# Interior Point Trajectories in Semidefinite Programming

@article{Goldfarb1998InteriorPT, title={Interior Point Trajectories in Semidefinite Programming}, author={Donald Goldfarb and Katya Scheinberg}, journal={SIAM J. Optim.}, year={1998}, volume={8}, pages={871-886} }

In this paper we study interior point trajectories in semidefinite programming (SDP) including the central path of an SDP. This work was inspired by the seminal work of Megiddo on linear programming trajectories [ Progress in Math. Programming: Interior-Point Algorithms and Related Methods, N. Megiddo, ed., Springer-Verlag, Berlin, 1989, pp. 131--158]. Under an assumption of primal and dual strict feasibility, we show that the primal and dual central paths exist and converge to the analytic…

## 89 Citations

THE CENTRAL PATH IN SMOOTH CONVEX SEMIDEFINITE PROGRAMS

- Mathematics
- 2002

Abstract In this paper we study the welldefinedness of the central path associated to a nonlinear convex semidefinite programming problem with smooth objective and constraint functions. Under…

On primal-dual path-following algorithms in semidefinite programming.

- Mathematics, Computer Science
- 1998

A weaker condition for a feasible full Newton step is established, and quadratic convergence to target points on the central path is shown, and it is shown how to compute large dynamic target updates which still allow full Newton steps.

Continuous methods for convex programming and convex semidefinite programming

- Mathematics
- 2017

The optimality and convergence of both interior point continuous trajectories are obtained for any interior feasible point under some mild conditions, and the convergence does not require the strict complementarity or the analyticity of the objective function.

On Two Interior-Point Mappings for Nonlinear Semidefinite Complementarity Problems

- MathematicsMath. Oper. Res.
- 1998

Extending our previous work Monteiro and Pang 1996, this paper studies properties of two fundamental mappings associated with the family of interior-point methods for solving monotone nonlinear…

Primal-Dual Affine-Scaling Algorithms Fail for Semidefinite Programming

- MathematicsMath. Oper. Res.
- 1999

It is shown that the primal-dual affine-scaling algorithm using the NT direction for the same semidefinite programming problem always generates a sequence converging to the optimal solution.

On the central path for nonlinear semidefinite programming

- MathematicsRAIRO Oper. Res.
- 2000

The well definedness of the central path associated to a given nonlinear (convex) semidefinite programming problem is studied and the existence and optimality of cluster points are established.

Primal-Dual Path-Following Algorithms for Semidefinite Programming

- Computer ScienceSIAM J. Optim.
- 1997

Two search directions within their family are characterized as being (unique) solutions of systems of linear equations in symmetric variables and, for the first time, a polynomially convergent long-step path-following algorithm for SDP which requires an extra $\sqrt{n}$ factor in its iteration-complexity order as compared to its linear programming counterpart.

An Interior-Point Perspective on Sensitivity Analysis in Semidefinite Programming

- MathematicsMath. Oper. Res.
- 2003

We study the asymptotic behavior of the interior-point bounds arising from the work of Yildirim and Todd on sensitivity analysis in semidefinite programming in comparison with the optimal partition…

On sensitivity of central solutions in semidefinite programming

- MathematicsMath. Program.
- 2001

The properties of the analytic central path of a semidefinite programming problem under perturbation of the right hand side of the constraints, including the limiting behavior when the central optimal solution, namely the analytic center of the optimal set, is approached are studied.

On Polynomial-time Path-following Interior-point Methods with Local Superlinear Convergence

- Mathematics
- 2016

Interior-point methods provide one of the most popular ways of solving convex optimization problems. Two advantages of modern interior-point methods over other approaches are: (i) robust global…

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