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362
SecondOrder Cone Programming
 MATHEMATICAL PROGRAMMING
, 2001
"... In this paper we survey the second order cone programming problem (SOCP). First we present several applications of the problem in various areas of engineering and robust optimization problems. We also give examples of optimization problems that can be cast as SOCPs. Next we review an algebraic struc ..."
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Cited by 233 (11 self)
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In this paper we survey the second order cone programming problem (SOCP). First we present several applications of the problem in various areas of engineering and robust optimization problems. We also give examples of optimization problems that can be cast as SOCPs. Next we review an algebraic structure that is connected to SOCP. This algebra is a special case of a Euclidean Jordan algebra. After presenting duality theory, complementary slackness conditions, and definitions and algebraic characterizations of primal and dual nondegeneracy and strict complementarity we review the logarithmic barrier function for the SOCP problem and survey the pathfollowing interior point algorithms for it. Next we examine numerically stable methods for solving the interior point methods and study ways that sparsity in the input data can be exploited. Finally we give some current and future research direction in SOCP.
Barrier Functions in Interior Point Methods
 MATHEMATICS OF OPERATIONS RESEARCH
, 1996
"... We show that the universal barrier function of a convex cone introduced by Nesterov and Nemirovskii is the logarithm of the characteristic function of the cone. This interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone. This provides ..."
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Cited by 57 (3 self)
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We show that the universal barrier function of a convex cone introduced by Nesterov and Nemirovskii is the logarithm of the characteristic function of the cone. This interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone. This provides a simple method for calculating the universal barrier for homogeneous cones. We identify some known barriers as the universal barrier scaled by an appropriate constant. We also calculate some new universal barrier functions. Our results connect the field of interior point methods to several branches of mathematics such as Lie groups, Jordan algebras, Siegel domains, differential geometry, complex analysis of several variables, etc.
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 54 (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, ...
Linear systems in Jordan algebras and primaldual interiorpoint algorithms
 Journal of Computational and Applied Mathematics
, 1997
"... We discuss a possibility of the extension of a primaldual interiorpoint algorithm suggested recently in [1]. We consider optimization problems defined on the intersection of a symmetric cone and an affine subspace. The question of solvability of a linear system arising in the implementation of the ..."
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Cited by 52 (4 self)
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We discuss a possibility of the extension of a primaldual interiorpoint algorithm suggested recently in [1]. We consider optimization problems defined on the intersection of a symmetric cone and an affine subspace. The question of solvability of a linear system arising in the implementation of the primaldual algorithm is analyzed. A nondegeneracy theory for the considered class of problems is developed. The Jordan algebra technique suggested in [5] plays major role in the present paper. 1 Introduction Recently F.Alizadeh, J.P. Haeberly and M. Overton suggested a primaldual interiorpoint algorithm for solving semidefinite problems [1] that shows extremely good convergence properties and a high degree of accuracy [2]. In the present paper we discuss a possibility of an extension of this algorithm to a broader class of optimization problems defined on the intersection of an affine subspace with a symmetric (i.e. selfdual, homogeneous ) cone. This class of problems includes linear ...
A Convergence Analysis of the Scalinginvariant Primaldual Pathfollowing Algorithms for Secondorder Cone Programming
 Optim. Methods Softw
, 1998
"... This paper is a continuation of our previous paper in which we studied a polynomial primaldual pathfollowing algorithm for SOCP using an analogue of the HRVW/KSH/M direction for SDP. We develop an improved and simplified complexity analysis which can be also applied to the algorithm using the NT di ..."
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Cited by 52 (5 self)
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This paper is a continuation of our previous paper in which we studied a polynomial primaldual pathfollowing algorithm for SOCP using an analogue of the HRVW/KSH/M direction for SDP. We develop an improved and simplified complexity analysis which can be also applied to the algorithm using the NT direction. Specifically, we show that the longstep algorithm using the NT direction has O(n log " 01 ) iterationcomplexity to reduce the duality gap by a factor of ", where n is the number of the secondorder cones. The complexity for the same algorithm using the HRVW/KSH/M direction is improved to O(n 3=2 log " 01 ) from O(n 3 log " 01 ) of the previous analysis. We also show that the short and semilongstep algorithms using the NT direction (and the HRVW/KSH/M direction) have O( p n log " 01 ) and O(n log " 01 ) iterationcomplexities, respectively. keywords: secondorder cone, interiorpoint methods, polynomial complexity, primaldual pathfollowing methods. 1 Introduction...
