Website under construction. For now only the first two parts of the course are online, Preliminaries and Optimization problems: the others will follow soon.

Convex Optimization

Teaching material conceived and developed by Fabio Furini, associate professor at DIAG, Sapienza University of Rome.

The course lecture notes, online: convex sets, functions of several variables, convex functions, quadratic functions, convex and concave optimization problems, with definitions, theorems with their proofs, examples worked out step by step and exercises with solutions.

Start with the preliminaries The exercises Try an interactive graph Download the lecture notes (PDF)

How each chapter is organized

Each chapter is one of the course notes, with the same numbers and the same colors as the lecture notes: on the website and in the PDF you find everything in the same place.

Definition

The precise meaning of a new concept.

Theorem, proposition, observation

A result to remember, with its hypotheses.

Example

A computation worked out in full, one step at a time.

In green the key point to take away.

Proof — opens with a click

All proofs are included, but folded: first read the statement, then open the proof when you want to study it. The same holds for the solutions of the exercises.

The parts of the course

• 1 · Preliminaries

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  Convex combinations and convex sets, functions of several variables (gradient, Hessian, level sets), norms, convex and concave functions, quadratic functions and their convexity.

  1. Convex sets · 2. Functions of several variables · 3. Convex and concave functions · 4. Quadratic functions

  The chapters

• 2 · Optimization problems

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  Optimization problems and their optimal solutions, local and global optima, convex and concave problems, where every local optimum is global.

  5. Optimization problems · 6. Convex and concave optimization problems

  The chapters

Don't miss

• Interactive lab

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  Move the parameters with the sliders, drag the points and watch sets, functions and optima change; then put yourself to the test with the challenges.

  The graphs

• Exercises with solutions

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  Exercises for every chapter, each with a worked solution that opens with a click.

  The exercises

• Lecture notes and exercise books in PDF

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  Two volumes of lecture notes and two exercise books, with the same numbering as the website.

  Downloads

• Full index

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  All chapters with their sections, on one page.

  The index

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By the same author:

• Linear Programming — polyhedra, the simplex algorithm, duality and the dual simplex, with the interactive lab;
• Algorithms — number representation, pseudo-code, sorting and searching, analysis and recursive algorithms;
• Integer Linear Programming — LP relaxation, valid inequalities, cutting planes, branch-and-bound and branch-and-cut;
• Combinatorial Optimization — knapsack, bin packing, coloring, TSP, scheduling and lot sizing, solved in every way;
• Network Optimization — graphs, spanning trees, shortest paths, network flows, the network simplex and project management;
• MIP Modelling — integer linear programming models, from the formulation to the solution with solvers;
• Operations Research Lab — modelling and solving optimization problems with Python and solvers;
• Mathematical Analysis 1 — real numbers, functions, limits, derivatives and series, with interactive graphs;
• Linear Algebra — vectors, matrices, determinants, linear systems and eigenvalues, with the computation lab.

Teaching material by Fabio Furini — DIAG, Sapienza University of Rome.

Questo sito è disponibile anche in italiano.