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Generalization of Primal—Dual Interior-Point Methods to Convex Optimization Problems in Conic Form

Rigorous Numerics for Partial Differential Equations: The Kuramoto—Sivashinsky Equation

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Generalization of Primal—Dual Interior-Point Methods to Convex Optimization Problems in Conic Form

Content Provider	Springer Nature Link
Author	Tunçel, Levent
Copyright Year	2000
Abstract	We generalize primal—dual interior-point methods for linear programming (LP) problems to the convex optimization problems in conic form. Previously, the most comprehensive theory of symmetric primal—dual interior-point algorithms was given by Nesterov and Todd for feasible regions expressed as the intersection of a symmetric cone with an affine subspace. In our setting, we allow an arbitrary convex cone in place of the symmetric cone. Even though some of the impressive properties attained by Nesterov—Todd algorithms are impossible in this general setting of convex optimization problems, we show that essentially all primal—dual interior-point algorithms for LP can be extended easily to the general setting. We provide three frameworks for primal—dual algorithms, each framework corresponding to a different level of sophistication in the algorithms. As the level of sophistication increases, we demand better formulations of the feasible solution sets. Our algorithms, in return, attain provably better theoretical properties. We also make a very strong connection to quasi-Newton methods by expressing the square of the symmetric primal—dual linear transformation (the so-called scaling) as a quasi-Newton update in the case of the least sophisticated framework.
Ending Page	254
Page Count	26
Starting Page	229
File Format	PDF
ISSN	16153375
Journal	Foundations of Computational Mathematics
Issue Number	3
Volume Number	1
Language	English
Publisher	Springer-Verlag
Publisher Date	2013-01-04
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Symmetric cones Semidefinite programming Quasi-Newton updates Variable metric methods Economics general Complexity and performance of numerical algorithms Linear programming Linear and Multilinear Algebras, Matrix Theory Interior-point methods Convex programming Primal—dual algorithms Self-concordant barriers Convex optimization Numerical Analysis Computer Science Convex functions and convex programs Polynomial iteration-complexity Nonlinear programming Applications of Mathematics Math Applications in Computer Science Abstract computational complexity for mathematical programming problems Methods of nonlinear programming type Interior-point method
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics

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