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A Mathematical View of Interior-Point Methods in Convex Optimization (MPS-SIAM Series on Optimization)
by James Renegar

A Mathematical View of Interior-Point Methods in Convex Optimization (MPS-SIAM Series on Optimization)
By James Renegar

Publisher: Society for Industrial Mathematics
Number Of Pages: 118
Publication Date: 1987-01-01
ISBN-10 / ASIN: 0898715024
ISBN-13 / EAN: 9780898715026
Binding: Paperback

This compact book, through the simplifying perspective it presents, will take a reader who knows little of interior-point methods to within sight of the research frontier, developing key ideas that were over a decade in the making by numerous interior-point method researchers. It aims at developing a thorough understanding of the most general theory for interior-point methods, a class of algorithms for convex optimization problems. The study of these algorithms has dominated the continuous optimization literature for nearly 15 years. In that time, the theory has matured tremendously, but much of the literature is difficult to understand, even for specialists. By focusing only on essential elements of the theory and emphasizing the underlying geometry, A Mathematical View of Interior-Point Methods in Convex Optimization makes the theory accessible to a wide audience, allowing them to quickly develop a fundamental understanding of the material.

Summary: A simplifying perspective of IPM’s in Convex Optimization
Rating: 5

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!

Summary: A simplifying perspective of IPM’s in Convex Optimization
Rating: 5

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!

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