Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere

We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere, when the probability density function is a sum-of-squares polynomial of degree at most 2r with respect to the surface measure. We show that the exact rate of convergence is Theta(1/r^2), and explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.Comment: 14 pages, 2 figure

Worst-case examples for Lasserre's measure--based hierarchy for polynomial optimization on the hypercube

We study the convergence rate of a hierarchy of upper bounds for polynomial optimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], and a related hierarchy by De Klerk, Hess and Laurent [SIAM J. Optim. 27(1), (2017) pp. 347-367]. For polynomial optimization over the hypercube, we show a refined convergence analysis for the first hierarchy. We also show lower bounds on the convergence rate for both hierarchies on a class of examples. These lower bounds match the upper bounds and thus establish the true rate of convergence on these examples. Interestingly, these convergence rates are determined by the distribution of extremal zeroes of certain families of orthogonal polynomials.Comment: 17 pages, no figure

An error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution

We study the minimization of fixed-degree polynomials over the simplex. This problem is well-known to be NP-hard, as it contains the maximum stable set problem in graph theory as a special case. In this paper, we consider a rational approximation by taking the minimum over the regular grid, which consists of rational points with denominator $r$ (for given $r$). We show that the associated convergence rate is $O(1/r^2)$ for quadratic polynomials. For general polynomials, if there exists a rational global minimizer over the simplex, we show that the convergence rate is also of the order $O(1/r^2)$. Our results answer a question posed by De Klerk et al. (2013) and improves on previously known $O(1/r)$ bounds in the quadratic case.Comment: 17 page

The matter distribution in z ~ 0.5 redshift clusters of galaxies. II : The link between dark and visible matter

We present an optical analysis of a sample of 11 clusters built from the EXCPRES sample of X-ray selected clusters at intermediate redshift (z ~ 0.5). With a careful selection of the background galaxies we provide the mass maps reconstructed from the weak lensing by the clusters. We compare them with the light distribution traced by the early-type galaxies selected along the red sequence for each cluster. The strong correlations between dark matter and galaxy distributions are confirmed, although some discrepancies arise, mostly for merging or perturbed clusters. The average M/L ratio of the clusters is found to be: M/L_r = 160 +/- 60 in solar units (with no evolutionary correction), in excellent agreement with similar previous studies. No strong evolutionary effects are identified even if the small sample size reduces the significance of the result. We also provide a individual analysis of each cluster in the sample with a comparison between the dark matter, the galaxies and the gas distributions. Some of the clusters are studied for the first time in the optical.Comment: 25 pages, 9 figues + 11 figures in Annex, 4 tables. Accepted for publication in A&A. 1 reference correcte

Erosion assessment in the middle Kali Gandaki (Nepal ) : A sediment budget approach

International audienceActive mountains supply the largest sediment fl uxes experienced on earth. At mountain range scale, remote sensingapproaches, sediments provenance or stream power law analyses, collectively provide rough long-term estimatesof total erosion. Erosion is indeed controlled by rock uplift and climate, hence by a wide range of processes(detachment, transport and deposition), all operating within drainage basin units, yet with time and spatial patternsthat are quite complex at local scale. We focus on the Kali Gandaki valley, along the gorge section across theHigher Himalaya (e.g. from Kagbeni down to Tatopani). Along this reach, we identify sediment sources, storesand sinks, and consider hillslope interactions with valley floor, in particular valley damming at short and longertime scales, and their impact on sediment budgets and fluxes. A detailed sediment budget is presented, constrainedby available dates and/or relative chronology, ranging from several 10 kyr to a few decades. Obtained resultsspan over two orders of magnitude that can best be explained by the type and magnitude of erosional processesinvolved. We show that if large landslides contribute signifi cantly to the denudation history of active mountainrange, more frequent, medium to small scales landslides are in fact of primary concern for Himalayan population

Robust Mobile Object Tracking Based on Multiple Feature Similarity and Trajectory Filtering

This paper presents a new algorithm to track mobile objects in different scene conditions. The main idea of the proposed tracker includes estimation, multi-features similarity measures and trajectory filtering. A feature set (distance, area, shape ratio, color histogram) is defined for each tracked object to search for the best matching object. Its best matching object and its state estimated by the Kalman filter are combined to update position and size of the tracked object. However, the mobile object trajectories are usually fragmented because of occlusions and misdetections. Therefore, we also propose a trajectory filtering, named global tracker, aims at removing the noisy trajectories and fusing the fragmented trajectories belonging to a same mobile object. The method has been tested with five videos of different scene conditions. Three of them are provided by the ETISEO benchmarking project (http://www-sop.inria.fr/orion/ETISEO) in which the proposed tracker performance has been compared with other seven tracking algorithms. The advantages of our approach over the existing state of the art ones are: (i) no prior knowledge information is required (e.g. no calibration and no contextual models are needed), (ii) the tracker is more reliable by combining multiple feature similarities, (iii) the tracker can perform in different scene conditions: single/several mobile objects, weak/strong illumination, indoor/outdoor scenes, (iv) a trajectory filtering is defined and applied to improve the tracker performance, (v) the tracker performance outperforms many algorithms of the state of the art

Worst-case examples for Lasserre's measure--based hierarchy for polynomial optimization on the hypercube

We study the convergence rate of a hierarchy of upper bounds for polynomial optimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], and a related hierarchy by De Klerk, Hess and Laurent [SIAM J. Optim. 27(1), (2017) pp. 347-367]. For polynomial optimization over the hypercube, we show a refined convergence analysis for the first hierarchy. We also show lower bounds on the convergence rate for both hierarchies on a class of examples. These lower bounds match the upper bounds and thus establish the true rate of convergence on these examples. Interestingly, these convergence rates are determined by the distribution of extremal zeroes of certain families of orthogonal polynomials.Comment: 17 pages, no figure

ACP-EEC CONSULTATIVE ASSEMBLY JOINT COMMITTEE RESOLUTION on cultural cooperation between the ACP States and the EEC.

We consider the problem of minimizing a continuous function f over a compact set K. We compare the hierarchy of upper bounds proposed by Lasserre [Lasserre JB (2011) A new look at nonnegativity on closed sets and polynomial optimization. SIAM J. Optim. 21(3):864–885] to bounds that may be obtained from simulated annealing. We show that, when f is a polynomial and K a convex body, this comparison yields a faster rate of convergence of the Lasserre hierarchy than what was previously known in the literature

Worst-case examples for Lasserre's measure-based hierarchy for polynomial optimization on the hypercube

We study the convergence rate of a hierarchy of upper bounds for polynomial optimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], and a related hierarchy by De Klerk, Hess and Laurent [SIAM J. Optim. 27(1), (2017) pp. 347-367]. For polynomial optimization over the hypercube, we show a refined convergence analysis for the first hierarchy. We also show lower bounds on the convergence rate for both hierarchies on a class of examples. These lower bounds match the upper bounds and thus establish the true rate of convergence on these examples. Interestingly, these convergence rates are determined by the distribution of extremal zeroes of certain families of orthogonal polynomials