Convex Optimization

A comprehensive graduate-level course and textbook covering the theory, applications, and numerical algorithms of convex optimization. It emphasizes recognizing and formulating convex problems in engineering and sciences.

Master the mathematical foundation and practical algorithms of convex optimization for engineering and data science.

Author: Stephen Boyd, Lieven Vandenberghe

Acknowledgments: Supported in part by NSF, and through contributions from students and colleagues at Stanford and UCLA, including special mentions of Arkadi Nemirovski and Kishan Baheti.

1. Define the components of a mathematical optimization problem, including the objective function, constraints, and variables.
2. Distinguish between least-squares, linear programming, and convex optimization problems based on their mathematical properties.
3. Compare local and global optimization strategies and evaluate the computational complexity associated with each.
4. Define and distinguish between affine sets, convex sets, and cones using formal combination notation.
5. Identify and represent standard convex sets including Euclidean balls, ellipsoids, polyhedra, and the positive semidefinite cone.
6. Apply operations that preserve convexity, such as intersection, affine transformations, and perspective functions, to verify set properties.
7. Identify and apply operations that preserve convexity, including the pointwise supremum of affine functions and vector composition rules.
8. Derive the Lagrange conjugate of various functions and apply Young's inequality.
9. Characterize quasiconvexity using sublevel sets and first/second-order differentiable conditions.
10. Formulate and Transform: Convert raw optimization problems into standard convex forms using slack variables and constraint elimination.