Self-Concordant Barriers for Convex Approximations of Structured Convex Sets

• Published in: Foundations of computational mathematics Vol. 10; no. 5; pp. 485 - 525
• Main Authors: Tunçel, Levent, Nemirovski, Arkadi
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.10.2010; Springer; Springer Nature B.V
• Subjects: Accuracy; Applications of Mathematics; Approximation; Barriers; Combinatorial analysis; Combinatorics; Computation; Computational mathematics; Computer Science; Economics; Foundations; Joints; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Optimization; Set theory; Semidefinite programming; 52A41; 90C06; Packing-covering problems; 90C59; Interior-point methods; 90C05; Self-concordant barriers; 90C51; Convex optimization; 90C25; 90C22; 49M37
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-010-9069-x

We show how to approximate the feasible region of structured convex optimization problems by a family of convex sets with explicitly given and efficient (if the accuracy of the approximation is moderate) self-concordant barriers. This approach extends the reach of the modern theory of interior-point methods, and lays the foundation for new ways to treat structured convex optimization problems with a very large number of constraints. Moreover, our approach provides a strong connection from the theory of self-concordant barriers to the combinatorial optimization literature on solving packing and covering problems.