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On the Boundary Behaviour of the Riemannian Structure of a SelfConcordant Barrier Function
, 1999
"... this paper we investigate the asymptotic behaviour of this Riemannian structure, and of its geodesics and its curvature, near points of the boundary where the boundary is smooth and strongly convex, which means that its curvature, described by its second fundamental form, is positive definite. In or ..."
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. In order to obtain good asymptotic expansions near such points, we introduce Assumption 2.1 below about the behaviour of the function f near the boundary, and argue that these assumptions are quite natural, cf. Remark 2.3. Under these assumptions we will show in Section 5 that, after suitable
Interiorpoint Methods
, 2000
"... The modern era of interiorpoint methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadrati ..."
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Cited by 603 (15 self)
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, monotone linear complementarity, and convex programming over sets that can be characterized by selfconcordant barrier functions.
On the SelfConcordance of the Universal Barrier Function
, 1995
"... Let K be a regular convex cone in R n and F (x) its universal barrier function. Let D k F (x)[h; : : : ; h] be kth order directional derivative at the point x 2 K 0 and direction h 2 R n . We show that for every m 3 there exists a constant c(m) ? 0 depending only on m such that jD m F (x) ..."
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)[h; : : : ; h]j c(m) D 2 F (x)[h; h] m=2 . For m = 3, this is the selfconcordance inequality of Nesterov and Nemirovskii. Our proof uses a powerful recent result of Bourgain.
SelfConcordant Barriers For Hyperbolic Means
, 2000
"... The geometric mean and the function (det()) 1=m (on the mbym positive denite matrices) are examples of \hyperbolic means": functions of the form p 1=m , where p is a hyperbolic polynomial of degree m. (A homogeneous polynomial p is \hyperbolic" with respect to a vector d if the pol ..."
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Cited by 5 (2 self)
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if the polynomial t 7! p(x + td) has only real roots for every vector x.) Any hyperbolic mean is positively homogeneous and concave (on a suitable domain): we present a selfconcordant barrier for its hypograph, with barrier parameter O(m 2 ). Our approach is direct, and shows, for example, that the function m
SELFCONCORDANT BARRIERS AND CHEBYSHEV SYSTEMS*
"... Abstract. We explicitly calculate characteristic functions of cones of generalized polynomials corresponding to Chebyshev systems on intervals of the real line and the circle. Thus, in principle, we calculate homogeneous selfconcordant barriers for this class of cones. This class includes almost al ..."
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Abstract. We explicitly calculate characteristic functions of cones of generalized polynomials corresponding to Chebyshev systems on intervals of the real line and the circle. Thus, in principle, we calculate homogeneous selfconcordant barriers for this class of cones. This class includes almost
SELFCONCORDANCE IS NPHARD
"... Abstract. We give an elementary proof of a somewhat curious result, namely, that deciding whether a convex function is selfconcordant is in general an intractable problem. 1. ..."
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Abstract. We give an elementary proof of a somewhat curious result, namely, that deciding whether a convex function is selfconcordant is in general an intractable problem. 1.
A Unifying Investigation of InteriorPoint Methods for Convex Programming
 FACULTY OF MATHEMATICS AND INFORMATICS, TU DELFT, NL2628 BL
, 1992
"... In the recent past a number of papers were written that present low complexity interiorpoint methods for different classes of convex programs. Goal of this article is to show that the logarithmic barrier function associated with these programs is selfconcordant, and that the analyses of interiorp ..."
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Cited by 5 (4 self)
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In the recent past a number of papers were written that present low complexity interiorpoint methods for different classes of convex programs. Goal of this article is to show that the logarithmic barrier function associated with these programs is selfconcordant, and that the analyses of interior
Composite SelfConcordant Minimization ∗
"... We propose a variable metric framework for minimizing the sum of a selfconcordant function and a possibly nonsmooth convex function endowed with a computable proximal operator. We theoretically establish the convergence of our framework without relying on the usual Lipschitz gradient assumption on ..."
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Cited by 6 (5 self)
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We propose a variable metric framework for minimizing the sum of a selfconcordant function and a possibly nonsmooth convex function endowed with a computable proximal operator. We theoretically establish the convergence of our framework without relying on the usual Lipschitz gradient assumption
A SUPERLINEARLY CONVERGENT ALGORITHM FOR LARGE SCALE MULTISTAGE STOCHASTIC NONLINEAR PROGRAMMING
, 2003
"... This paper presents an algorithm for solving a class of large scale nonlinear programming problem which is originally derived from the multistage stochastic convex nonlinear programming. Using the Lagrangiandual method and the MoreauYosida regularization, the primal problem is neatly transformed ..."
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into a smooth convex problem. By introducing a selfconcordant barrier function, an approximate generalized Newton method is then designed to solve the problem. The algorithm is shown to be of superlinear convergence. Some numerical results are presented to demonstrate the viability of the proposed
Two InteriorPoint Algorithms for a Class of Convex Programming Problems
, 1994
"... This paper describes two algorithms for the problem of minimizing a linear function over the intersection of an affine set and a convex set which is required to be the closure of the domain of a strongly selfconcordant barrier function. One algorithm is a pathfollowing method, while the other is a ..."
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This paper describes two algorithms for the problem of minimizing a linear function over the intersection of an affine set and a convex set which is required to be the closure of the domain of a strongly selfconcordant barrier function. One algorithm is a pathfollowing method, while the other
Results 1  10
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706,678