Semidefinite Programming (SDP), which the author remarks is linear programming for the 21st century, has lately been one of the most exciting and active areas of research in the mathematical programming community. This tremendous excitement was spurred in part by the development of efficient interior point methods (IPMs) for the solution of SDPs, and important applications of the SDP especially in combinatorial optimization. I believe Etienne De Klerk gives an excellent introduction to these two topics, in this short, but concise monograph published by Kluwer Academic Publishers.

Topics covered include theory (duality, degeneracy, complementarity, properties of central path), algorithms (primal and primal dual affine scaling, path following, potential reduction algorithms), and finally applications (approximating the stable set and coloring number of a graph, the satisfiability problem, and quadratic programming).

Most of the material presented is based on the personal research of the author with other colloborators, and reflect his personal taste, and various insights on the subject. The monograph is probably the first textbook exclusively devoted to the SDP, and can be used in a graduate course on the subject. Personally, I enjoyed it immensely!.

David Hilbert was arguably one of the greatest mathematicians
ever!. He contributed to several branches of mathematics,
including functional analysis, mathematical physics,
calculus of variations, and algebraic number theory.
(Everyone knows what a Hilbert space is right!)

At the turn of the 20th century, Hilbert enumerated
23 unsolved problems of mathematics that he considered worthy
of further investigation. To this day, very few of these, including
the 10th problem, on the finite solvability of Diophantine
equations, have been resolved! (thanks to
Yuri Matiyasevich, Martin Davis and Julia Robinson!).
Besides, Hilbert was also a character (read the section
when Norbert Weiner of cybernetics fame, came to give
a talk at Gottingen, and .... :-)).

Incidentally the author Constance Reid is the sister of
Julia Robinson (of Hilbert's 10th problem fame!),
hence there can no one better to write about
Hilbert!. Besides Constance Reid is a well known chronicler
of mathematicians lives (this one is a tour de force and
her best!).

No one can can call himself/herself a mathematician without
having Reid's book on his/her bookshelf. Strongly
recommended!

David Hilbert was arguably one of the greatest mathematicians
ever!. He contributed to several branches of mathematics,
including functional analysis, mathematical physics,
calculus of variations, and algebraic number theory.
(Everyone knows what a Hilbert space is right!)

At the turn of the 20th century, Hilbert enumerated
23 unsolved problems of mathematics that he considered worthy
of further investigation. To this day, very few of these, including
the 10th problem, on the finite solvability of Diophantine
equations, have been resolved! (thanks to
Yuri Matiyasevich, Martin Davis and Julia Robinson!).
Besides, Hilbert was also a character (read the section
when Norbert Weiner of cybernetics fame, came to give
a talk at Gottingen, and .... :-)).

Incidentally the author Constance Reid is the sister of
Julia Robinson (of Hilbert's 10th problem fame!),
hence there can no one better to write about
Hilbert!. Besides Constance Reid is a well known chronicler
of mathematicians lives (this one is a tour de force and
her best!).

No one can can call himself/herself a mathematician without
having Reid's book on his/her bookshelf. Strongly
recommended!

At last, we find a book which develops a thorough understanding
of the most general theory of interior point methods for
convex optimization, and is easily accessible. As the author
himself remarks "Much of the literature on the general
theory of interior point methods is difficult to understand,
even for specialists. My hope is that this book will make
the most general theory accessible to a wide audience -
especially Ph.D. students, the next generation of optimizers".
The book covers basic interior point theory including
the theory of self concordant functionals. There is a chapter
on conic programming covering the relationship between interior point
methods and duality theory, and the development
of primal dual interior point algorithms for solving conic
optimization problems (Conic programming includes linear,
semidefinite and second order cone programming as special
cases!).

One can then "perhaps" take on Nesterov and Nemirovskii's
seminal treatise on Interior Point Polynomial Algorithms
in Convex Programming, one of the most widely cited references
in optimization, which I must confess is not exactly
an easy read.

To summarize, conic optimization and efficient interior point
methods to solve them are certainly one of the most exciting
areas in optimization recently, and Renegar's excellent,
intuitive and short book is a welcome addition to the
bookshelf of any serious optimizer!. Strongly recommended!

At last, we find a book which develops a thorough understanding
of the most general theory of interior point methods for
convex optimization, and is easily accessible. As the author
himself remarks "Much of the literature on the general
theory of interior point methods is difficult to understand,
even for specialists. My hope is that this book will make
the most general theory accessible to a wide audience -
especially Ph.D. students, the next generation of optimizers".
The book covers basic interior point theory including
the theory of self concordant functionals. There is a chapter
on conic programming covering the relationship between interior point
methods and duality theory, and the development
of primal dual interior point algorithms for solving conic
optimization problems (Conic programming includes linear,
semidefinite and second order cone programming as special
cases!).

