Renato D. C. Monteiro

A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization

Abstract. In this paper, we present a nonlinear programming algorithm for solving semidefinite programs (SDPs) in standard form. The algorithms distinguishing feature is a change of variables that replaces the symmetric, positive semidefinite variable X of the SDP with a rectangular variable R according to the factorization X=RRT. The rank of the factorization, i.e., the number of columns of R, is chosen minimally so as to enhance computational speed while maintaining equivalence with the SDP. Fundamental results concerning the convergence of the algorithm are derived, and encouraging computational results on some large-scale test problems are also presented.

Interior path following primal-dual algorithms. Part I: Linear programming

AbstractWe describe a primal-dual interior point algorithm for linear programming problems which requires a total of

Interior path following primal-dual algorithms. Part II: Convex quadratic programming

O\left( {\sqrt n L} \right)

Local Minima and Convergence in Low-Rank Semidefinite Programming

number of iterations, whereL is the input size. Each iteration updates a penalty parameter and finds the Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the logarithmic barrier function problem. The algorithm is based on the path following idea.

Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions

This paper deals with a class of primal--dual interior-point algorithms for semidefinite programming (SDP) which was recently introduced by Kojima, Shindoh, and Hara [SIAM J. Optim., 7 (1997), pp. 86--125]. These authors proposed a family of primal-dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix X1/2S-1/2. They study three primal--dual algorithms based on this family of search directions: a short-step path-following method, a feasible potential-reduction method, and an infeasible potential-reduction method. However, they were not able to provide an algorithm which generalizes the long-step path-following algorithm introduced by Kojima, Mizuno, and Yoshise [Progress in Mathematical Programming: Interior Point and Related Methods, N. Megiddor, ed., Springer-Verlag, Berlin, New York, 1989, pp. 29--47]. In this paper, we characterize two search directions within their family as being (unique) solutions of systems of linear equations in symmetric variables. Based on this characterization, we present a simplified polynomial convergence proof for one of their short-step path-following algorithms and, for the first time, a polynomially convergent long-step path-following algorithm for SDP which requires an extra

Rank-Two Relaxation Heuristics for MAX-CUT and Other Binary Quadratic Programs

\sqrt{n}

Iteration-Complexity of Block-Decomposition Algorithms and the Alternating Direction Method of Multipliers

factor in its iteration-complexity order as compared to its linear programming counterpart, where n is the number of rows (or columns) of the matrices involved.

Limiting behavior of the affine scaling continuous trajectories for linear programming problems

AbstractWe describe a primal-dual interior point algorithm for convex quadratic programming problems which requires a total of