Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming

This paper deals with an analysis of the Conjugate Gradient (CG) method (Hestenes and Stiefel in J Res Nat Bur Stand 49:409-436, 1952), in the presence of degenerates on indefinite linear systems. Several approaches have been proposed in the literature to issue the latter drawback in optimization frameworks, including reformulating the original linear system or recurring to approximately solving it. All the proposed alternatives seem to rely on algebraic considerations, and basically pursue the idea of improving numerical efficiency. In this regard, here we sketch two separate analyses for the possible CG degeneracy. First, we start detailing a more standard algebraic viewpoint of the problem, suggested by planar methods. Then, another algebraic perspective is detailed, relying on a novel recently proposed theory, which includes an additional number, namely grossone. The use of grossone allows to work numerically with infinities and infinitesimals. The results obtained using the two proposed approaches perfectly match, showing that grossone may represent a fruitful and promising tool to be exploited within Nonlinear Programming

How grossone can be helpful to iteratively compute negative curvature directions

We consider an iterative computation of negative curvature directions, in large scale optimization frameworks. We show that to the latter purpose, borrowing the ideas in [1, 3] and [4], we can fruitfully pair the Conjugate Gradient (CG) method with a recently introduced numerical approach involving the use of grossone [5]. In particular, though in principle the CG method is well-posed only on positive definite linear systems, the use of grossone can enhance the performance of the CG, allowing the computation of negative curvature directions, too. The overall method in our proposal significantly generalizes the theory proposed for [1] and [3], and straightforwardly allows the use of a CG-based method on indefinite Newton’s equations

Bound state vector solitons with locked and precessing states of polarization

We report experimental observation of new tightly and loosely bound state vector solitons with locked and precessing states of polarization in a carbon nanotube mode locked fiber laser in the anomalous dispersion regime

SIZE-CONTROLLED PREPARATION OF GOLD NANOPARTICLES BY USING GUANOSINE MONOPHOSPHATE

This study was financially supported by the Russian Foundation for Basic Research (project No 20-33-90296\20)

Slow deterministic vector rogue waves

For an erbium-doped fiber laser mode-locked by carbon nanotubes, we demonstrate experimentally and theoretically a new type of the vector rogue waves emerging as a result of the chaotic evolution of the trajectories between two orthogonal states of polarization on the Poincare sphere. In terms of fluctuation induced phenomena, by tuning polarization controller for the pump wave and in-cavity polarization controller, we are able to control the Kramers time, i.e. the residence time of the trajectory in vicinity of each orthogonal state of polarization, and so can cause the rare events satisfying rogue wave criteria and having the form of transitions from the state with the long residence time to the state with a short residence time

Objective and violation upper bounds on a DIRECT-filter method for global optimization

This paper addresses the problem of solving a constrained global optimization problem using a modification of the DIRECT method that incorporates the filter methodology to simultaneously minimize the objective function and the constraints violation. Thus, in the “Selection” step of the herein proposed DIRECT-filter algorithm, the hyperrectangles are classified in four categories and subsequently handled separately. The new algorithm also imposes upper bounds on the objective function and constraints violation aiming to discard some hyperrectangles from the process of identifying the potentially optimal ones. A heuristic to avoid the exploration of the hyperrectangles that have been mostly divided is also implemented. Preliminary numerical experiments are carried out to show the effectiveness of the imposed upper bounds on the objective and violation as well as the goodness of the heuristic.The authors wish to thank two anonymous referees for theircomments and suggestions to improve the paper. This work has been supported by FCT{ Fundação para a Ciência e Tecnologia within the Projects Scope: UID/CEC/00319/2019 and UID/MAT/00013/2013

Penalty-based heuristic direct method for constrained global optimization

This paper is concerned with an extension of the heuristic DIRECT method, presented in[8], to solve nonlinear constrained global optimization (CGO) problems. Using a penalty strategy based on a penalty auxiliary function, the CGO problem is transformed into a bound constrained problem. We have analyzed the performance of the proposed algorithm using fixed values of the penalty parameter, and we may conclude that the algorithm competes favourably with other DIRECT-type algorithms in the literature.The authors wish to thank two anonymous referees for their comments and suggestions to improve the paper. This work has been supported by FCT – Fundação para a Ciência e Tecnologia within the R&D Units Project Scope: UIDB/00319/2020, UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM

KiDS-SQuaD: The KiDS Strongly lensed Quasar Detection project

New methods have been recently developed to search for strong gravitational lenses, in particular lensed quasars, in wide-field imaging surveys. Here, we compare the performance of three different, morphology- and photometry- based methods to find lens candidates over the Kilo-Degree Survey (KiDS) DR3 footprint (440 deg$^2$). The three methods are: i) a multiplet detection in KiDS-DR3 and/or Gaia-DR1, ii) direct modeling of KiDS cutouts and iii) positional offsets between different surveys (KiDS-vs-Gaia, Gaia-vs-2MASS), with purpose-built astrometric recalibrations. The first benchmark for the methods has been set by the recovery of known lenses. We are able to recover seven out of ten known lenses and pairs of quasars observed in the KiDS DR3 footprint, or eight out of ten with improved selection criteria and looser colour pre-selection. This success rate reflects the combination of all methods together, which, taken individually, performed significantly worse (four lenses each). One movelty of our analysis is that the comparison of the performances of the different methods has revealed the pros and cons of the approaches and, most of all, the complementarities. We finally provide a list of high-grade candidates found by one or more methods, awaiting spectroscopic follow-up for confirmation. Of these, KiDS 1042+0023 is to our knowledge the first confirmed lensed quasar from KiDS, exhibiting two quasar spectra at the same source redshift at either sides of a red galaxy, with uniform flux-ratio $f\approx1.25$ over the wavelength range $0.45\mu\mathrm{m}<\lambda<0.75\mu\mathrm{m}.$Comment: 12 pages, 4 figures, 4 tables, accepted for publication in MNRA

Mulitiscale spatiotemporal structures in mode-locked fiber lasers

Using an Er-doped fiber laser as a test bed, here we for the first time experimentally demonstrate the simultaneous effect of the fast scale (round-trip time) and slow scale (thousands round-trip time) instabilities on the emergence of breathers similar to the Akhmediev breathers, Peregrine solitons, and partially mode-locked chaotic solitons. The anomalous statistics of the laser output power justifies the connection of the observed spatiotemporal structures with bright and dark rogue waves. Apart from the interest in laser physics for revealing mechanisms of the multiscale dynamics, the obtained results can be of fundamental interest for studying spatiotemporal patterns induced by the interplay of the mechanisms mentioned above in various distributed systems

Unrepeatered 240-km 64-QAM transmission using distributed raman amplification over SMF fiber

We present a theoretical and experimental investigation of unrepeatered transmission over standard single-mode fiber (SMF-28) using several schemes of distributed Raman amplification, including first, second, and dual order. In order to further extend the transmission distance, we utilize advanced bidirectional higher-order ultra-long Raman fiber laser-based amplification, where we use fiber Bragg gratings (FBGs) to reflect Stokes-shifted light from the secondary pumps. Our work demonstrates the possibility of transmission up to 240-km span length with a total span loss of 52.7 dB. Here, we use a 28-Gbaud signal using a 64-quadrature amplitude modulation (QAM) modulation format. Our results highlight the contribution of nonlinear compensation using digital back propagation in a digital signal processor (DSP) code at the receiver