# Primal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results

@article{Alizadeh1998PrimalDualIM, title={Primal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results}, author={Farid Alizadeh and Jean Pierre Haeberly and Michael L. Overton}, journal={SIAM J. Optim.}, year={1998}, volume={8}, pages={746-768} }

Primal-dual interior-point path-following methods for semidefinite programming are considered. Several variants are discussed, based on Newton's method applied to three equations: primal feasibility, dual feasibility, and some form of centering condition. The focus is on three such algorithms, called the XZ, XZ+ZX, and Q methods. For the XZ+ZX and Q algorithms, the Newton system is well defined and its Jacobian is nonsingular at the solution, under nondegeneracy assumptions. The associated…

## 457 Citations

Some New Search Directions for Primal-Dual Interior Point Methods in Semidefinite Programming

- MathematicsSIAM J. Optim.
- 2000

These directions imply that a path-following method using the proposed directions can achieve the high accuracy typically attained by methods employing the direction proposed by Alizadeh, Haeberly, and Overton, but each iteration requires at most half the amount of flops.

A primal–dual regularized interior-point method for semidefinite programming

- Mathematics, Computer ScienceOptim. Methods Softw.
- 2017

It is shown that it is possible to recover an optimal solution of the original primal–dual pair by taking one step of Newton method to a sequence of regularized SDPs at each iteration for both the Nesterov–Todd and dual Helmberg–Kojima–Monteiro (HKM) directions.

PRIMAL-DUAL INTERIOR-POINT METHODS FOR SEMIDEFINITE PROGRAMMING IN FINITE PRECISION

- Computer Science
- 2000

The error analysis indicates that primal-dual interior-point methods for semidenite program- ming could be numerically stable if certain coecient matrices associated with the iterations are well-conditioned, but are unstable otherwise.

Superlinear Convergence of a Symmetric Primal-Dual Path Following Algorithm for Semidefinite Programming

- MathematicsSIAM J. Optim.
- 1998

This paper establishes the superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming (SDP) under the assumptions that the semidefinite program has a…

Strengthened existence and uniqueness conditions for search directions in semidefinite programming

- Mathematics
- 2005

Superlinear Convergence of an Infeasible Predictor-Corrector Path-Following Interior Point Algorithm for a Semidefinite Linear Complementarity Problem Using the Helmberg--Kojima--Monteiro Direction

- Mathematics, Computer ScienceSIAM J. Optim.
- 2011

It is shown that for any linear semidefinite feasibility problem, superlinear convergence using the interior point algorithm, with the Helmberg-Kojima-Monteiro (HKM) direction, can be achieved for a suitable starting point.

A Scaled Gauss--Newton Primal-Dual Search Direction for Semidefinite Optimization

- Mathematics, Computer ScienceSIAM J. Optim.
- 2001

A "Gauss--Newton" direction is presented where a local norm is used in the least squares formulation, and a polynomial complexity analysis and computational evaluation of the resulting primal-dual algorithm is given.

Polynomial Convergence of Primal-Dual Algorithms for Semidefinite Programming Based on the Monteiro and Zhang Family of Directions

- Computer Science, MathematicsSIAM J. Optim.
- 1998

The polynomial convergence of the class of primal-dual feasible interior-point algorithms for semidefinite programming (SDP) based on the Monteiro and Zhang family of search directions is established for the first time.

A Superlinearly Convergent Primal-Dual Infeasible-Interior-Point Algorithm for Semidefinite Programming

- Mathematics, Computer ScienceSIAM J. Optim.
- 1998

A primal-dual infeasible-interior-point path-following algorithm for solving semidefinite programming (SDP) problems and a sufficient condition for the superlinear convergence of the algorithm is proposed.

NEW SEARCH DIRECTIONS FOR SEMIDEFINITE PROGRAMMING 225 Given the strengths and weaknesses of the AHO , HKM

- Mathematics
- 2000

Search directions for primal-dual path-following methods for semidefinite programming (SDP) are proposed. These directions have the properties that (1) under certain nondegeneracy and strict…

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