---
title: Convex Optimization
description: Convex optimisation solves problems where local optima are global. You will learn convex sets and functions, duality, algorithms and how to model engineering and machine-learning problems as convex programs.
category: mathematics
subcategory: applied-mathematics
difficulty: beginner, intermediate, advanced
url: /subject/convex-optimization
---

# Convex Optimization

Convex optimisation solves problems where local optima are global. You will learn convex sets and functions, duality, algorithms and how to model engineering and machine-learning problems as convex programs.

## Available Resources

2 Books • 1 Courses • 1 Websites • 1 Papers

## Books

### 1. First-Order Methods in Optimization

**Author:** Amir Beck

Rigorous reference on first-order algorithms for convex problems: subgradient, proximal gradient, FISTA, mirror descent, smoothing, primal-dual methods, ADMM and conditional gradient, with full convergence proofs and a detailed treatment of proximal operators and conjugate functions.

**Difficulty:** Advanced | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/1611974984?tag=edmonddante07-20

**Tags:** convex-optimization, first-order-methods, proximal-gradient, fista, admm

### 2. Geometric Algorithms and Combinatorial Optimization

**Author:** Martin Grötschel, László Lovász, Alexander Schrijver

Research monograph showing how the ellipsoid method and lattice basis reduction yield polynomial-time algorithms for combinatorial optimization. Establishes the equivalence of separation and optimization over polyhedra, then applies it to matroids, submodular functions, perfect graphs and other structured problems.

**Difficulty:** Advanced | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/3540567402?tag=edmonddante07-20

**Tags:** combinatorial-optimization, ellipsoid-method, polyhedral-combinatorics, lattice-basis-reduction, linear-programming

## Websites

### 1. CVXPY Documentation and Tutorials

**Author:** Steven Diamond, Akshay Agrawal, Stephen Boyd et al.

Official documentation for CVXPY, an open-source Python library for modeling convex problems. It includes tutorials on disciplined convex programming, supported solvers, parameters, and worked examples in machine learning, finance, control and signal processing.

**Difficulty:** Beginner | **Language:** English | **Price:** Free

**Link:** https://www.cvxpy.org/

**Tags:** cvxpy, python, disciplined-convex-programming, convex-optimization, optimization-modeling

## Papers

### 1. Convex Optimization: Algorithms and Complexity

**Author:** Sébastien Bubeck

Monograph on the complexity theory of convex optimization: ellipsoid and cutting-plane methods, gradient descent in smooth and non-smooth settings, Nesterov acceleration with lower bounds, mirror descent, interior-point methods, and stochastic and randomized algorithms. Published in Foundations and Trends in Machine Learning.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://arxiv.org/abs/1405.4980

**Tags:** convex-optimization, complexity-theory, gradient-methods, mirror-descent, first-order-methods

## Courses

### 1. CMU 10-725: Convex Optimization (Ryan Tibshirani)

**Author:** Ryan Tibshirani

Carnegie Mellon graduate course site with slides, scribed notes, homework and linked lecture videos. It focuses on algorithms used in statistics and machine learning: gradient and subgradient methods, proximal methods, Newton's method, duality, ADMM, coordinate descent and stochastic methods.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://www.stat.cmu.edu/~ryantibs/convexopt/

**Tags:** convex-optimization, proximal-methods, duality, admm, machine-learning

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