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28
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 205 (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.
Implementation of interior point methods for mixed semidefinite and second order cone optimization problems
 Optimization Methods and Software
"... There is a large number of implementational choices to be made for the primaldual interior point method in the context of mixed semidefinite and second order cone optimization. This paper presents such implementational issues in a unified framework, and compares the choices made by different resear ..."
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Cited by 40 (0 self)
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There is a large number of implementational choices to be made for the primaldual interior point method in the context of mixed semidefinite and second order cone optimization. This paper presents such implementational issues in a unified framework, and compares the choices made by different research groups. This is also the first paper to provide an elaborate discussion of the implementation in SeDuMi.
A Study of Search Directions in PrimalDual InteriorPoint Methods for Semidefinite Programming
, 1998
"... We discuss several di#erent search directions which can be used in primaldual interiorpoint methods for semidefinite programming problems and investigate their theoretical properties, including scale invariance, primaldual symmetry, and whether they always generate welldefined directions. Among ..."
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Cited by 35 (1 self)
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We discuss several di#erent search directions which can be used in primaldual interiorpoint methods for semidefinite programming problems and investigate their theoretical properties, including scale invariance, primaldual symmetry, and whether they always generate welldefined directions. Among the directions satisfying all but at most two of these desirable properties are the AlizadehHaeberlyOverton, HelmbergRendl VanderbeiWolkowicz/KojimaShindohHara/Monteiro, NesterovTodd, Gu, and Toh directions, as well as directions we will call the MTW and Half directions. The first five of these appear to be the best in our limited computational testing also. Key words: semidefinite programming, search direction, invariance properties. AMS Subject classification: 90C05. Abbreviated title: Search directions in SDP 1 Introduction This paper is concerned with interiorpoint methods for semidefinite programming (SDP) problems and in particular the various search directions they use and ...
Fixing Variables in Semidefinite Relaxations
 SIAM J. MATRIX ANAL. APPL
, 1996
"... The standard technique of reduced cost fixing from linear programming is not trivially extensible to semidefinite relaxations as the corresponding Lagrange multipliers are usually not available. We propose a general technique for computing reasonable Lagrange multipliers to constraints which are not ..."
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Cited by 27 (2 self)
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The standard technique of reduced cost fixing from linear programming is not trivially extensible to semidefinite relaxations as the corresponding Lagrange multipliers are usually not available. We propose a general technique for computing reasonable Lagrange multipliers to constraints which are not part of the problem description. Its specialization to the semidefinite f\Gamma1; 1g relaxation of quadratic 01 programming yields an efficient routine for fixing variables. The routine offers the possibility to exploit problem structure. We extend the traditional bijective map between f0; 1g and f\Gamma1; 1g formulations to the constraints such that the dual variables remain the same and structural properties are preserved. In consequence the fixing routine can efficiently be applied to optimal solutions of the semidefinite f0; 1g relaxation of constrained quadratic 01 programming, as well. We provide numerical results showing the efficacy of the approach.
Generalization Of PrimalDual InteriorPoint Methods To Convex Optimization Problems In Conic Form
, 1999
"... We generalize primaldual interiorpoint methods for linear programming problems to the convex optimization problems in conic form. Previously, the most comprehensive theory of symmetric primaldual interiorpoint algorithms was given by Nesterov and Todd [8, 9] for the feasible regions expressed as ..."
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Cited by 20 (4 self)
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We generalize primaldual interiorpoint methods for linear programming problems to the convex optimization problems in conic form. Previously, the most comprehensive theory of symmetric primaldual interiorpoint algorithms was given by Nesterov and Todd [8, 9] for the 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 NesterovTodd algorithms is impossible in this general setting of convex optimization problems, we show that essentially all primaldual interiorpointalgorithms for LP can be extended easily to the general setting. We provide three frameworks for primaldual 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, but our algorithms, in return, atta...
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 19 (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...
