Algorithm For Approximate Solution Research Articles

An Improved Driving Training-Based Optimization Algorithm for the Minimum Gap Graph Connected Partition Problem

In this paper, we consider the minimum gap partition problem in an undirected connected graph with nonnegative vertex weights. This problem involves in dividing the given graph into a specified number of connected subgraphs such that the total difference between the largest and the smallest vertex weights in each subgraph is minimized. Namely, let G = (V, E, W) be a given undirected connected graph with vertex set V , edge set E, and nonnegative vertex weight function W : V → R+, and let k ≥ 2 be a given positive integer. The minimum gap partition problem consists of constructing a partition P = {V1, V2, . . . , Vk} of non-empty and pairwise disjoint subsets of V such that each vertex set Vi, i = 1, 2, . . . , k, induces a connected subgraph G[Vi] of G. The objective of the problem is to find such a partition of V that minimizes the total difference 1≤i≤k maxv∈Viw(v) − minv∈Viw(v). This problem, as many other variants of graph partition problems, is known to be NP-hard problem. We designed an efficient metaheuristic algorithm for finding approximate solution of large-scale instances of this problem. The quality of the proposed algorithm is assessed comparing with the previous algorithms proposed for the problem.

Deep reinforcement learning algorithms for dynamic pricing and inventory management of perishable products

A perishable product has a limited shelf life, and inefficient management often leads to waste. This paper focuses on dynamic pricing and inventory management strategies for perishable products. By implementing effective inventory control and the right pricing policy, it is possible to maximize expected revenue. However, the exponential growth of the problem size due to the shelf life of the products makes it impractical to use methods that guarantee optimal solutions, such as Dynamic Programming (DP). Therefore, approximate solution algorithms become necessary. We use Deep Reinforcement Learning (DRL) algorithms to address the dynamic pricing and ordering problem for perishable products, considering price and age-dependent stochastic demand. We investigate Deep Q Learning (DQL) solutions for discrete action spaces and Soft Actor-Critic (SAC) solutions for continuous action spaces. To mitigate the negative impact of the stochastic environment inherent in the problem, we propose two different DQL approaches. Our results show that the proposed DQL and SAC algorithms effectively address inventory control and dynamic pricing for perishable products, even when products of different ages are offered simultaneously. Compared to dynamic programming, our proposed DQL approaches achieve an average approximation of 95.5 % and 96.6 %, and reduce solution times by 71.5 % and 79.9 %, respectively, for the largest problem. In addition, the SAC algorithm achieves on average 4.6 % and 1.7 % better results and completes the task 56.1 % and 48.2 % faster than the proposed DQL algorithms.

Virtual network function reconfiguration in 5G networks: An optimization perspective

AbstractOne of the major challenges in managing 5G networks is the reconfiguration of network slices. The task covers in particular reconfiguration and relocation of virtual network functions (VNFs) so as to match the service requirements of the slices and the availability of resources of the data centers. In this article, we study in deep the problem of optimal VNFs reconfiguration analyzing a number of its variants. We define two exact, compact integer linear programming formulations of the VNFs reconfiguration problem, and derive two approximate solution algorithms, one of them based on the column generation method. Numerical tests and comparisons illustrate the performance of the algorithms and the quality of the provided solutions.

On the “Onion Husk” Algorithm for Approximate Solution of the Traveling Salesman Problem

On the “Onion Husk” Algorithm for Approximate Solution of the Traveling Salesman Problem

AN ALGORITHM FOR APPROXIMATING SOLUTIONS OF SPLIT HAMMERSTEIN INTEGRAL EQUATIONS

In this paper, we introduce split Hammerstein integral equations of the form: u ∈ H_1 such that u + K_1F_1u = 0 and A(u) + K_2F_2(Au) = 0, where K_1, F_1 are maximal monotone maps defined on a real Hilbert space H_1, with D(K_1) = D(F_1) = H_1; K_2, F_2 are maximal monotone maps defined on a real Hilbert space H_2, with D(K_2) = D(F_2) = H_2 and A, a bounded linear map from H_1 to H_2. The sequence of the algorithm is proved to converge strongly to a solution of the split Hammerstein integral equation. As far as we know, this is the first algorithm whose sequence approximates a solution of a split Hammerstein integral equation in this direction. Finally, the theorem proved, improves and complements some important related recent results in the literature.

Scheduling of Software Test to Minimize the Total Completion Time

This paper investigates a single-machine scheduling problem of a software test with shared common setup operations. Each job has a corresponding set of setup operations, and the job cannot be executed unless its setups are completed. If two jobs have the same supporting setups, the common setups are performed only once. No preemption of any processing is allowed. This problem is known to be computationally intractable. In this study, we propose sequence-based and position-based integer programming models and a branch-and-bound algorithm for finding optimal solutions. We also propose an ant colony optimization algorithm for finding approximate solutions, which will be used as the initial upper bound of the branch-and-bound algorithm. The computational experiments are designed and conducted to numerically appraise all of the proposed methods.

