On the centralization of the circumcentered-reflection method

This paper is devoted to deriving the first circumcenter iteration scheme that does not employ a product space reformulation for finding a point in the intersection of two closed convex sets. We introduce a so-called centralized version of the circumcentered-reflection method (CRM). Developed with the aim of accelerating classical projection algorithms, CRM is successful for tracking a common point of a finite number of affine sets. In the case of general convex sets, CRM was shown to possibly diverge if Pierra's product space reformulation is not used. In this work, we prove that there exists an easily reachable region consisting of what we refer to as centralized points, where pure circumcenter steps possess properties yielding convergence. The resulting algorithm is called centralized CRM (cCRM). In addition to having global convergence, cCRM converges linearly under an error bound condition, and superlinearly if the two target sets are so that their intersection have nonempty interior and their boundaries are locally differentiable manifolds. We also run numerical experiments with successful results.Comment: 29 pages with 7 figure

A successive centralized circumcenter reflection method for the convex feasibility problem

In this paper we present the successive centralization of the circumcenter reflection scheme with several control sequences for solving the convex feasibility problem in Euclidean space. Assuming that a standard error bound holds, we prove the linear convergence of the method with the most violated constraint control sequence. Under additional smoothness assumptions, we prove the superlinear convergence. Numerical experiments confirm the efficiency of our method

Circumcentric directions of cones

Generalized circumcenters have been recently introduced and employed to speed up classical projection-type methods for solving feasibility problems. In this note, circumcenters are enforced in a new setting; they are proven to provide inward directions to sets given by convex inequalities. In particular, we show that circumcentric directions of finitely generated cones belong to the interior of their polars. We also derive a measure of interiorness of the circumcentric direction, which then provides a special cone of search directions, all being feasible to the convex region under consideration.Comment: 1

Completamento de Matrizes usando Métodos de Projeção

TCC(graduação) - Universidade Federal de Santa Catarina. Campus Blumenau. Licenciatura em MatemáticaNeste estudo buscamos encontrar soluções para o problema de completamento de matrizes de distância Euclidianas, utilizando métodos de projeções. As matrizes de distância são muito importantes no estudo de problemas de geometria de distâncias, pois permitem descrever as distâncias entre os pontos como entradas de uma matriz. O completamento de uma matriz é a sua reconstituição a partir de um subconjunto de seus elementos conhecidos. Este também pode ser compreendido como um problema de viabilidade, em que a solução encontra-se na intersecção de conjuntos. Para encontrar a solução deste problema, foram estudados e implementados os seguintes métodos de projeções: o método das projeções alternadas, Douglas-Rachford, reflexões circuncentradas e o método de reflexões circuncentradas centralizado. Estes métodos foram minuciosamente comparados, quando resolvendo o problema de completamento de matrizes, por uma série de experimentos numéricos.In this study we seek to find solutions to the problem of completing Euclidean distance matrices using projection methods. Distance matrices are very important in the study of distance geometry problems, as they allow us to describe the distances between points as entries of a matrix. The completion of a matrix is its reconstruction from a subset of its elements. It can also be seen as a feasibility problem. To find the solution of this feasibility problem, the following projection methods were studied and implemented: the method of alternating projections, Douglas-Rachford method , and the circumcentered-reflections and the centralized circumcetered-reflections methods. Those methods were thoroughly compared, when solving the matrix completing problem, through a series of numerical experiments

Strategies for Enhancing Product Yield: Design of Experiments (DOE) for <em>Escherichia coli</em> Cultivation

E. coli is considered one of the best model organism for biopharmaceutical production by fermentation. Its utility in process development is employed to develop various vaccines, metabolites, biofuels, antibiotics and synthetic molecules in large amounts based on the amount of yield in shake flasks, bioreactors utilised by batch, fed-batch and continuous mode. Production of the desired molecule is facilitated in the bioreactor by employing strategies to increase biomass and optimised yield. The fermentation is a controlled process utilising media buffers, micronutrients and macronutrients, which is not available in a shake flask. To maximise the production temperature, dissolved oxygen (aerobic), dissolved nitrogen (anaerobic), inducer concentration, feed or supplementation of nutrients is the key to achieving exponential growth rate and biomass. Design of experiments (DOE) is critical for attaining maximum gain, in cost-effective manner. DOE comprises of several strategies likewise Plakett-Burman., Box–Behnken, Artificial Neural Network, combination of these strategies leads to reduction of cost of production by 2–8 times depending on molecules to be produced. Further minimising downstream process for quickly isolation, purification and enrichment of the final product