A primal-dual interior point method whose running time depends only on the constraint matrix

Vavasis, Stephen A.; Ye, Yinyu

doi:10.1007/bf02592148

Cited by 95 publications

(125 citation statements)

References 28 publications

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“…It is worthwhile to note that in this simple problem, the infeasible central path has a sharp turn in the first iteration which may happen a number of times for general problem as discussed in [38]. The arc-search method is expected to perform better than Mehrotra's method in iterations that are close to the sharp turns.…”

Section: A Simple Illustrative Examplementioning

confidence: 92%

CurveLP-A MATLAB implementation of an infeasible interior-point algorithm for linear programming

Yang¹

2016

Numer Algor

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Mehrotra's algorithm has been the most successful infeasible interior-point algorithm for linear programming since 1990. Most popular interior-point software packages for linear programming are based on Mehrotra's algorithm. This paper proposes an alternative algorithm, arc-search infeasible interior-point algorithm. We will demonstrate, by testing Netlib problems and comparing the test results obtained by arc-search infeasible interior-point algorithm and Mehrotra's algorithm, that the proposed arc-search infeasible interior-point algorithm is a more efficient algorithm than Mehrotra's algorithm.

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Section: A Simple Illustrative Examplementioning

confidence: 92%

CurveLP-A MATLAB implementation of an infeasible interior-point algorithm for linear programming

Yang¹

2016

Numer Algor

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“…The event described in Theorem 1 can be viewed as a crossover event of Vavasis and Ye [25,28]: a state-action, although we don't know which one it is, was in the initial policy but it will never stay in or return to the policies after a certain number of iterations, during the iterative process of the Simplex or simple policy-iteration method.…”

Section: Proof Of Strong Polynomialitymentioning

confidence: 99%

The Simplex and Policy-Iteration Methods Are Strongly Polynomial for the Markov Decision Problem with a Fixed Discount Rate

2011

Mathematics of OR

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140

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We prove that the classic policy-iteration method (Howard 1960), including the Simplex method (Dantzig 1947) with the most-negative-reduced-cost pivoting rule, is a strongly polynomial-time algorithm for solving the Markov decision problem (MDP) with a fixed discount rate. Furthermore, the computational complexity of the policyiteration method (including the Simplex method) is superior to that of the only known strongly polynomial-time interior-point algorithm ([28] 2005) for solving this problem. The result is surprising since the Simplex method with the same pivoting rule was shown to be exponential for solving a general linear programming (LP) problem, the Simplex (or simple policy-iteration) method with the smallest-index pivoting rule was shown to be exponential for solving an MDP regardless of discount rates, and the policy-iteration method was recently shown to be exponential for solving a undiscounted MDP. We also extend the result to solving MDPs with sub-stochastic and transient state transition probability matrices.

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“…strongly polynomial-time algorithms or polynomial arithmetic complexity) remain unknown. Furthermore, the arithmetic complexity of LP appears to be linked with a fundamental invariant measuring the intrinsic complexity of numerical solutions: the condition number (see, e.g., [VY93,CCP03]). Our results reveal a class of non-linear problems where similar subtleties arise when comparing discrete and continuous complexity.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Optimization and NP_R-completeness of certain fewnomials

Pébaÿ

Rojas

Thompson

2009

Proceedings of the 2009 Conference on Symbolic Numeric Computation

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We dedicate this paper to Tien-Yien Li on the occasion of his 65th birthday. Happy 65, TY!. ABSTRACTWe give a high precision polynomial-time approximation scheme for the supremum of any honest n-variate (n + 2)-nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and inequality checks, and are polynomial in n and the logarithm of a certain condition number. For the special case of polynomials (i.e., integer exponents), the log of our condition number is sub-quadratic in the sparse size. The best previous complexity bounds were exponential in the sparse size, even for n fixed. Along the way, we partially extend the theory of A-discriminants to real exponents and exponential sums, and find new and natural NP R -complete problems.

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A primal-dual interior point method whose running time depends only on the constraint matrix

Cited by 95 publications

References 28 publications

CurveLP-A MATLAB implementation of an infeasible interior-point algorithm for linear programming

CurveLP-A MATLAB implementation of an infeasible interior-point algorithm for linear programming

The Simplex and Policy-Iteration Methods Are Strongly Polynomial for the Markov Decision Problem with a Fixed Discount Rate

Optimization and NP_R-completeness of certain fewnomials

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