and F (x) its universal barrier function. Let D k.pdf

and F (x) its universal barrier function. Let D k.pdf

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and F (x) its universal barrier function. Let D k

On the Self{Concordance of the Universal BarrierFunctionOsman GulerTechnical Report 95{02, August 1995AbstractLet K be a regular convex cone in Rn and F (x) its universal barrier function.Let DkF (x)[h; : : : ; h] be kth order directional derivative at the point x 2 K0 anddirection h 2 Rn . We show that for every m  3 there exists a constant c(m) 0depending only on m such that jDmF (x)[h; : : : ; h]j  c(m)D2F (x)[h; h]m=2. Form = 3, this is the self{concordance inequality of Nesterov and Nemirovskii. Ourproof uses a powerful recent result of Bourgain. Key words. Universal barrier function, self{concordance, interior point methods.Abbreviated title. Universal Barrier Function.AMS(MOS) subject classi cations: primary 90C25, 90C60, 52A41; secondary 90C06,52A40.Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore,Maryland 21228, USA. (e-mail: guler@math.umbc.edu). Research partially supported by the NationalScience Foundation under grant DMS{9306318. 1 1 IntroductionInterior point methods have occupied a prominent place in continuous optimization eversince Karmarkar [7] introduced his polynomial{time projective algorithm for linear pro-gramming in 1984. Although much of the early activities has been in linear programmingand monotone linear complementarity problems, Nesterov and Nemirovskii [11] have suc-cessfully developed a theory of interior point methods for general nonlinear convex pro-gramming problems and monotone variational inequalities. One of the key ideas of thistheory is the notion of a self{concordant barrier function for a convex set.We recall some relevant concepts from [11]. Let Q  IRn be an open convex set. Afunction F : Q! IR is called a self{concordant barrier function if it is at least three timesdi erentiable, convex, and satis es the propertiesjD3F (x)[h; h; h]j  2(D2F (x)[h; h])3=2; (1)jDF (x)[h]j2  #D2F (x)[h; h]; (2)and F (x)!1 as x! @Q:Here DkF (x)[h; : : : ; h] is the kth directional of F at x a

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