Primal-Dual Affine-Scaling Algorithms Fail for Semidefinite Programming

@article{Muramatsu1999PrimalDualAA,
  title={Primal-Dual Affine-Scaling Algorithms Fail for Semidefinite Programming},
  author={Masakazu Muramatsu and Robert J. Vanderbei},
  journal={Math. Oper. Res.},
  year={1999},
  volume={24},
  pages={149-175}
}
In this paper, we give an example of a semidefinite programming problem in which primal-dual affine-scaling algorithms using the HRVW/KSH/M, MT, and AHO directions fail. We prove that each of these algorithms can generate a sequence converging to a non-optimal solution and that, for the AHO direction, even its associated continuous trajectory can converge to a non-optimal point. In contrast with these directions, we show that the primal-dual affine-scaling algorithm using the NT direction for… 

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References

SHOWING 1-10 OF 68 REFERENCES
Polynomial Primal-Dual Affine Scaling Algorithms in Semidefinite Programming
TLDR
Two primal-dual affine scaling algorithms for linear programming are extended to semidefinite programming, where τ0 is a measure of centrality of the the starting solution, and the latter a bound of O(τ0nL2) iterations.
Superlinear Convergence of a Symmetric Primal-Dual Path Following Algorithm for Semidefinite Programming
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
Affine scaling algorithm fails for semidefinite programming
TLDR
An affine scaling algorithm for semidefinite programming (SDP) is introduced, and an example of a semideFinite program such that the affine scaled algorithm converges to a non-optimal point is given.
Interior Point Trajectories in Semidefinite Programming
TLDR
This work considers a class of trajectories that are similar to the central path but can be constructed to pass through any given interior feasible or infeasible point, and study their convergence.
A Polynomial Primal-Dual Dikin-Type Algorithm for Linear Programming
TLDR
A new primal-dual affine scaling method for linear programming that yields a strictly complementary optimal solution pair, and also allows a polynomial-time convergence proof.
Primal-Dual Path-Following Algorithms for Semidefinite Programming
TLDR
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.
Primal-Dual Interior-Point Methods for Self-Scaled Cones
TLDR
This paper presents efficiency estimates for several symmetric primal-dual methods that can loosely be classified as path-following methods for convex programming problems expressed in conic form when the cone and its associated barrier are self-scaled.
A Superlinearly Convergent Primal-Dual Infeasible-Interior-Point Algorithm for Semidefinite Programming
TLDR
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.
Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization
TLDR
It is argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion.
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