A mathematical view of interior-point methods in convex optimization.

*(English)*Zbl 0986.90075
MPS/SIAM Series on Optimization 3. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. Philadelphia, PA: MPS, Mathematical Programming Society, vi, 116 p. (2001).

The book is devoted to the most general theory of interior-point methods. It focusses on essential elements of the theory and emphasizes the underlying geometry in order to make the results accessible to a wide audience.

Chapter 1 provides a review of the more important results pertinent to continuous optimization theory. In Chapter 2 the intrinsic inner products are introduced and the theory of basic interior-point methods is presented. The results are written in terms of this concept that simplifies the operator manipulation in the proofs. Self-concordant and barrier functionals are studied and some primal path-following methods are discussed. Chapter 3 presents some basic results in the classical duality theory and their connections with interior-point methods. Primal-dual path-following methods and potential-reduction methods are presented and analyzed.

The book might be used for graduate courses in optimization. It would be of interest to both students and researchers who wish to better assimilate the most general theory of interior-point methods.

Chapter 1 provides a review of the more important results pertinent to continuous optimization theory. In Chapter 2 the intrinsic inner products are introduced and the theory of basic interior-point methods is presented. The results are written in terms of this concept that simplifies the operator manipulation in the proofs. Self-concordant and barrier functionals are studied and some primal path-following methods are discussed. Chapter 3 presents some basic results in the classical duality theory and their connections with interior-point methods. Primal-dual path-following methods and potential-reduction methods are presented and analyzed.

The book might be used for graduate courses in optimization. It would be of interest to both students and researchers who wish to better assimilate the most general theory of interior-point methods.

Reviewer: Juan Manuel Otero (Habana)

##### MSC:

90C51 | Interior-point methods |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |