Abstract
Although it is a very old theme, unconstrained optimization is an area which is always actual for many scientists. Today, the results of unconstrained optimization are applied in different branches of science, as well as generally in practice. Here, we present the line search techniques. Further, in this chapter we consider some unconstrained optimization methods. We try to present these methods but also to present some contemporary results in this area.
Highlights
IntroductionOptimization is a very old subject of a great interest; we can search deep into a human history to find important examples of applying optimization in the usual life of a human being, for example, the need of finding the best way to produce food yielded finding the best piece of land for producing, as well as (later on, how the time was going) the best ways of treatment of the chosen land and the chosen seedlings to get the best results
The classical steepest descent method which is designed by Cauchy [24] can be considered as one among the most important procedures for minimization of real-valued function defined on Rn
The classical and the oldest steepest descent step size tk, which was designed by Cauchy, is computed as [26]
Summary
Optimization is a very old subject of a great interest; we can search deep into a human history to find important examples of applying optimization in the usual life of a human being, for example, the need of finding the best way to produce food yielded finding the best piece of land for producing, as well as (later on, how the time was going) the best ways of treatment of the chosen land and the chosen seedlings to get the best results. Optimization task is to find the minimum (maximum) of the objective function f ðxÞ 1⁄4 f ðx1; ...; xnÞ, under the conditions (1), (2), and (3). Definition 1.1.2 x∗ is called a weak local minimizer of f if there exists a neighborhood N of x∗, such that f ðx∗Þ ≤ f ðxÞ for all x ∈ N. Definition 1.1.3 x∗ is called a strict (strong) local minimizer of f if there exists a neighborhood N of x∗, such that f ðx∗Þ , f ðxÞ for all x ∈ N. Considering backward definitions 1.1.2 and 1.1.3, the procedure of finding local minimizer (weak or strict) does not seem such easy; it seems that we should examine all points from the neighborhood of x∗, and it looks like a very difficult task. There exist two important classes of iterative methods—line search methods and trust-region methods—made in the aim to solve the unconstrained optimization problem (4). We try to give some of the most recent results in these areas
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
