Probability Bounds for Polynomial Functions in Random Variables

Abstract

Random sampling is a simple but powerful method in statistics and the design of randomized algorithms. In a typical application, random sampling can be applied to estimate an extreme value, say maximum, of a function f over a set S ⊆ R n . To do so, one may select a simpler (even finite) subset S 0 ⊆ S, randomly take some samples over S 0 for a number of times, and pick the best sample. The hope is to find a good approximate solution with reasonable chance. This paper sets out to present a number of scenarios for f , S and S 0 where certain probability bounds can be established, leading to a quality assurance of the procedure. In our setting, f is a multivariate polynomial function. We prove that if f is d-th order homogeneous polynomial in n variables and F is its corresponding super-symmetric tensor, and ξ i (1 ≤ i ≤ n) are i.i.d. Bernoulli random variables taking 1 or −1 with equal probability, then Prob f (ξ 1 , ξ 2 , · · · , ξ n ) ≥ c 1 n − d 2 F 1 ≥ c 2 , where c 1 , c 2 > 0 are two universal constants. Several new inequalities concerning probabilities of the above nature are presented in this paper. Moreover, we show that the bounds are tight in most cases. Applications of our results in optimization are discussed as well.

Approximation methods for complex polynomial optimization

“…uniform distribution on Ω m . As each component of x ξ defined by (8) belongs to conv (Ω m ), by Lemma 5.2, there exists y ∈ Ω n m such that…”

“…So [23] further considered spherically constrained homogeneous polynomial optimization and proposed a deterministic algorithm with an improved approximation ratio. For most recent development on approximation algorithms for homogeneous polynomial optimization, we refer the interested readers to [8,13].…”

“…To the best of our knowledge, there is no result on approximation methods for general degree complex polynomial optimization as such, except for the practice of transforming a general high degree complex polynomial to the real case by doubling the problem dimension, and then resorting to the existing approximation algorithms for the real-valued polynomials [9,10,11,23,8,13]. The latter approach, however, may lose the handle on the structure of the problem, hence misses nice properties of the complex polynomial functions.…”

A note on semidefinite programming relaxations for polynomial optimization over a single sphere

“…Constructx = ξ/ ξ 2 , then we have Prob(f (x) 3 2/θf min (HQQS)) θ/2 with θ := 1/960. As far as we know, the best approximation ratio for quartic polynomial optimization with a single sphere constraint is O(n/ ln n) in [12,25], and the ratio under consideration is relative ratio. The merit of our approximation scheme is that both approximation ratios obtained by our two algorithms are absolute ones.…”

Approximation algorithms for optimization of real-valued general conjugate complex forms

“…Since polynomial optimization problems are generally NP-hard, various polynomialtime approximation algorithms have been proposed for solving certain classes of high-degree polynomial optimization models-a summary of research can be found in the monograph of Li et al [21]. Improvements on approximation ratios of these polynomial optimization models have been recently made by He et al [8] and Hou and So [12]. In the context of complex polynomial optimization, approximation algorithms are mostly proposed for the quadratic models.…”