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51
PrimalDual InteriorPoint Methods for SelfScaled Cones
 SIAM Journal on Optimization
, 1995
"... In this paper we continue the development of a theoretical foundation for efficient primaldual interiorpoint algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are selfscaled (see [9]). The class of problems under consideration includes li ..."
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Cited by 192 (12 self)
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In this paper we continue the development of a theoretical foundation for efficient primaldual interiorpoint algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are selfscaled (see [9]). The class of problems under consideration includes linear programming, semidefinite programming and quadratically constrained quadratic programming problems. For such problems we introduce a new definition of affinescaling and centering directions. We present efficiency estimates for several symmetric primaldual methods that can loosely be classified as pathfollowing methods. Because of the special properties of these cones and barriers, two of our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.
On implementing a primaldual interiorpoint method for conic quadratic optimization
 MATHEMATICAL PROGRAMMING SER. B
, 2000
"... Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linea ..."
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Cited by 43 (5 self)
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Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linear and convex quadratic optimization is a special case. Conic quadratic optimization problems can in theory be solved efficiently using interiorpoint methods. In particular it has been shown by Nesterov and Todd that primaldual interiorpoint methods developed for linear optimization can be generalized to the conic quadratic case while maintaining their efficiency. Therefore, based on the work of Nesterov and Todd, we discuss an implementation of a primaldual interiorpoint method for solution of largescale sparse conic quadratic optimization problems. The main features of the implementation are it is based on a homogeneous and selfdual model, handles the rotated quadratic cone directly, employs a Mehrotra type predictorcorrector
Hyperbolic Polynomials and Interior Point Methods for Convex Programming
 Mathematics of Operations Research
, 1996
"... Hyperbolic polynomials have their origins in partial differential equations. We show in this paper that they have applications in interior point methods for convex programming. Each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone. We give an e ..."
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Cited by 40 (3 self)
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Hyperbolic polynomials have their origins in partial differential equations. We show in this paper that they have applications in interior point methods for convex programming. Each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone. We give an explicit representation of this cone in terms of polynomial inequalities. The function F (x) = \Gamma log p(x) is a logarithmically homogeneous selfconcordant barrier function for the hyperbolicity cone with barrier parameter equal to the degree of p. The function F (x) possesses striking additional properties that are useful in designing longstep interior point methods. For example, we show that the longstep primal potential reduction methods of Nesterov and Todd and the surfacefollowing methods of Nesterov and Nemirovskii extend to hyperbolic barrier functions. We also show that there exists a hyperbolic barrier function on every homogeneous cone. Key words. hyperbolic polynomials, ...
On the Riemannian geometry defined by selfconcordant barriers and interiorpoint methods
 Found. Comput. Math
"... We consider the Riemannian geometry defined on a convex set by the Hessian of a selfconcordant barrier function, and its associated geodesic curves. These provide guidance for the construction of efficient interiorpoint methods for optimizing a linear function over the intersection of the set with ..."
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Cited by 33 (0 self)
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We consider the Riemannian geometry defined on a convex set by the Hessian of a selfconcordant barrier function, and its associated geodesic curves. These provide guidance for the construction of efficient interiorpoint methods for optimizing a linear function over the intersection of the set with an affine manifold. We show that algorithms that follow the primaldual central path are in some sense close to optimal. The same is true for methods that follow the shifted primaldual central path among certain infeasibleinteriorpoint methods. We also compute the geodesics in several simple sets. ∗ Copyright (C) by SpringerVerlag. Foundations of Computational Mathewmatics 2 (2002), 333–361.
Approximation Algorithms for MAX3CUT and Other Problems via Complex Semidefinite Programming
, 2002
"... ..."
A Polynomial PrimalDual PathFollowing Algorithm for Secondorder Cone Programming
 Research Memorandum No. 649, The Institute of Statistical Mathematics
, 1997
"... Secondorder cone programming (SOCP) is the problem of minimizing linear objective function over crosssection of secondorder cones and an affine space. Recently this problem gets more attention because of its various important applications including quadratically constrained convex quadratic progr ..."
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Cited by 16 (1 self)
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Secondorder cone programming (SOCP) is the problem of minimizing linear objective function over crosssection of secondorder cones and an affine space. Recently this problem gets more attention because of its various important applications including quadratically constrained convex quadratic programming. In this paper we deal with a primaldual pathfollowing algorithm for SOCP to show many of the ideas developed for primaldual algorithms for LP and SDP carry over to this problem. We define neighborhoods of the central trajectory in terms of the "eigenvalues" of the secondorder cone, and develop an analogue of HRVW/KSH/M direction, and establish O( p n log " 01 ), O(n log " 01 ) and O(n 3 log " 01 ) iterationcomplexity bounds for shortstep, semilongstep and longstep pathfollowing algorithms, respectively, to reduce the duality gap by a factor of ". keywords: secondorder cone, interiorpoint methods, polynomial complexity, primaldual pathfollowing methods. 1 Intro...
SelfScaled Barrier Functions: Decomposition and Classification
, 1999
"... The theory of selfscaled conic programming provides a unified framework for the theories of linear programming, semidefinite programming and convex quadratic programming with convex quadratic constraints. Nesterov and Todd's concept of selfscaled barrier functionals allows the exploitation of ..."
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Cited by 12 (10 self)
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The theory of selfscaled conic programming provides a unified framework for the theories of linear programming, semidefinite programming and convex quadratic programming with convex quadratic constraints. Nesterov and Todd's concept of selfscaled barrier functionals allows the exploitation of algebraic and geometric properties of symmetric cones in certain variants of the barrier method applied to selfscaled conic programming problems. In a first part of this article we show that selfscaled barrier functionals can be decomposed into direct sums of selfscaled barrier functionals over the irreducible components of the underlying symmetric cone. Applying this decomposition theory in a second part, we give a complete classification of the set of selfscaled barrier functionals that are invariant under the action of the orthogonal group of their conic domain of definition (we call such functionals isotropic). Key words. selfconcordant barriers, symmetric cones, selfscaled cones, inte...
SELF–SCALED BARRIER FUNCTIONS ON SYMMETRIC CONES AND THEIR CLASSIFICATION
, 2008
"... Self–scaled barrier functions on self–scaled cones were introduced through a set of axioms in 1994 by Y. E. Nesterov and M. J. Todd as a tool for the construction of long–step interior point algorithms. This paper provides firm foundation for these objects by exhibiting their symmetry properties, th ..."
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Cited by 11 (3 self)
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Self–scaled barrier functions on self–scaled cones were introduced through a set of axioms in 1994 by Y. E. Nesterov and M. J. Todd as a tool for the construction of long–step interior point algorithms. This paper provides firm foundation for these objects by exhibiting their symmetry properties, their intimate ties with the symmetry groups of their domains of definition, and subsequently their decomposition into irreducible parts and algebraic classification theory. In a first part we recall the characterisation of the family of self–scaled cones as the set of symmetric cones and develop a primal–dual symmetric viewpoint on selfscaled barriers, results that were first discovered by the second author. We then show in a short, simple proof that any pointed, convex cone decomposes into a direct sum of irreducible components in a unique way, a result which can also be of independent interest. We then show that any self–scaled barrier function decomposes in an essentially unique way into a direct sum of self–scaled barriers defined on the irreducible components of the underlying symmetric cone. Finally, we present a complete algebraic classification of self–scaled barrier functions using the correspondence between symmetric cones and Euclidean Jordan algebras.