A new polynomial-time algorithm for linear programming
Combinatorica
Recovering optimal dual solutions on Karmarkar's polynomial algorithm for linear programming
Mathematical Programming: Series A and B
A polynomial-time algorithm, based on Newton's method, for linear programming
Mathematical Programming: Series A and B
An algorithm for linear programming which requires O((m+n)n2 + (m+n)1.5n)L) arithmetic operations
Mathematical Programming: Series A and B
A random polynomial-time algorithm for approximating the volume of convex bodies
Journal of the ACM (JACM)
A new algorithm for minimizing convex functions over convex sets
Mathematical Programming: Series A and B
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Random walks and an O*(n5) volume algorithm for convex bodies
Random Structures & Algorithms
Self-scaled barriers and interior-point methods for convex programming
Mathematics of Operations Research
Improved bounds for sampling contingency tables
Random Structures & Algorithms
SIAM Journal on Computing
Fast Algorithms for Logconcave Functions: Sampling, Rounding, Integration and Optimization
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
The geometry of logconcave functions and sampling algorithms
Random Structures & Algorithms
Projective re-normalization for improving the behavior of a homogeneous conic linear system
Mathematical Programming: Series A and B
A Randomized Cutting Plane Method with Probabilistic Geometric Convergence
SIAM Journal on Optimization
An empirical evaluation of walk-and-round heuristics for mixed integer linear programs
Computational Optimization and Applications
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Let K be a polytope in Rn defined by m linear inequalities. We give a new Markov Chain algorithm to draw a nearly uniform sample from K. The underlying Markov Chain is the first to have a mixing time that is strongly polynomial when started from a "central" point x0. If s is the supremum over all chords pq passing through x0 of (|p-x0|)/(|q-x0|) and ε is an upper bound on the desired total variation distance from the uniform, it is sufficient to take O(m n( n log (s m) + log 1/ε)) steps of the random walk. We use this result to design an affine interior point algorithm that does a single random walk to solve linear programs approximately. More precisely, suppose Q = {z | Bz ≤ 1} contains a point z such that cT z ≥ d and r := supz ∈ Q |Bz| + 1, where B is an m x n matrix. Then, after τ = O(mn (n ln(mr/ε) + ln 1/δ)) steps, the random walk is at a point xτ for which cT xτ ≥ d(1-ε) with probability greater than 1-δ. The fact that this algorithm has a run-time that is provably polynomial is notable since the analogous deterministic affine algorithm analyzed by Dikin has no known polynomial guarantees.