Copyright information to be inserted by the Publishers A GENERAL FRAMEWORK FOR ESTABLISHING POLYNOMIAL CONVERGENCE OF LONG-STEP METHODS FOR SEMIDEFINITE PROGRAMMING SAMUEL BURER ∗ Department of Management Sciences, University of Iowa, Iowa City, Iowa 52242-1000, USA RENATO D. C. MONTEIRO † School of ISyE, Georgia Tech, Atlanta, Georgia 30332, USA This paper considers feasible long-step primal-dual path-following methods for semidefinite programming based on Newton direc- tions associated with central path equations of the form Φ(PXP T ,P -T SP -1 ) - νI = 0, where the map Φ and the nonsingular matrix P satisfy several key properties. An iteration-complexity bound for the long-step method is derived in terms of an upper bound on a certain scaled norm of the second derivative of Φ. As a consequence of our general framework, we derive polynomial iteration-complexity bounds for long-step algorithms based on the following four maps: Φ(X, S)=(XS + SX)/2, Φ(X, S)= L T x SL x , Φ(X, S)= X 1/2 SX 1/2 , and Φ(X, S)= W 1/2 XSW -1/2 , where L x is the lower Cholesky factor of X and W is the unique symmetric matrix satisfying S = WXW . KEY WORDS: semidefinite programming, interior-point methods, path-following methods, long- step methods, Newton directions, central path 1 Introduction Semidefinite programming (SDP) is a generalization of linear programming (LP) in which a linear function of a symmetric matrix variable X is minimized over an affine subspace of real symmetric matrices subject to the constraint that X be positive semidefinite. Semidefinite programming shares many features of linear programming, including a large number of applications, a rich duality theory, and the ability to be solved (more precisely, approximated) in polynomial time. In the past several years, a major part of the research into SDP has focused on both the theoretical and practical solution of SDP problems using extensions of interior-point methods for LP. Many authors have proposed interior-point algorithms for solving SDP problems (see for example [1, 2, 4, 6, 7, 8, 9, 10, 11, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26]). Many of the recent works on interior-point algorithms for SDP are concentrated on primal-dual methods. Feasible primal-dual path-following algorithms for SDP simultaneously solve the primal and dual SDP problems by maintaining primal feasibility in X and dual feasibility in (S, y) while iteratively solving the system XS = 0. The key idea is to follow the central path by moving in the direction obtained by the application of Newton’s method to the central path equation XS = νI . Newton’s method, however, results in an equation of the form XΔS +ΔXS = νI − XS, (1) which in general yields nonsymmetric directions. Many authors have investigated alternate yet equivalent equations of the central path for which Newton’s method does yield symmetric directions (see for example [2, 4, 7, 10, 11, 13, 14, 17, 20, 24]). * The work of this author was based on research supported bythe National Science Foundation under grants INT-9600343, INT-9910084, CCR-9700448, CCR-9902010 and CCR-0203426. † The work of this author was based on research supported bythe National Science Foundation under grants INT-9600343, INT-9910084,CCR-9700448, CCR-9902010, and CCR-0203113. 1