An optimal algorithm for convex minimization problems with nonconstant step-sizes

In [1], Nesterov has introduced an optimal algorithm with constant step-size, with is the Lipschitz constant of objective function. The algorithm is proved to converge with optimal rate . In this paper, we propose a new algorithm, which is allowed nonconstant step-sizes . We prove the convergence and convergence rate of the new algorithm. It is proved to have the convergence rate as the original one. The advance of our algorithm is that it is allowed nonconstant step-sizes and give us more free choices of step-sizes, which convergence rate is still optimal. This is a generalization of Nesterov's algorithm. We have applied the new algorithm to solve the problem of finding an approximate solution to the integral equation.

To the development of optimal methods for licorice seeds growing (Glycyrrhiza glabra L.) in irrigated lands of the Republic of Karakalpakstan

The research was carried out at the Nukus branch of the Tashkent State Agrarian University.The article presents the results of research on the development of optimal methods for growing licorice (Glycyrrhiza glabra L.) from seeds on salinity lands of the Republic of Karakalpakstan. In particular, the optimal rates of application of potassium geohumate for soaking licorice seeds, the rate of washing during soil desalination, the rates of mineral fertilizers and irrigation have been determined. The study of the effect of potassium geohumate on seed germination showed that soaking licorice seeds in a 0.05% solution of potassium geohumate for 3 days followed by drying the seeds in the open air provided the best germination on the 30th day after sowing and amounted to 72.6%, while the germination of licorice in the variant with the use of 0.03% potassium geohumate solution was 38.6%. The germination of the control variant was only 22.3%. It has also been established that the optimal rate of mineral fertilization for licorice is the application of 300 kg/ha of superphosphate, 150 kg/ha of potassium and 100 kg/ha of ammonium nitrate for 8 irrigations with a rate of 600 m3/ha.

Synchronization Strings: Codes for Insertions and Deletions Approaching the Singleton Bound

We introduce synchronization strings , which provide a novel way to efficiently deal with synchronization errors , i.e., insertions and deletions. Synchronization errors are strictly more general and much harder to cope with than more commonly considered Hamming-type errors , i.e., symbol substitutions and erasures. For every ε > 0, synchronization strings allow us to index a sequence with an ε -O(1) -size alphabet, such that one can efficiently transform k synchronization errors into (1 + ε)k Hamming-type errors . This powerful new technique has many applications. In this article, we focus on designing insdel codes , i.e., error correcting block codes (ECCs) for insertion-deletion channels. While ECCs for both Hamming-type errors and synchronization errors have been intensely studied, the latter has largely resisted progress. As Mitzenmacher puts it in his 2009 survey [30]: “ Channels with synchronization errors...are simply not adequately understood by current theory. Given the near-complete knowledge, we have for channels with erasures and errors...our lack of understanding about channels with synchronization errors is truly remarkable. ” Indeed, it took until 1999 for the first insdel codes with constant rate, constant distance, and constant alphabet size to be constructed and only since 2016 are there constructions of constant rate insdel codes for asymptotically large noise rates. Even in the asymptotically large or small noise regimes, these codes are polynomially far from the optimal rate-distance tradeoff. This makes the understanding of insdel codes up to this work equivalent to what was known for regular ECCs after Forney introduced concatenated codes in his doctoral thesis 50 years ago. A straightforward application of our synchronization strings-based indexing method gives a simple black-box construction that transforms any ECC into an equally efficient insdel code with only a small increase in the alphabet size. This instantly transfers much of the highly developed understanding for regular ECCs into the realm of insdel codes. Most notably, for the complete noise spectrum, we obtain efficient “near-MDS” insdel codes, which get arbitrarily close to the optimal rate-distance tradeoff given by the Singleton bound. In particular, for any δ ∈ (0,1) and ε > 0, we give a family of insdel codes achieving a rate of 1 - δ - ε over a constant-size alphabet that efficiently corrects a δ fraction of insertions or deletions.

Online regularized pairwise learning with non-i.i.d. observations

In this paper, we consider the online regularized pairwise learning (ORPL) algorithm with least squares loss function for non-independently and identically distribution (non-i.i.d.) observations. We first establish new Bennett’s inequalities for [Formula: see text]-mixing sequence, geometrically [Formula: see text]-mixing sequence, [Formula: see text]-geometrically ergodic Markov chain and uniformly ergodic Markov chain. Then we establish the convergence rates for the last iterate of the ORPL algorithm with the polynomially decaying step sizes and varying regularization parameters for non-i.i.d. observations. These established results in this paper extend the previously known results of ORPL from i.i.d. observations to the case of non-i.i.d. observations, and the established result of ORPL for [Formula: see text]-mixing can be nearly optimal rate of ORPL for i.i.d. observations with [Formula: see text]-norm.

