My recommendation would be

Convex Optimization – Boyd and Vandenberghe - Cambridge University Press

Which is downloadable for free if you cannot afford the book itself. It also has an associated MOOC (open course), see

https://www.stanford.edu/~boyd/cvxbook/

for more information. It is fairly complete, so you should find what you are interested in.

Conventionally, the definition of the convex optimization problem (we recall) requires that the objective function f to be minimized and the feasible set be convex. In the special case of linear programming (LP), the objective function is both concave and convex, and so LP can also consider the problem of maximizing an objective function without confusion. However, for most convex minimization problems, the objective function is not concave, and therefore a problem and then such problems are formulated in the standard form of convex optimization problems, that is, minimizing the convex objective function.

For nonlinear convex minimization, the associated maximization problem obtained by substituting the supremum operator for the infimum operator is not a problem of convex optimization, as conventionally defined. However, it is studied in the larger field of convex optimization as a problem of convex maximization.[9]

The convex maximization problem is especially important for studying the existence of maxima. Consider the restriction of a convex function to a compact convex set: Then, on that set, the function attains its constrained maximum only on the boundary.[10] Such results, called "maximum principles", are useful in the theory of harmonic functions, potential theory, and partial differential equations.

The problem of minimizing a quadratic multivariate polynomial on a cube is NP-hard.[11] In fact, in the quadratic minimization problem, if the matrix has only one negative eigenvalue, is NP-hard.[12]

My recommendation would be

Convex Optimization – Boyd and Vandenberghe - Cambridge University Press

Which is downloadable for free if you cannot afford the book itself. It also has an associated MOOC (open course), see

https://www.stanford.edu/~boyd/cvxbook/

for more information. It is fairly complete, so you should find what you are interested in.

And to continue from Sebastien, if you like Matlab there is also a toolbox:

http://cvxr.com/cvx/

from the same authors.

Thank you Fariborz, and every one!

I do not think there is one book which covers all of what you need in great depth. If focus lies on Lagrange, Slater, Primal/Dual and optimality conditions, then I would definitely recommend the book "Nonlinear Programming" by Bazaraa, Sherali and Shetty, published by Wiley and in its third edition from 2006.

"Optimization methods" Gabasov and Kirilova (1976). The book (1983) is difficult for the beginers!

I heard that there has been an online course given by Prof. Stephen Boyd on the same topic ?

For the basics of convex optimization, i recommend the book

A Unified Algebraic Approach to Control Design, [1998 Springer ISBN 0-7484-0592-5].

The first chapters and the appendices are fundamental to convex optimization. the other chapters apply these principles to linear control design.

I recommend the following books:

1- Introductory Lectures on Convex Optimization: A Basic Course

Y. Nesterov

2- A Course in Convexity

Alexander Barvinok

3- Convex analysis and nonlinear optimization: theory and examples

Jonathan M. Borwein, Adrian S. Lewis

4-Convex Optimization & Euclidean Distance Geometry

Jon Dattorro

5-Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications (MPS-SIAM Series on Optimization)

Aharon Ben-Tal, Arkadi Nemirovski

Thanks all !

A good book is: Bazara, M., & Shetty, C. (1979). Nonlinear programming:

Theory and algorithms. New York, NY: Wiley.

If you don't have a background in Linear Programming, I would recommend that you first read "Linear Programming and Network Flows", by Bazaraa, Jarvis and Sherali, before you embark on the more advanced topics of Nonlinear Programming. The first chapters, in particular, give a mild introduction to convexity analysis concepts.

Stanford course is also available through iTunes University.....

Jorge Nocedal & Stephan J. Wright, Numerical optimization, Springer

An interesting book is: aubin,J.P., Optima and Equilibria, An Introduction to Nonlinear Analysis, Springer (2003).

1. Rockafellar R.: Convex Analysis, Princeton University Press, New Jersey (1970)

2. Bonnans J.F., J.C. Gilbert, C. Lemaréchal, and C.A. Sagastizábal: Numerical Optimization. Theoretical and Practical Aspects, Springer, (2006)

3. Browien J., Lewis A.: Convex analysis and nonlinear optimization, Canada, (1999)

4. Sun W., Yuan Y.: Optimization theory and methods, Nonlinear programming, ISBN-10:0-387-24975-3 Springer Science+Business Media, LLC, (2006)

What about:

Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications

By Aharon Ben-Tal and Arkadi Nemirovski

The book provides convex optimization problems, with emphasis on conic quadratic and semidefinite programming.

Contents: Preface; Lecture 1: Linear Programming; Lecture 2: From Linear Programming to Conic Programming; Lecture 3: Conic Quadratic Programming; Lecture 4: Semidefinite Programming; Lecture 5: Computational Tractability of Convex Programs; Lecture 6: Interior Point Polynomial Time Methods for Linear Programming, Conic Quadratic Programming, and Semidefinite Programming; Solutions to Selected Exercises; Index.

2001 / xvi + 488 pages / Softcover / ISBN-10: 0-89871-491-5 | ISBN-13: 978-0-898714-91-3

For me Boyd was difficult to follow and after several failed attempts to learn optimization from Boyd, i found this amazing book

An introduction to optimization by Edwin P K Chong.

This book starts with very basics and then gradually goes into optimization. Also exercise problems are very much in the scope with what is covered in the text (not easy though) and hence they are encouraging to be solved.

Anh Nguyen Duc

I'd go for Nocedal and Wright - the newest edition, whenever it was published. It's not as technical as some of the ones mentioned already, and the style of writing is nice. It's rather important that the book provides discussions on everything concerning the crucial topic "optimality" and "optimality conditions", before the theorems come in - and their style is good.