Convex Optimization

Notes based on Convex Optimization by Stephen Boyd and Lieven Vandenberghe.

Convex optimization lies at the heart of modern machine learning, control, signal processing, and finance. Its power comes from a simple geometric fact: every local minimum of a convex problem is also global — and efficient algorithms exist to find it.

Chapters

• Chapter 1: Mathematical Background — Inner products, norms, matrix algebra, calculus preliminaries
• Chapter 2: Convex Sets — Affine sets, convex sets, cones, hyperplanes, separating hyperplane theorem
• Chapter 3: Convex Functions — Definitions, properties, first/second-order conditions, smooth and strongly convex functions
• Chapter 4: Convex Optimization Problems — Standard form, LP, QP, QCQP, SOCP, SDP, problem hierarchy
• Chapter 5: Duality — Lagrangian, dual function, weak/strong duality, KKT conditions, sensitivity analysis
• Chapter 11: Interior-Point Methods — Logarithmic barrier, barrier method, central path

Quick Reference

• Problem Forms & Transformations — Standard formulations hierarchy and reductions
• Convexity, Smoothness & Strong Convexity Equivalences — Characterizations and convergence rates