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 FS 16FS 15
 MAT932 Convex Optimization Michel Baes FS 16 Do 15.00-17.00 Y35F32
 Exercises 1Exercises 2Exercises 3Exercises 4Exercises 5Exercises 6Exercises 7Exercises 8Exercises 9Exercises 10Exercises 11Exercise 12Exercise 13 objective.mNesterov matlab

### Overview

Convexity plays a central role in the design and analysis of modern and highly successful algorithms for solving real-world optimization problems. The lecture (in English) on convex optimization will treat in a balanced manner theory (convex analysis, optimality conditions) and algorithms for convex optimization.
Starting with basic results on convex sets and functions, with a particular emphasis on semidefinite sets, we develop a self-contained duality theory for convex optimization problems. As instrumental tools in the theory of Lagrange multipliers and optimality conditions for convex optimization problems, we introduce the notions of subdifferential and conjugate functions. The course will be illustrated by many applications in accordance with the interests of the audience.
On the algorithmic part, we cover several classes of optimization methods, from the simple subgradient method to mirror-descent schemes and Interior-Point Methods. Several exercise sessions will be devoted to the familiarization of software to solve efficiently some semidefinite optimization problems.

The lecture will follow mostly albeit loosely the textbook by S. Boyd and L. Vandenberghe, Convex Optimization, made available on the net (download)

### Lecture-by-lecture references

These slides are designed to be self-contained; I consider them as an adequate substitute to a reference book.
Many thanks to all the students who spotted typos and inaccuracies in earlier versions of those slides. Their questions and comments also helped a lot.
You can find below a non-exhaustive list of references that inspired me, and that can be used as further reading material. Do not hesitate to contact me if you need further references in optimization, more related to your field of primary interest.

Lecture 1

Sparse signal recovery

G. Pope, C. Studer, and M. Baes, "Coherence-based recovery guarantees for generalized basis-pursuit de-quantizing", Proceedings of the 37th IEEE International Conference on Acoustics, Speech, and Signal Processing, Kyoto.

Portfolio optimization

S. Boyd, L. Vandenberghe, "Convex Optimization", Section 4.4.1.

Positron Emission Tomography:

A. Ben-Tal, T. Margalit, and A. Nemirovski, "The Ordered Subsets Mirror Descent Optimization Method with Applications to Tomography", SIAM Journal on Optimization, 12(2001), pp. 79-108.

Lecture 2

Semidefinite Optimization

L. Vandenberghe and S. Boyd, "Semidefinite programming", SIAM Review, 38(1996), n. 1, pp. 49-95.

Separation Theorems

Laurent Schwartz, "Analyse 1: Théorie des ensembles et topologie", Hermann, 1997.

Lectures 3, 4

J. Tind and L. Wolsey, "An elementary survey of general duality theory in Mathematical Programming", Mathematical Programming 21(1981), pp. 241-261.

A. Ben-Tal, A. Nemirovski, "Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications", MPS/SIAM Series on Optimization, n. 2, SIAM, 2001. Lecture 2.

Support vector machines

S. Boyd, L. Vandenberghe, "Convex Optimization", Section 8.6.

Lectures 5, 6

R. Rockafellar, "Convex Analysis", Princeton Mathematics Series, n. 28, Princeton University Press, 1970.

Y. Nesterov, "Introductory lectures on convex optimization: a basic course", Applied Optimization Series, n. 87, Kluwer Academic Publishers, 2003. Section 3.1.

Lecture 7

Introduction and applications: S. Boyd, L. Vandenberghe, "Convex Optimization", Chapter 1 and Section 4.4.2.

CQr and SDr: A. Ben-Tal, A. Nemirovski, "Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications", MPS/SIAM Series on Optimization, n. 2, SIAM, 2001. Lectures 3 and 4.

### References

A. Barvinok, A Course in Convexity. American Mathematical Society, 2003.
This well-written book covers convex analysis, with a particular emphasis on the interactions between convex analysis and combinatorial optimization problems.

A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization - Analysis, Algorithms, and Engineering Applications, MPS-SIAM Series on Optimization, MPS-SIAM.
This authoritative work displays a great variety of applications of convex optimization. It constitutes an ideal complement to the book of Boyd and Vandenberghe. Available upon request.

D. P. Bertsekas, A. Nedic and A. E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003.

D. Bertsimas and J. N. Tsitsiklis, Introduction to Linear Optimization, Athena Scientific, 1997.

S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
This thick book serves as one of the main references for the course. The authors deal with a number of applications of convex optimization in an impressive variety of fields.

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. SIAM, 1994.

E. de Klerk, Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications, Book Series: APPLIED OPTIMIZATION, Vol. 65. Kluwer Academic Publishers.

Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Book Series: APPLIED OPTIMIZATION, Vol. 87. Kluwer Academic Publishers.
This important book emerged from the lecture notes of Pr. Yurii Nesterov. It focuses on the study of algorithms for convex optimization, and, among others, interior-point methods. Available upon request..

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985.
This well-known book is an excellent reference on linear algebra. Its sequel "Topics in Matrix Analysis" is also a classic. Available upon request.

J. Renegar, A Mathematical View of Interior-Point Methods in Convex Optimization, MPS-SIAM Series on Optimization.
This short book has been developed from a graduate course of Jim Renegar. It contains an original analysis of interior-point methods, which complements beautifully the book of Nesterov. Available upon request.

R.T. Rockafellar, Convex Analysis, Princeton University Press, 1997.
This is a classical and rather complete reference on convexity. Available upon request.

H. Wolkowicz, R. Saigal and L. Vandenberghe, Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, Kluwer Academic Publishers.
This book is a collection of well-written reviews on a variety of aspects of semidefinite programming, including a discussion on its numerical behavior and the development of a robustness analysis.

A. Nemirovski and D. Yudin, Problem Complexity and Method Efficiency in Optimization, John Wiley, 1983.
This dense work focuses on the question "How efficient can optimization algorithms be?". Available upon request..

Y. Nesterov and A. Nemirovski, Interior Point Polynomial Algorithms in Convex Programming, Studies in Applied Mathematics Vol. 13, SIAM, 1993.
This authoritative but hard book was the first to introduce the concept of self-concordance. It would be difficult to overestimate its importance and its influence on Convex Optimization during the last 20 years.