A Review of Analytical Algorithms for Approximate Solutions of Ordinary Differential Equation

This review article aims to provide an in-depth analysis of a wide range of analytical methods used for solving ordinary differential equations (ODEs). ODEs are fundamental in numerous scientific and engineering disciplines, making the development, and understanding of effective solution techniques crucial. We explore various approaches, including the Adomian decomposition method, homotopy perturbation method, homotopy analysis method, variational iteration method, Daftardar-Jafari method, successive approximation method, power series method, and modified Adomian decomposition method. Each method is discussed in terms of its principles, applications, advantages, limitations, and computational considerations. This comprehensive overview will serve as a valuable resource for researchers, practitioners, and students interested in solving ODEs.

New three-term conjugate gradient algorithm for solving monotone nonlinear equations and signal recovery problems

This work presents a new three-term projection algorithm for solving nonlinear monotone equations. The paper is aimed at constructing an efficient and competitive algorithm for finding approximate solutions of nonlinear monotone equations. This is based on a new choice of the conjugate gradient direction which satisfies the sufficient descent condition. The convergence of the algorithm is shown under Lipschitz continuity and monotonicity of the involved operator. Numerical experiments presented in the paper show that the algorithm needs a less number of iterations in comparison with existing algorithms. Furthermore, the proposed algorithm is applied to solve signal recovery problems.

On the Uniqueness of the Bounded Solution for the Fractional Nonlinear Partial Integro-Differential Equation with Approximations

This paper studies the uniqueness of the bounded solution to a new Cauchy problem of the fractional nonlinear partial integro-differential equation based on the multivariate Mittag–Leffler function as well as Banach’s contractive principle in a complete function space. Applying Babenko’s approach, we convert the fractional nonlinear equation with variable coefficients to an implicit integral equation, which is a powerful method of studying the uniqueness of solutions. Furthermore, we construct algorithms for finding analytic and approximate solutions using Adomian’s decomposition method and recurrence relation with the order convergence analysis. Finally, an illustrative example is presented to demonstrate constructions for the numerical solution using MATHEMATICA.

Strong convergence of two regularized relaxed extragradient schemes for solving the split feasibility and fixed point problem with multiple output sets

We study the split feasibility and fixed point problem with multiple output sets in real Hilbert spaces. Using Tikhonov's regularization and viscosity approximation techniques, we propose two relaxed extragradient algorithms for finding approximate solutions to this problem and prove that the sequences they generate converge strongly. Finally, we give some consequences and an application of our results, and also present numerical examples to validate the efficiency of our algorithms.

Numerical algorithms for solutions of nonlinear problems in some distance spaces

<abstract><p>This paper introduces some numerical algorithms for finding solutions of nonlinear problems like functional equations, split feasibility problems (SFPs) and variational inequality problems (VIPs) in the setting of Hilbert and Banach spaces. Our approach is based on the Thakur-Thakur-Postolache (TTP) iterative algorithm and the class of mean nonexpansive mappings. First we provide some convergence results (including weak and strong convergence) in the setting of Banach space. To support these results, we provide a numerical example and prove that our TTP algorithm in this case converges faster to fixed point compared to other iterative algorithms of the literature. After that, we consider two new TTP type projection iterative algorithms to solve SFPs and VIPs on the Hilbert space setting. Our result are new in analysis and suggest new type effective numerical algorithms for finding approximate solutions of some nonlinear problems.</p></abstract>

Coordinated Scheduling of Two-Agent Production and Transportation Based on Non-Cooperative Game

A two-agent production and transportation coordinated scheduling problem in a single-machine environment is suggested to compete for one machine from different downstream production links or various consumers. The jobs of two agents compete for the processing position on a machine, and after the processed, they compete for the transport position on a transport vehicle to be transported to two agents. The two agents have different objective functions. The objective function of the first agent is the sum of the makespan and the total transportation time, whereas the objective function of the second agent is the sum of the total completion time and the total transportation time. Given the competition between two agents for machine resources and transportation resources, a non-cooperative game model with agents as game players is established. The job processing position and transportation position corresponding to the two agents are mapped as strategies, and the corresponding objective function is the utility function. To solve the game model, an approximate Nash equilibrium solution algorithm based on an improved genetic algorithm (NE-IGA) is proposed. The genetic operation based on processing sequence and transportation sequence, as well as the fitness function based on Nash equilibrium definition, are designed based on the features of the two-agent production and transportation coordination scheduling problem. The effectiveness of the proposed algorithm is demonstrated through numerical experiments of various sizes. When compared to heuristic rules such as the Longest Processing Time first (LPT) and the Shortest Processing Time first (SPT), the objective function values of the two agents are reduced by 4.3% and 2.6% on average.

