Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization
@article{Alizadeh1995InteriorPM,
title={Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization},
author={Farid Alizadeh},
journal={SIAM J. Optim.},
year={1995},
volume={5},
pages={13-51}
}This paper studies the semidefinite programming SDP problem, i.e., the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First the classical cone duality is reviewed as it is specialized to SDP is reviewed. Next an interior point algorithm is presented that converges to the optimal solution in polynomial time. The approach is a direct extension of Ye’s projective method…
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References
SHOWING 1-10 OF 98 REFERENCES
A primal—dual potential reduction method for problems involving matrix inequalities
- MathematicsMath. Program.
- 1995
The method is based on the theory developed by Nesterov and Nemirovsky and generalizes Gonzaga and Todd's method for linear programming and proves an overallworst-case operation count of O(m5.5L1.5).
Interior path following primal-dual algorithms
- Mathematics, Computer Science
- 1988
This work describes a primal-dual interior point algorithm for linear programming and convex quadratic programming problems which requires a total of O(n$\sp3$L) arithmetic operations and shows that the duality gap is reduced at each iteration by a factor of 1, where $\delta$ is positive and depends on some parameters associated with the objective function.
A primal-dual interior point algorithm for linear programming
- Mathematics
- 1988
This chapter presents an algorithm that works simultaneously on primal and dual linear programming problems and generates a sequence of pairs of their interior feasible solutions. Along the sequence…
The ellipsoid method and its consequences in combinatorial optimization
- Mathematics, Computer ScienceComb.
- 1981
The method yields polynomial algorithms for vertex packing in perfect graphs, for the matching and matroid intersection problems, for optimum covering of directed cuts of a digraph, and for the minimum value of a submodular set function.
Semi-Definite Matrix Constraints in Optimization
- Computer Science
- 1985
Modifications of various iterative schemes in nonlinear programming are considered in order to develop an effective algorithm for the educational testing problem, and comparative numerical experiments are described.
A Polynomial-Time Primal-Dual Affine Scaling Algorithm for Linear and Convex Quadratic Programming and Its Power Series Extension
- Mathematics, Computer ScienceMath. Oper. Res.
- 1990
An algorithm for linear and convex quadratic programming problems that uses power series approximation of the weighted barrier path that passes through the current iterate in order to find the next iterate that can be interpreted as an affine scaling algorithm in the primal-dual setup is described.
A new polynomial-time algorithm for linear programming
- Mathematics, Computer ScienceComb.
- 1984
It is proved that given a polytopeP and a strictly interior point a εP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property: the ratio of the radius of the smallest sphere with center a′, containingP′ to theradius of the largest sphere withCenter a′ contained inP′ isO(n).
Interior-point methods in parallel computation
- Mathematics30th Annual Symposium on Foundations of Computer Science
- 1989
The results extend to the weighted bipartite matching problem and to the zero-one minimum-cost flow problem, yielding O*( square root m log C) algorithms, which improves previous bounds on these problems and illustrates the importance of interior-point methods in parallel algorithm design.
Interior path following primal-dual algorithms. part I: Linear programming
- MathematicsMath. Program.
- 1989
A primal-dual interior point algorithm for linear programming problems which requires a total of O(n L) number of iterations to find the Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the logarithmic barrier function problem.