convexity: the property that decides what is solvable
Convexity is what separates optimization problems you can solve with a guarantee from ones you can only hope about. This lesson defines convex sets and functions, proves why every local minimum is global, and gives you the operations that let you recognise convexity without touching a Hessian.
Applications, Geodesic Convexity, and Why Non-Convex Can Still Be Solvable
Where this machinery earns its place: low-rank matrix completion, synchronization, and PCA. Plus the two ideas that explain why non-convex manifold problems are often solved to global optimality anyway.
Lagrangian Duality: From Primal to Dual
Every constrained optimization problem has a twin. Learn how to build the Lagrangian, derive the dual problem, and use weak duality, strong duality, and the KKT conditions to certify optima — with worked examples from linear programming and SVMs.