Convex Optimization: Theory, Algorithms, and Modeling

Everyone 'knows' linear problems are easy and nonlinear ones are hard — so why does the same solver crush a million-variable nonlinear problem and get hopelessly lost on one with two variables?

Solvers for LP, SOCP, and SDP compare things that have no natural order — vectors, even whole matrices — and the secret turns out to be a single geometric object hiding inside every one of these problems.

A one-line image-denoising energy defeats the standard Hessian test entirely — and the shortcut every expert uses instead hides a composition trap that fools almost everyone who meets it.

Compress an entire function into one number per slope — the best gap a line can open against it — and you've either thrown almost everything away or built a perfect secret twin. Which is it, and why does the answer secretly power every likelihood in computer vision?

Photoshop's seamless paste solves a million-variable optimization problem before your finger leaves the mouse — what hidden structure makes a problem that huge feel that easy, and why does the same trick fail the moment a few outliers sneak in?

Step beyond polyhedra into second-order cones and semidefinite matrices, unlocking a far richer language of constraints.

What actually happens at a constrained optimum? A geometric picture that makes dual problems feel intuitive.

Uncover the fundamental conditions that certify optimality — and see how they unlock closed-form dual solutions.

Examine condition numbers, gradient descent trajectories, and why ill-conditioned problems choke simple solvers.

See how modern solvers transform hard inequality constraints into smooth logarithmic barrier functions.

Gradient descent fails at kinks. Meet the proximal operator that handles L1 norms and non-smooth terms effortlessly.

Learn the operator-splitting trick behind modern imaging solvers, and build intuition for consensus optimization.

Some inverse problems have infinitely many solutions. Discover how adding penalty terms restores well-posedness.

Rotations and correspondences live on non-convex sets. See how semidefinite relaxations give provable bounds.

Take messy graphics and vision formulations and reformulate them into disciplined convex programs.