# A New Primal-Dual Interior-Point Method for Semidefinite Programming

@inproceedings{Alizadeh1994ANP, title={A New Primal-Dual Interior-Point Method for Semidefinite Programming}, author={Farid Alizadeh and Jean Pierre Haeberly and Michael L. Overton}, year={1994} }

The semidefinite programming problem (SDP) is: min tr CX s.t. tr A{sub i}X = b{sub i}, i = 1, {hor_ellipsis}, m, and X {>=} 0. Here C and A{sub i} are fixed symmetric matrices and X {>=} 0 is a semidefinite constraint on the unknown symmetric matrix variable X. The dual of SDP is: max b{sup T} y s.t. Z + {Sigma}{sub i=1}{sup m} y{sub i}A{sub i} = C and Z {>=} 0. Interior point methods for SDP have been developed by Nesterov and Nemirovskii, Alizadeh, Vandenberghe and Boyd, and others. Primal…

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## 59 Citations

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

- MathematicsSIAM J. Optim.
- 1998

The XZ+ZX method is more robust with respect to its ability to step close to the boundary, converges more rapidly, and achieves higher accuracy than other methods considered, including Mehrotra predictor-corrector variants and issues of numerical stability.

Primal-Dual Interior-Point Algorithms for Semidefinite Optimization Based on a Simple Kernel Function

- Computer Science, MathematicsJ. Math. Model. Algorithms
- 2005

This paper presents a primal-dual interior-point algorithm for SDO problems based on a simple kernel function which was first presented at the Proceedings of Industrial Symposium and Optimization Day, Australia, November 2002; the function is not self-regular.

Primal-Dual Interior-Point Methods for Second-Order Conic Optimization Based on Self-Regular Proximities

- Computer ScienceSIAM J. Optim.
- 2002

Based on the so-called self-regular proximity functions, new primal-dual Newton methods for solving SOCO problems are proposed and it will be shown that these new large-update IPMs for SOCO enjoy polynomial iteration bounds analogous to those of their LO and SDO cousins.

Strengthened existence and uniqueness conditions for search directions in semidefinite programming

- Mathematics
- 2005

A Primitive Interior Point Algorithm for Semidefinite Programs in

- Computer Science
- 1994

The aim of this working paper is to present a simple and basic version of Kojima, Shindoh and Hara's primal-dual interior-point algorithm for SDPs together with a computer program of the algorithm in Mathematica, placing the emphasis on easy understanding of the fundamental idea of the algorithms without going into its theoretical details.

Semidefinite programming for assignment and partitioning problems

- Computer Science
- 1998

An efficient "partial infeasible" primal-dual interior-point algorithm is developed by using a conjugate gradient method and by taking advantage of the special data structure of the SDP relaxation, which plays a significant role in simplifying the problem.

An Interior Point Method for Semidefinite Programming and Max-Cut Bounds

- Computer Science
- 1994

The eigenvalue upper bound based on the maximal eigen value of the Laplacian of the graph with the polyhedral upper bound delivered by triangle inequalities is combined and the bounds show considerable improvements to previous results on complete graphs.

A Simplified / Improved HKM Direction for Certain Classes of Semidefinite Programming

- Mathematics
- 2002

Semidefinite Programming (SDP) provides strong bounds for many NP-hard combinatorial problems. Arguably the most popular/efficient search direction for solving SDPs using a primal-dual interior point…

Semidefinite Programming Relaxations for the Quadratic Assignment Problem

- MathematicsJ. Comb. Optim.
- 1998

These new relaxations, and their duals, satisfy the Slater constraint qualification, and so can be solved numerically using primal-dual interior-point methods.

A Short Course on Semidefinite Programming ( in order of appearance )

- Mathematics
- 2004

Convex Program general abstract convex program is (CP ) μ := inf f(x) subject to G(x) L 0 x K 0, f : X → real valued convex function on K; X,Y Banach spaces; x K y partial order induced convex cone…

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