Democracy in the Firm

The problem of collective decision-making arising from market failures is addressed using the democratic principle applied within the assembly of shareholders. A basic requirement is imposed (the Pareto principle): collective choices should not be at odds with the interests of all shareholders, as expressed by their preferences. This requirement puts bounds on what the collective can choose: it should remain within the set of averages of what the shareholders want. Further refining these bounds, a notion of political stability is proposed; it is defined with respect to (super) majority voting. One searches for the smallest rate of super majority for which a stable collective choice exists. This optimal rate is reviewed under classical assumptions from the social choice literature. It is shown how the dimensionality of the collective decision-making problem and the polarization of the electorate critically impact political stability, and hence the optimal rate of super majority.

Multi-class Multipath Routing Protocol for Low Power and Lossy Networks, with Energy Balanced Optimal Rate Assignment

The wireless sensor networks (WSNs) are special network which has purpose of gathering information in certain area. Multipath routing is the paramount path of addressing QoS and energy balancing concerns in low power and lossy networks (LLNs) especially in IoT technology. Most of the extant efforts bring the limited number of disjoint paths into play, and intersect the traffic among them pursuant to a compound metric or centralized optimization problem. This paper proposes a multi-class multipath routing protocol for LLNs (called M2RPL), that construct a braided multipath routing graph based on the standard RPL (IPv6 Routing Protocol for Low-Power and Lossy Networks) protocol, and an energy balanced optimal rate assignment mechanism (EBORA) that solves a local multi class optimization problem for minimum cost traffic rate assignment. Moreover the algorithm tries to maximize network lifetime by equalizing the energy dissipation rates of equi-level nodes. The simulation results expose the efficiency of the proposed framework, leading to an average 15% increase in lifetime, and improved QoS performance in terms of delay and reliability, compared to some well-known algorithms.

Caesarean section versus vaginal birth in the perception of woman who gave birth by both methods

The increase in the number of births by Caesarean section is a phenomenon whose global expansion is generated by numerous factors and especially by the contemporary perceptions of women regarding childbirth meeting the interests of the professionals in the field. However, the opinion of many women towards the benefits of Caesarean delivery is often not based on the experience or information from reliable sources. This study aimed at sharing the experience of women who gave birth both vaginally and by Caesarean section, focusing on their perception of these events. The study included 26 women and the conclusion of the vast majority (77%) was that natural birth is preferable and they would recommend it as the first option to future mothers. In addition, the analysis of the cases in which, on the contrary, they would recommend birth by Caesarean section (23%) revealed that they objectively had births that had not been optimally managed and hence, the recommendation for careful, professional evaluation of the conditions of birth for each case. Reaching an optimal rate of Caesarean sections is an objective that can be achieved through correct information, health education and the correct management of the cases.

Convergence and quasi-optimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver

We consider a second-order elliptic boundary value problem with strongly monotone and Lipschitz-continuous nonlinearity. We design and study its adaptive numerical approximation interconnecting a finite element discretization, the Banach–Picard linearization, and a contractive linear algebraic solver. In particular, we identify stopping criteria for the algebraic solver that on the one hand do not request an overly tight tolerance but on the other hand are sufficient for the inexact (perturbed) Banach–Picard linearization to remain contractive. Similarly, we identify suitable stopping criteria for the Banach–Picard iteration that leave an amount of linearization error that is not harmful for the residual a posteriori error estimate to steer reliably the adaptive mesh-refinement. For the resulting algorithm, we prove a contraction of the (doubly) inexact iterates after some amount of steps of mesh-refinement/linearization/algebraic solver, leading to its linear convergence. Moreover, for usual mesh-refinement rules, we also prove that the overall error decays at the optimal rate with respect to the number of elements (degrees of freedom) added with respect to the initial mesh. Finally, we prove that our fully adaptive algorithm drives the overall error down with the same optimal rate also with respect to the overall algorithmic cost expressed as the cumulated sum of the number of mesh elements over all mesh-refinement, linearization, and algebraic solver steps. Numerical experiments support these theoretical findings and illustrate the optimal overall algorithmic cost of the fully adaptive algorithm on several test cases.