One can then "perhaps" take on Nesterov and Nemirovskii's
seminal treatise on Interior Point Polynomial Algorithms
in Convex Programming, one of the most widely cited references
in optimization, which I must confess is not exactly
an easy read.

To summarize, conic optimization and efficient interior point
methods to solve them are certainly one of the most exciting
areas in optimization recently, and Renegar's excellent,
intuitive and short book is a welcome addition to the
bookshelf of any serious optimizer!. Strongly recommended!

I am a big fan of Robert Heinlein, although I find all his Hugo award winning novels especially "Stranger in a Strange Land" and the "The Moon is a Harsh Mistress" pretty hard to digest. However "The Fantasies of Robert Heinlein" collects 8 intriguing stories from one of the true masters of SF.

"All you Zombies", where a man in a bar reveals that he is his every relation (father, mother et al) through a series of time paradoxes and "And he built a crooked house" where an architect constructs a house in the shape of a tesseract (a four dimensional hypercube) in which he and the new occupants of the house find themselves trapped are simply masterpieces. Check out "They --" as well.

A must read for any Heinlein and serious SF fan.

Combinatorial Optimisation : Networks and Matroids by Eugene Lawler examines shortest paths, network flows, bipartite matching, non bipartite matching. More importantly there is an excellent introduction to matroid theory including matroids and the greedy algorithm, matroid intersections and matroid parity problems, some of these Lawler's own results.

However there is not much on NP completeness, since this book was published in 1976. For a more to date version of events in combinatorial optimisation one might want to look at Papadimitriou and Steglitz's book on combinatorial optimisation (quite old too, considering this was published in 1982), Ahuja, Magnanti and Orlin's book on Network algorithms, Hochbaum's book on approximation algorithms and Cook, Cunnigham,Pulleyblank and Schrijver's book on combinatorial optimisation (listed in the order they were published).

Lawler's book is extremely well written and I am delighted that this book is now published by Dover, and hence easily affordable.

This book is a great sequel to Garey and Johnson. The appendix of this book gives a list of all NP optimisation problems together with their current approximability (or inapproximability results) in a Garey Johnson fashion.

Developing approximation algorithms for NP hard problems is now a very active field in Mathematical Programming and Theoretical Computer Science. There have been a number of exciting developments like semidefinite programming , the Goemans Williamson algorithm for max cut et al.

On the other hand, from a theoretical computer science point of view, we now have a proof that many of these problems cannot have polynomial approximation algorithms unless P=NP.

This book provides an excellent introduction to both areas. A worthy supplement to Garey and Johnson, Papadimitriou's books on combinatorial optimisation and computational complexity, Hochbaum's book on approximation algorithms, Alon and Spencer's book on the probabilistic method and finally Motwani and Raghavan's book on randomised algorithms.

This book is a great sequel to Garey and Johnson. The appendix of this book gives a list of all NP optimisation problems together with their current approximability (or inapproximability results) in a Garey Johnson fashion.

Developing approximation algorithms for NP hard problems is now a very active field in Mathematical Programming and Theoretical Computer Science. There have been a number of exciting developments like semidefinite programming , the Goemans Williamson algorithm for max cut et al.

On the other hand, from a theoretical computer science point of view, we now have a proof that many of these problems cannot have polynomial approximation algorithms unless P=NP.

This book provides an excellent introduction to both areas. A worthy supplement to Garey and Johnson, Papadimitriou's books on combinatorial optimisation and computational complexity, Hochbaum's book on approximation algorithms, Alon and Spencer's book on the probabilistic method and finally Motwani and Raghavan's book on randomised algorithms.

Paul Erdos once remarked that you need not believe in God, but you certainly have to believe in the book in which God maintains the "perfect" mathematical proofs. Martin Aigner and Gunter Ziegler have certainly done a great job with this book, a fitting tribute to the great Erdos himself.

I had purchased a copy of the 1st edition of this book and was plesantly surprised that the authors had come up with a 2nd edition, with a few more "perfect" proofs.

My personal favorites are "The Shannon capacity of a graph". where the Lovasz theta number would eventually lead to semidefinite programming, Erdos' probabilistic method where probability makes counting sometimes easy, computing the number of trees in a graph, how many guards it takes to guard a museum, and the section on Turan's theorem.

This book deserves to be on the bookshelves of both amateur and professional mathematicians.