Smale’s 17th problem: Average polynomial time to compute affine and projective solutions

Beltrán, Carlos; Pardo, Luis Miguel

doi:10.1090/s0894-0347-08-00630-9

Cited by 62 publications

(112 citation statements)

References 32 publications

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“…And in the case at hand this departure turned out to pay off. The average (over f ) of the expected (over (g, ζ)) number of iterations of the algorithm proposed in [5] is O(n 5 N 2 D 3 log D). One of the most notable features of the ideas introduced by Beltrán and Pardo is the use of a measure on the space of pairs (g, ζ) which is friendly enough to perform a probabilistic analysis while, at the same time, does allow for efficient sampling.…”

Section: Introductionmentioning

confidence: 99%

On a problem posed by Steve Smale

2011

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Abstract. The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial, on the average, in the size N of the input system. A partial solution to this problem was given by Carlos Beltrán and Luis Miguel Pardo who exhibited a randomized algorithm doing so. In this paper we further extend this result in several directions. Firstly, we exhibit a linear homotopy algorithm that efficiently implements a non-constructive idea of Mike Shub. This algorithm is then used in a randomized algorithm, call it LV,à la Beltrán-Pardo. Secondly, we perform a smoothed analysis (in the sense of Spielman and Teng) of algorithm LV and prove that its smoothed complexity is polynomial in the input size and σ −1 , where σ controls the size of of the random perturbation of the input systems. Thirdly, we perform a condition-based analysis of LV. That is, we give a bound, for each system f , of the expected running time of LV with input f . In addition to its dependence on N this bound also depends on the condition of f . Fourthly, and to conclude, we return to Smale's 17th problem as originally formulated for deterministic algorithms. We exhibit such an algorithm and show that its average complexity is N O(log log N) . This is nearly a solution to Smale's 17th problem.

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Section: Introductionmentioning

confidence: 99%

On a problem posed by Steve Smale

2011

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“…In [5,6], this result is used to prove that randomized linear homotopy paths require a small (polynomial in the size of the input) number of homotopy steps to run, on average. The algorithm in [6] works as follows: first, an initial pair (g, η 0 ) is chosen using a certain randomized procedure. Then, the path-following method of [27, Theorem 6.1] is used to approximate the solution path η t associated to the linear homotopy f t = (1 − t)g + t f and thus find an approximate zero of the input system f .…”

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confidence: 99%

“…The average running time of the algorithm in [6] is already polynomial in the size of the input. However, Mike Shub has pointed out in [22] that the path-following method can be done much faster.…”

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confidence: 99%

A continuation method to solve polynomial systems and its complexity

Beltrán

2010

Numer. Math.

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In a recent work Shub (Found. Comput. Math. 9:171-178, 2009), Shub obtained a new upper bound for the number of steps needed to continue a known zero η 0 of a system f 0 , to a zero η T of an input system f T , following the path of pairsHe proved that if one can choose the step-size in an optimal way, then the number of steps is essentially bounded by the length of the path of ( f t , η t ) in the so-called condition metric. However, the proof of that result in Shub (Found. Comput. Math. 9:171-178, 2009) is not constructive. We give an explicit description of an algorithm which attains that complexity bound, including the choice of step-size.

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“…x, v R = 0 which is equivalent to the complex equation x, v C = 0 based on the observation from Equation (9). Therefore the horizontal space can be characterized as…”

Section: Path Tracking Methodsmentioning

confidence: 99%

“…In our algorithm, we chose the predictor-corrector scheme for the path tracking. There is a similar path tracking algorithm using "Projective Newton's iterations" alone which has been intensively studied theoretically such as in [5], [6], [7], [8], [9], [10], [11], [12], [15], [25], [26], [28], [29], [30], and [31], to list a few. In Section 11, numerical results are presented in comparing these two approaches.…”

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confidence: 99%

Spherical projective path tracking for homotopy continuation methods

Chen

2012

Communications in Information and Systems

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Abstract. Solving systems of polynomial equations is an important problem in mathematics with a wide range of applications in many fields. The homotopy continuation method is a large class of reliable and efficient numerical methods for solving systems of polynomial equations. An essential component in the homotopy continuation method is the path tracking algorithm for tracking smooth paths of one real dimension. In this regard, "divergent paths" pose a tough challenge as the tracking of such paths is generally impossible. The existence of such paths is, in part, caused by C n , the space in which homotopy methods usually operate, being non-compact. A well known remedy is to operate inside the complex projective space CP n instead. Path tracking inside CP n is the focus of this article. Taking the Riemannian geometry of CP n into account, we derive the basic algorithm for projective path tracking using the sphere, S 2n+1 , as the model of computation. Remarkable results from numerical experiments using this method are presented.

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Smale’s 17th problem: Average polynomial time to compute affine and projective solutions

Cited by 62 publications

References 32 publications

On a problem posed by Steve Smale

On a problem posed by Steve Smale

A continuation method to solve polynomial systems and its complexity

Spherical projective path tracking for homotopy continuation methods

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