Interiorpoint methods for optimization
, 2008
"... This article describes the current state of the art of interiorpoint methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twen ..."
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Cited by 17 (0 self)
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This article describes the current state of the art of interiorpoint methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twenty years.
Similarity and Other Spectral Relations for Symmetric Cones
 Linear Algebra And Its Applications 312
, 1998
"... A onetoone relation is established between the nonnegative spectral values of a vector in a primitive symmetric cone and the eigenvalues of its quadratic representation. This result is then exploited to derive similarity relations for vectors with respect to a general symmetric cone. For two pos ..."
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Cited by 11 (2 self)
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A onetoone relation is established between the nonnegative spectral values of a vector in a primitive symmetric cone and the eigenvalues of its quadratic representation. This result is then exploited to derive similarity relations for vectors with respect to a general symmetric cone. For two positive definite matrices X and Y , the square of the spectral geometric mean is similar to the matrix product XY , and it is shown that this property carries over to symmetric cones. We also extend the result that the eigenvalues of a matrix product XY are less dispersed than the eigenvalues of the Jordan product (XY +YX)=2. The paper further contains a number of inequalities that are useful in the context of interior point methods, and an extension of Stein's theorem to symmetric cones. Key words. Symmetric cone, Euclidean Jordan algebra, optimization. There are two symmetric cones that are widely used in almost any area of applied mathematics, namely the nonnegative orthant and the cone o...
Pattern Separation Via Ellipsoids and Conic Programming
, 1998
"... this document. The first chapter is about mathematical programming. We will start by describing how and why researchers were led to study special types of mathematical programs, namely convex programs and conic programs. We will also provide a detailed discussion about conic duality and give a class ..."
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Cited by 11 (0 self)
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this document. The first chapter is about mathematical programming. We will start by describing how and why researchers were led to study special types of mathematical programs, namely convex programs and conic programs. We will also provide a detailed discussion about conic duality and give a classification of conic programs. We will then describe what are selfscaled cones and why they are so useful in conic programming. Finally, we will give an overview of what can be modelled using a SQL conic program, keeping in mind our pattern separation problem. Since most of the material in the chapter is standard, many of the proofs are omitted. The second chapter will concentrate on pattern separation. After a short description of the problem, we will successively describe four different separation methods using SQL conic programming. For each method, various properties are investigated. Each algorithm has in fact been successively designed with the objective of eliminating the drawbacks of the previous one, CONTENTS 3 while keeping its good properties. We conclude this chapter with a small section describing the state of the art in pattern separation with ellipsoids. The third chapter reports some computational experiments with our four methods, and provides a comparison with other separation procedures. Finally, we conclude this work by providing a short summary, highlighting the author's personal contribution and giving some interesting perspectives for further research. Chapter 1 Conic programming 1.1 Introduction
Geometry of Homogeneous Convex Cones, Duality Mapping, and Optimal SelfConcordant Barriers
, 2002
"... We study homogeneous convex cones. We rst characterize the extreme rays of such cones in the context of their primal construction (due to Vinberg) and also in the context of their dual construction (due to Rothaus). Then, using these results, we prove that every homogeneous cone is facially exposed ..."
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Cited by 5 (0 self)
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We study homogeneous convex cones. We rst characterize the extreme rays of such cones in the context of their primal construction (due to Vinberg) and also in the context of their dual construction (due to Rothaus). Then, using these results, we prove that every homogeneous cone is facially exposed. We provide an alternative proof of a result of Guler and Tuncel that the Siegel rank of a symmetric cone is equal to its Caratheodory number. Our proof does not use the Jordanvon NeumannWigner characterization of the symmetric cones but it easily follows from the primal construction of the homogeneous cones and our results on the geometry of homogeneous cones in primal and dual forms. We study optimal selfconcordant barriers in this context. We briey discuss the duality mapping in the context of automorphisms of convex cones and prove, using numerical integration, that the duality mapping is not an involution on certain selfdual cones.