Shifted Jack Polynomials, Binomial Formula, And Applications
, 1996
"... In this note we prove an explicit binomial formula for Jack polynomials and discuss some applications of it. ..."
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Cited by 49 (8 self)
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In this note we prove an explicit binomial formula for Jack polynomials and discuss some applications of it.
Hyperbolic Polynomials and Convex Analysis
, 1998
"... Abstract. A homogeneous real polynomial p is hyperbolic with respect to a given vector d if the univariate polynomial t ↦ → p(x − td) has all real roots for all vectors x. Motivated by partial differential equations, G˚arding proved in 1951 that the largest such root is a convex function of x, and s ..."
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Cited by 38 (4 self)
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Abstract. A homogeneous real polynomial p is hyperbolic with respect to a given vector d if the univariate polynomial t ↦ → p(x − td) has all real roots for all vectors x. Motivated by partial differential equations, G˚arding proved in 1951 that the largest such root is a convex function of x, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize G˚arding’s result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convexanalytic tools for such symmetric functions, of interest in interiorpoint methods for optimization problems over related cones. 1
Complementarity Functions and Numerical Experiments on Some Smoothing Newton Methods for Secondordercone Complementarity Problems
, 2003
"... Two results on the secondordercone complementarity problem are presented. We show that the squared smoothing function is strongly semismooth. Under monotonicity and strict feasibility we provide a new proof, based on a penalized natural complementarity function, for the solution set of the secon ..."
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Cited by 37 (13 self)
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Two results on the secondordercone complementarity problem are presented. We show that the squared smoothing function is strongly semismooth. Under monotonicity and strict feasibility we provide a new proof, based on a penalized natural complementarity function, for the solution set of the secondordercone complementarity problem being bounded. Numerical results of squared smoothing Newton algorithms are reported.
The complex Wishart distribution and the symmetric group
"... Let V be the space of (r; r) Hermitian matrices and let\Omega be the cone of the positive definite ones. We say that the random variable S; taking its values has the complex Wishart distribution fl p;oe if IE(exp trace (`S)) = (det(I r \Gamma oe`)) where oe and oe \Gamma ` are ; and where p ..."
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Cited by 34 (2 self)
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Let V be the space of (r; r) Hermitian matrices and let\Omega be the cone of the positive definite ones. We say that the random variable S; taking its values has the complex Wishart distribution fl p;oe if IE(exp trace (`S)) = (det(I r \Gamma oe`)) where oe and oe \Gamma ` are ; and where p = 1; 2; : : : ; r \Gamma 1 or p ? r \Gamma 1. In this paper, we compute all moments of S: More specifically, if h = (h 1 ; : : : ; h n ) is any word of (r; r) complex matrices and if we introduce the complex random variables S(h) = trace (Sh 1 : : : Sh n ) and S (h) = trace (S h 1 \Delta \Delta \Delta S h n ); we are able to compute in a simple way the expressions IE(S(h ) \Delta \Delta \Delta S(h )) for any set of words fh ; \Delta \Delta \Delta ; h g: Similarly we compute IE(S ) \Delta \Delta \Delta S whenever it exists. This provides a general answer to the questions raised by D. Maiwald and D. Kraus (2000) for moments of order 4. Our technique is to use the multilinear forms r (oe)(h 1 ; : : : ; h k ); where belongs to the group S k of permutations of f1; : : : ; kg. For instance, if = (4)(2; 5)(6; 3; 1); the form r (oe) is defined by r (oe)(h 1 ; : : : ; h 6 ) = trace (oeh 4 )trace (oeh 2 oeh 5 )trace (oeh 6 oeh 3 oeh 1 ): Denote by m() the number of cycles of ; and write q for p \Gamma r: Our theorems 2 and 3 can be presented in the following compact way by using the convolution product associated to the group algebra A(S k ): Our proofs of these formulas are elementary. The remainder of the paper is devoted to the inversion of the second formula, i.e. to the computation of )); for any . To this end, we use the irreducible characters of S k
Maximal surface group representations in isometry groups of classical hermitian symmetric spaces
 GROUP REPRESENTATIONS WITH MAXIMAL TOLEDO INVARIANT 85
, 2005
"... We present a survey on the problem of finding the connected components of moduli spaces for representations of surface groups in isometry groups of classical Hermitian symmetric spaces of noncompact type. ..."
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Cited by 31 (4 self)
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We present a survey on the problem of finding the connected components of moduli spaces for representations of surface groups in isometry groups of classical Hermitian symmetric spaces of noncompact type.