Perceived Integrity of Distributed Streaming Media Based on AWTC-TT Algorithm Optimization

With the development of economy, more and more attention has been paid to the monitoring system, which provides a reliable and powerful guarantee for people’s daily life, property security, and national security. The intelligent video surveillance introduces computer vision-related technologies into traditional video surveillance and realizes the analysis and understanding of video data without artificial dependence to obtain valuable target information in the perceived video data. On this basis, functions such as abnormal event monitoring and real-time alarm are realized. Distributed streaming media monitoring has changed the manual-based monitoring and content analysis modes of traditional monitoring, but the high-complexity calculations such as motion estimation and motion compensation in the encoding process increase the burden of monitoring and sensing equipment. Especially with the development of wireless multimedia technology, the traditional video coding has been unable to meet the requirements of monitoring and sensing equipment in the monitoring system based on wireless technology. This paper proposes an adaptive weighted tensor completion algorithm to complete the repair of streaming media data perceived by ordinary sensing devices. In the proposed algorithm, considering the unbalanced information distribution and data redundancy problems that may exist in the data, the tensor data is adjusted according to the approximate solution algorithm to obtain tensor data that only retains important information and the information distribution is more balanced and reasonable. In the iterative solution process, in order to better map the impact of each dimension of data in the repair process, an adaptive weighting mechanism is proposed according to the data characteristics to obtain the corresponding weight value of each dimension of data. Finally, the proposed approximate tensor solving algorithm and adaptive weighting mechanism are applied to a simple low-rank tensor completeness algorithm based on tensor columns to form the algorithm of this paper, and it is used to repair perceptual streaming media data with data missing problems. The experimental results show that the algorithm in this paper can improve the perceived streaming media data quality by 3% based on the known data information and maintain an advantage of 2% in average processing time. It avoids the replacement of sensing equipment and also provides data quality assurance for subsequent sensing streaming media content analysis. It has certain research significance for the development of monitoring system with artificial intelligence management for target perception and tracking.

Comparison of the ant colony optimization algorithm and its two modifications

The ant optimization algorithm is one of the effective modern algorithms for finding ap-proximate solutions of the salesman problem and similar problems of finding routes on graphs. The first version of this metaheuristic optimization algorithm was proposed by Marco Dorigo in 1992 [1]. After some time, several modifications of this algorithm have been proposed in the literature. The aim of the study is to conduct a comparative analysis of the ant optimization algo-rithm (Ant Colony Optimization, ASO) [1] and its most successful modifications: Ant Colony System (ACS) [2] and Max-Min Ant System (MMAS) [3]. To do this, the system features of information exchange in the ant colony during the search for food are analyzed. A step-by-step algorithm that simulates the natural behavior of forage ants in finding the shortest path to deliver food to the anthill is presented. A software implementation of the three listed ant algorithms for solving the travelling salesman problem has been developed. Through the interface window, you can enter the number of cities, the number of ants, and the maximum number of iterations, fix the settings of the algorithm and select any of the three algorithms. The program randomly locates cities and selects the starting city for each ant. The software product is multi-threaded, i.e. during the calculations the interface is not blocked, which allows you to control the process of program execution, namely: start, pause, stop, resume work. The results of the program are: vis-ualization of the shortest route found, the length of this route and the smallest iteration number, which achieves the shortest route. Comparative analysis of the results allowed us to draw the following conclusions: 1) With well-chosen algorithm settings, iterative methods usually give a result close to optimal, however, the number of iterations required for this may differ significantly. 2) The study of the travelling salesman problem by ant algorithms is experimental rather than theoretical. The result very much depends on the parameters of the algorithm settings, but the theoretical study of these dependencies remains relevant and unresolved.

Weapon-target assignment problem: exact and approximate solution algorithms

The Weapon-Target Assignment (WTA) problem aims to assign a set of weapons to a number of assets (targets), such that the expected value of survived targets is minimized. The WTA problem is a nonlinear combinatorial optimization problem known to be NP-hard. This paper applies several existing techniques to linearize the WTA problem. One linearization technique (Camm et al. in Oper Res 50(6):946–955, 2002) approximates the nonlinear terms of the WTA problem via convex piecewise linear functions and provides heuristic solutions to the WTA problem. Such approximation problems are, though, relatively easy to solve from the computational point of view even for large-scale problem instances. Another approach proposed by O’Hanley et al. (Eur J Oper Res 230(1):63–75, 2013) linearizes the WTA problem exactly at the expense of incorporating a significant number of additional variables and constraints, which makes many large-scale problem instances intractable. Motivated by the results of computational experiments with these existing solution approaches, a specialized new exact solution approach is developed, which is called branch-and-adjust. The proposed solution approach involves the compact piecewise linear convex under-approximation of the WTA objective function and solves the WTA problem exactly. The algorithm builds on top of any existing branch-and-cut or branch-and-bound algorithm and can be implemented using the tools provided by state-of-the-art mixed integer linear programming solvers. Numerical experiments demonstrate that the proposed specialized algorithm is capable of handling very large scale problem instances with up to 1500 weapons and 1000 targets, obtaining solutions with optimality gaps of up to \(2.0\%\) within 2 h of computational runtime.

