Optimization becomes dramatically more predictable when convexity is present. Convex problems admit a precise notion of “no hidden traps”: any locally optimal point is globally optimal, and the geometry of feasible sets allows constraints to be handled by algebraic certificates rather than guesswork. Convex duality is the formal mechanism that turns these geometric facts into […]

Gradient-based methods are the default workhorses of continuous optimization because they are simple, scalable, and broadly applicable. Their behavior, however, depends decisively on one choice: how far to move each step. Step-size selection is not an implementation detail; it is the mechanism that turns a descent direction into a convergent algorithm. The core theory of […]

Many modern optimization problems have the form “smooth loss plus nonsmooth structure.” The loss measures fit to data or agreement with constraints; the nonsmooth term enforces sparsity, robustness, or other desired behavior. These problems are often convex and highly structured, but that structure is invisible to plain gradient descent because nonsmooth terms break differentiability. Proximal […]

Optimization is full of techniques, but proofs in optimization are mostly built from a small set of reusable ideas. If you can recognize which idea is supposed to fire in a given problem, you can read papers faster, debug your own arguments, and avoid chasing the wrong kind of estimate. Convexity is the best entry […]

It is easy to learn optimization by absorbing a list of methods and examples that somebody else chose. It is harder, and far more valuable, \to learn how to manufacture examples on demand: examples that isolate a single phenomenon, expose a hidden hypothesis, or stress-test a theorem at its boundary. This article is a recipe […]

Optimization is a subject where small misunderstandings propagate into large errors. A single misapplied condition can turn a true theorem into a false claim, or can make an algorithm look correct while quietly solving the wrong problem. This article collects common mistakes that appear in coursework, research writing, and implementation. The emphasis is not on […]