Quadratic Interpolation Algorithm for Minimizing Tabulated Function

Y.A El-Rewaily,S.Z Hassan,Youness E.A

doi:10.3844/jmssp.2008.217.221

Abstract

Problem statement: The problem of finding the minimum value of objective function, when we know only some values of it, is needed in more practical fields. Quadratic interpolation algorithms are the famous tools deal with this kind of these problems. These algorithms interested with the polynomial space in which the objective function is approximated. Approach: In this study we approximated the objective function by a one dimensional quadratic polynomial. This approach saved the time and the effort to get the best point at which the objective is minimized. Results: The quadratic polynomial in each one of the steps of the proposed algorithm, accelerate the convergent to the best value of the objective function without taking into account all points of the interpolation set. Conclusion: Any n-dimensional problem of finding a minimal value of a function, given by some values, can be converted to one dimensional problem easier in deal.

Highlights

Many optimization problems can be occur in practice, for example, in most of labs, one obtains data for certain phenomena and wants know that the point at which this phenomena is minimized
Algorithms are based on the progressive building and updating of a model of the objective function[9,10]
In the following discussion we present analysis on which the proposed algorithm for finding the minimal point of tabulated function is based

Summary

Introduction

Many optimization problems can be occur in practice, for example, in most of labs, one obtains data for certain phenomena and wants know that the point at which this phenomena is minimized. Let I be denote to the set of points x ∈ X ⊂ Rn at which the function f : X ⊂ Rn → R is given in the Table 1. If f is known only at one point of α, s, we can consider one of the other two values (say φ(α2 )) is less than φ(α1 ) and φ(α3 )

Results

Conclusion