ON AN ALGORITHM OF FINDING AN APPROXIMATE SOLUTION OF A PERIODIC PROBLEM FOR A THIRD-ORDER DIFFERENTIAL EQUATION

ON AN ALGORITHM OF FINDING AN APPROXIMATE SOLUTION OF A PERIODIC PROBLEM FOR A THIRD-ORDER DIFFERENTIAL EQUATION

NUMERICAL SOLUTION OF THE DYNAMIC PROBLEM OF AXISYMMETRIC VIBRATIONS OF REINFORCED SHELLS

The reliability of the results obtained in the work is determined by the rigor and correctness of the statements of the initial problems; theoretical substantiation of the finite-difference schemes used; controlled accuracy of numerical calculations; conducting test calculations; compliance of the established regularities with the general properties of oscillations of thin-walled structural elements. The correctness of the formulation of the problems is achieved by using the well-known equations of the theory of shells and rods of the Tymoshenko type, which are approximations of the original equations of the three-dimensional theory of elasticity. When deriving the equations, the equations of oscillations of the multilayer shell in the smooth region and the equations of oscillations of reinforcing ribbed elements (transverse ribs) were obtained. It is not difficult to show that the indicated equations by the classification of equations in partial derivatives are equations of the hyperbolic type, which are an approximation of the oscillating equations of three-dimensional elastic bodies and sufficiently correctly reproduce wave processes in non-homogeneous shell structures, taking into account spatial gaps. Numerical algorithms for approximate solutions of the original equations are based on the use of the integro-interpolation method of constructing difference schemes. When constructing difference schemes, kinematic quantities refer to difference points with integer indices, and the values of deformations and moment forces refer to difference points with half-integer indices. This approximation of the initial kinematic and static values allows the fulfillment of the law of conservation of the total mechanical energy of the elastic structure at the difference level. The numerical algorithm is based on the use of separate finite-difference relations in the smooth domain and on the lines of spatial discontinuities with the second order of accuracy in spatial and temporal coordinates.

Scalable grid‐based approximation algorithms for partially observable Markov decision processes

AbstractPartially observable Markov decision processes (POMDPs) are a well‐established sequential decision making framework. Once a problem is modeled using this framework, a suitable POMDP solution algorithm is employed to obtain a policy that guides the user throughout the decision making process. However, POMDPs are notoriously difficult to solve to optimality. Therefore, there exist many approximate solution algorithms that are designed to generate policies for large‐scale POMDP models. On the other hand, many such approaches lack performance guarantees in terms of the solution quality. In this article, we focus on exact solution methods as well as approximate methods that provide bounds on the optimal value. Specifically, we investigate the performance improvements for the POMDP solution algorithms obtained through distributed implementations. We provide a detailed empirical analysis on various test problems, which highlights the benefits of the proposed approach.

A new fast and accurate heuristic for the Automatic Scene Detection Problem

The Automatic Scene Detection Problem (ASDP) is a combinatorial optimization problem that arises in the context of video processing and that has a central role in the management, storing and content retrieval of videos. The problem consists of partitioning the shots of a given video into scenes by optimizing a measure related to the similarity between the given shots. In this article, we build up upon the results from the literature on the ASDP in order to design a new approximate solution algorithm able to outperform the current state-of-the-art both in terms of speed and quality of the solution.

New hybrid three-term spectral-conjugate gradient method for finding solutions of nonlinear monotone operator equations with applications

In this paper, we present a new hybrid spectral-conjugate gradient (SCG) algorithm for finding approximate solutions to nonlinear monotone operator equations. The hybrid conjugate gradient parameter has the Polak–Ribière–Polyak (PRP), Dai–Yuan (DY), Hestenes–Stiefel (HS) and Fletcher–Reeves (FR) as special cases. Moreover, the spectral parameter is selected such that the search direction has the descent property. Also, the search directions are bounded and the sequence of iterates generated by the new hybrid algorithm converge globally. Furthermore, numerical experiments were conducted on some benchmark nonlinear monotone operator equations to assess the efficiency of the proposed algorithm. Finally, the algorithm is shown to have the ability to recover disturbed signals.