An Investigation of the Common Solutions for Coupled Systems of Functional Equations Arising in Dynamic Programming

Işık, Hüseyin; Sintunavarat, Wutiphol

doi:10.3390/math7100977

Mathematics

2019

DOI: 10.3390/math7100977

|View full text |Cite

An Investigation of the Common Solutions for Coupled Systems of Functional Equations Arising in Dynamic Programming

Hüseyin Işık

Wutiphol Sintunavarat

Abstract: The purpose of this paper is to introduce the new notion of a specific point in the space of the bounded real-valued functions on a given non-empty set and present a result based on the existence and uniqueness of such points. As a consequence of our results, we discuss the existence of a unique common solution to coupled systems of functional equations arising in dynamic programming.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2020

2022

Publication Types

Select...

Article3

Relationship

Self Cite0

Independent3

Authors

Journals

Cited by 3 publications

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Dynamic programming in applied tasks which are allowing to reduce the options selection

Karpov

Struchenkov

2020

Rossijskij tehnologičeskij žurnal

View full text Add to dashboard Cite

The article discusses the dynamic programming algorithm developed by R. Bellman, based on the search for the optimal trajectory connecting the nodes of a predefined regular grid of states. Possibilities are analyzed for a sharp increase in the effectiveness of using dynamic programming in solving applied problems with specific features, which allows us to refuse to split a regular grid of states and implement an algorithm for finding the optimal trajectory when rejecting not only unpromising options for paths leading to each of the states, and all of them continuations, as in R. Bellmanʼs algorithm, but also actually hopeless states and all variants of paths emanating from them. The conditions are formulated and justified under which the rejection of hopeless states is possible. It has been established that many applied problems satisfy these conditions. To solve such problems, a new dynamic programming algorithm described in the article is proposed and implemented. Concrete examples of such applied problems are given: the optimal distribution of a homogeneous resource between several consumers, the optimal loading of vehicles, the optimal distribution of finances when choosing investment projects. To solve these problems, dynamic programming algorithms with rejecting unpromising paths, but without rejecting states, were previously proposed. The number of hopeless states that appear at various stages of dynamic programming and, accordingly, the effectiveness of the new algorithm depends on the specific numerical values of the source data. For the two-parameter problem of optimal loading of vehicles with weight and volume constraints, the results of comparative calculations by the R. Bellman algorithm and the new dynamic programming algorithm are presented. As a source of data for a series of calculations, pseudorandom numbers were used. As a result of the analysis, it was shown that the comparative efficiency of the algorithm with rejection of states increases with increasing dimension of the problem. So, in the problem of the optimal choice of items for loading a vehicle of a given carrying capacity with a number of items of 150, the number of memorized states and the counting time are reduced by 50 and 57 times, respectively, when using the new algorithm compared to the classical algorithm of R. Bellman. And for 15 items, the corresponding numbers are 13 and 4.

show abstract

Dynamic programming in applied tasks which are allowing to reduce the options selection

Karpov

Struchenkov

2020

Rossijskij tehnologičeskij žurnal

View full text Add to dashboard Cite

show abstract

Two-stage spline-approximation in linear structure routing

Karpov

Struchenkov

2021

Rossijskij tehnologičeskij žurnal

View full text Add to dashboard Cite

In the article, computer design of routes of linear structures is considered as a spline approximation problem. A fundamental feature of the corresponding design tasks is that the plan and longitudinal profile of the route consist of elements of a given type. Depending on the type of linear structure, line segments, arcs of circles, parabolas of the second degree, clothoids, etc. are used. In any case, the design result is a curve consisting of the required sequence of elements of a given type. At the points of conjugation, the elements have a common tangent, and in the most difficult case, a common curvature. Such curves are usually called splines. In contrast to other applications of splines in the design of routes of linear structures, it is necessary to take into account numerous restrictions on the parameters of spline elements arising from the need to comply with technical standards in order to ensure the normal operation of the future structure. Technical constraints are formalized as a system of inequalities. The main distinguishing feature of the considered design problems is that the number of elements of the required spline is usually unknown and must be determined in the process of solving the problem. This circumstance fundamentally complicates the problem and does not allow using mathematical models and nonlinear programming algorithms to solve it, since the dimension of the problem is unknown. The article proposes a two-stage scheme for spline approximation of a plane curve. The curve is given by a sequence of points, and the number of spline elements is unknown. At the first stage, the number of spline elements and an approximate solution to the approximation problem are determined. The method of dynamic programming with minimization of the sum of squares of deviations at the initial points is used. At the second stage, the parameters of the spline element are optimized. The algorithms of nonlinear programming are used. They were developed taking into account the peculiarities of the system of constraints. Moreover, at each iteration of the optimization process for the corresponding set of active constraints, a basis is constructed in the null space of the constraint matrix and in the subspace – its complement. This makes it possible to find the direction of descent and solve the problem of excluding constraints from the active set without solving systems of linear equations. As an objective function, along with the traditionally used sum of squares of the deviations of the initial points from the spline, the article proposes other functions taking into account the specificity of a particular project task.

show abstract

On common fixed point results in bicomplex valued metric spaces with application

Tassaddiq

Ahmad

Al-Mazrooei

et al. 2022

MATH

View full text Add to dashboard Cite

<abstract><p>Metric fixed-point theory has become an essential tool in computer science, communication engineering and complex systems to validate the processes and algorithms by using functional equations and iterative procedures. The aim of this article is to obtain common fixed point results in a bicomplex valued metric space for rational contractions involving control functions of two variables. Our theorems generalize some famous results from literature. We supply an example to show the originality of our main result. As an application, we develop common fixed point results for rational contractions involving control functions of one variable in the context of bicomplex valued metric space.</p></abstract>

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

An Investigation of the Common Solutions for Coupled Systems of Functional Equations Arising in Dynamic Programming

Cited by 3 publications

References 7 publications

Dynamic programming in applied tasks which are allowing to reduce the options selection

Dynamic programming in applied tasks which are allowing to reduce the options selection

Two-stage spline-approximation in linear structure routing

On common fixed point results in bicomplex valued metric spaces with application

Contact Info

Product

Resources

About