scholarly journals Approximation of the Constant in a Markov-Type Inequality on a Simplex Using Meta-Heuristics

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 264
Author(s):  
Grzegorz Sroka ◽  
Mariusz Oszust
Keyword(s):  
Optimization Problem ◽  
Numerical Solutions ◽  
Minimal Polynomial ◽  
Equilibrium Measure ◽  
Type Inequality ◽  
Bilevel Optimization ◽  
Energy Integral ◽  
Upper Approximation ◽  
Markov Type ◽  
Extreme Problem

Markov-type inequalities are often used in numerical solutions of differential equations, and their constants improve error bounds. In this paper, the upper approximation of the constant in a Markov-type inequality on a simplex is considered. To determine the constant, the minimal polynomial and pluripotential theories were employed. They include a complex equilibrium measure that solves the extreme problem by minimizing the energy integral. Consequently, examples of polynomials of the second degree are introduced. Then, a challenging bilevel optimization problem that uses the polynomials for the approximation was formulated. Finally, three popular meta-heuristics were applied to the problem, and their results were investigated.

Design Optimization of an L-Shaped Extrusion Die

2009 ◽  
Author(s):  
Oktay Yilmaz ◽  
Hasan Gunes ◽  
Kadir Kirkkopru
Keyword(s):  
Simulated Annealing ◽  
Optimization Problem ◽  
Numerical Solutions ◽  
Latin Hypercube Sampling ◽  
Kriging Model ◽  
Narrow Channel ◽  
Geometric Parameters ◽  
Die Design ◽  
Dimensional Parameter ◽  
Large Channel

It is an important problem in the polymer extrusion of complex profiles to balance the flow at the die exit. In this paper, we employ simulated annealing-kriging meta-algorithm to optimize the geometric parameters of a die channel to obtain a uniform exit velocity distribution. Design variables for our optimization problem involve the suitable geometric parameters for the die design, which are the thickness of the large channel and the length of the narrow channel. Die balance is based on the deviation of the velocity with respect to the average velocity at the die exit. So the cost function for the optimization problem involves the minimization of this deviation. For the design of numerical experiments, we use Latin Hypercube Sampling (LHS) to construct the kriging model. Then, based on the LHS points, the numerical solutions are performed using Polyflow, a commercial software based on the finite element method and is specifically designed to simulate the flow and heat transfer of non-newtonian, viscoelastic fluids. In our simulations, a HDPE (high density polyethylene) is used as extrusion material. Having obtained numerical simulations for N = 60 LHS points in two-dimensional parameter space (t and L), the optimization of these parameters is carried out by Simulated Annealing (SA) method in conjunction with kriging model. We show that kriging model employed in SA algorithm can be used to optimize the die geometry.

scholarly journals Lupaş-type inequality and applications to Markov-type inequalities in weighted Sobolev spaces

2020 ◽  
pp. 1950022
Author(s):  
Francisco Marcellán ◽  
José M. Rodríguez
Keyword(s):  
Sobolev Spaces ◽  
Jacobi Polynomials ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
The Other ◽  
Main Role ◽  
Markov Type ◽  
Sobolev Orthogonal Polynomials ◽  
Approximation Problems ◽  
Connection Formulas

Weighted Sobolev spaces play a main role in the study of Sobolev orthogonal polynomials. In particular, analytic properties of such polynomials have been extensively studied, mainly focused on their asymptotic behavior and the location of their zeros. On the other hand, the behavior of the Fourier–Sobolev projector allows to deal with very interesting approximation problems. The aim of this paper is twofold. First, we improve a well-known inequality by Lupaş by using connection formulas for Jacobi polynomials with different parameters. In the next step, we deduce Markov-type inequalities in weighted Sobolev spaces associated with generalized Laguerre and generalized Hermite weights.

Optimality results for a specific bilevel optimization problem

Optimization ◽  
2011 ◽  
Vol 60 (7) ◽  
pp. 813-822 ◽  
Author(s):  
S. Dempe ◽  
N. Gadhi
Keyword(s):  
Optimization Problem ◽  
Bilevel Optimization

Anticipatory Optimization of Traffic Control

2000 ◽  
Vol 1725 (1) ◽  
pp. 109-115 ◽  
Author(s):  
Henk J. van Zuylen ◽  
Henk Taale
Keyword(s):  
Traffic Control ◽  
Optimization Problem ◽  
Bilevel Optimization ◽  
Simulation Studies ◽  
Traffic Flows ◽  
System Optimum ◽  
Time Model ◽  
Traffic Conditions ◽  
Made In ◽  
Small Disturbances

Traffic control and travelers’ behavior are two mutually influential processes with different objectives. Decisions made in traffic control influence travelers’ possibilities in choosing their preferred mode, route, and time of departure; and the choices made by travelers influence the optimization possibilities for traffic control. This research presents the results of simulation studies and a mathematical analysis of this bilevel optimization problem. Under certain conditions, multiple stable situations are possible, but some of these situations are sensitive to small disturbances by which the system moves away from the original equilibrium state. There appears to be a nonlinear relationship between system parameters and the character and location of the equilibrium situations. The details of the travel time model appear to have a large influence. If road authorities want to optimize traffic control, they have to anticipate the reaction of travelers. This makes the optimization process much more complicated. Iterative optimization, where traffic control is adjusted as soon as traffic conditions change, generally does not lead to a system optimum. Methods are therefore necessary that allow for the optimization of traffic control while taking into account that traffic flows will change as a result of traffic control.

A V. A. Markov type inequality in Lp

1983 ◽  
Vol 22 (2) ◽  
pp. 1226-1231
Author(s):  
A. N. Podkorytov ◽  
E. M. Dyn'kin
Keyword(s):  
Type Inequality ◽  
Markov Type
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