Algorithm For Finding Approximate Solutions Research Articles

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.

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.

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>

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.

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.

FR-type algorithm for finding approximate solutions to nonlinear monotone operator equations

This paper focuses on the problem of convex constraint nonlinear equations involving monotone operators in Euclidean space. A Fletcher and Reeves type derivative-free conjugate gradient method is proposed. The proposed method is designed to ensure the descent property of the search direction at each iteration. Furthermore, the convergence of the proposed method is proved under the assumption that the underlying operator is monotone and Lipschitz continuous. The numerical results show that the method is efficient for the given test problems.

DELAY MODELING OF MATHEMATICAL MODELS OF BIOLOGY AND IMMUNOLOGY

Systems of differential-difference equations are mathematical models of many applied problems of biology, ecology, medicine, economics. The variety of mathematical models of real dynamic processes is due to the fact that their evolution does not occur instantaneously, but with some delays that have different biological interpretations. The introduction of delay allows you to build adequate mathematical models and describe new effects and phenomena in physics, ecology, immunology and other sciences. The exact solution of differential-difference equations can be found only in the simplest cases, so algorithms for finding approximate solutions of such equations are important. In this paper, a family of difference schemes is constructed for the approximate finding of solutions to initial problems with delay. Special cases are generalized Euler difference schemes. The conditions for the convergence of the generalized explicit Euler difference scheme are established. To automate the numerical simulation of systems with delays, an application program has been developed, which is used to approximate the solutions of SIR models with two delays.

A Fragmented Model for the Problem of Land Use on Hypergraphs

The paper considers a mathematical model of the land use problem on hypergraphs. It is shown that, within the framework of this model, the problem can be formulated as an optimization problem on a fragmented structure. Moreover, the problem of finding the optimal solution itself reduces to the problem of unconditional combinatorial optimization on a set of permutations. A variant of a hybrid algorithm for finding approximate solutions to the problem based on a combination of a fragmented algorithm and an ant colony algorithm is proposed.

Application of Ant Algorithm for Software Optimization

Different technologies, methods and algorithms are used when developing high-quality software systems. There are various ways to create optimal software. One of them is an ant algorithm. The ant algorithm can be attributed to the field of biomimetic. Ant algorithm is one of the most effective polynomial algorithms for finding approximate solutions, and it also solves similar route search problems in graphs. Various methods, including the ant algorithm, are used to improve software efficiency and optimize it. The essence of the concept is to analyze, use, and meta-heuristically optimize the behavior of ants in search of paths to the food source of the colony. This article analyzes the studies in this area, and explains the idea of the algorithm used, compares different ant systems, shows program code operators as ants, and applies the ant algorithm to them. As a result of applying the algorithm, the shortest path to some operators (cycle, condition) contained in the program code is found. The experiments perform good results.

Hybrid Beamforming for Active Sensing Using Sparse Arrays

This paper studies hybrid beamforming for active sensing applications, such as millimeter-wave or ultrasound imaging. Hybrid beamforming can substantially lower the cost and power consumption of fully digital sensor arrays by reducing the number of active front ends. Sparse arrays can be used to further reduce hardware costs. We consider phased arrays and employ linear beamforming with possibly sparse array configurations at both the transmitter and receiver. The quality of the acquired images is improved by adding together several component images corresponding to different transmissions and receptions. In order to limit the acquisition time of an image, we formulate an optimization problem for minimizing the number of component images subject to achieving a desired point spread function. Since this problem is not convex, we propose algorithms for finding approximate solutions in the fully digital beamforming case, as well as in the more challenging hybrid and analog beamforming cases that employ quantized phase shifters. We also determine upper bounds on the number of component images needed for achieving the fully digital solution using fully analog and hybrid architectures, and derive closed-form expressions for the beamforming weights in these cases. Simulations demonstrate that a hybrid sparse array with very few elements, and even fewer front ends, can achieve the resolution of a fully digital uniform array at the expense of a longer image acquisition time.

On the well‐posedness of periodic problems for the system of hyperbolic equations with finite time delay

AbstractA periodic problem for the system of hyperbolic equations with finite time delay is investigated. The investigated problem is reduced to an equivalent problem, consisting the family of periodic problems for a system of ordinary differential equations with finite delay and integral equations using the method of a new functions introduction. Relationship of periodic problem for the system of hyperbolic equations with finite time delay and the family of periodic problems for the system of ordinary differential equations with finite delay is established. Algorithms for finding approximate solutions of the equivalent problem are constructed, and their convergence is proved. Criteria of well‐posedness of periodic problem for the system of hyperbolic equations with finite time delay are obtained.

One Approach to the Solution of a Nonlocal Problem for Systems of Hyperbolic Equations with Integral Conditions

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations with two independent variables. We study the solvability of the considered problem and construct algorithms aimed at finding its approximate solutions by introducing additional functional parameters. The investigated problem is reduced to an equivalent problem that consists of the Goursat problem for a system of hyperbolic equations with parameters and a boundary-value problem with integral condition for a system of ordinary differential equations for the introduced parameters. We propose algorithms for finding approximate solutions of the problem based on the algorithms used for the solution of the equivalent problem and prove their convergence to the exact solution.

Быстрый алгоритм приближенного нахождения решения задачи Коши для ОДУ

Быстрый алгоритм приближенного нахождения решения задачи Коши для ОДУ

Numerical methods for solution of Volterra and Fredholm integral equations for functions with values in L-spaces

We consider Volterra and Fredholm integral equations for functions with values in L-spaces. This includes corresponding problems for set-valued functions, fuzzy-valued functions and many others. We prove theorems of existence and uniqueness of the solution for such equations and suggest some algorithms for finding approximate solutions. We get initial results in the approximation of functions with values in L-spaces by piecewise linear functions and we also get the error estimates of trapezoidal quadrature formulas. We apply these results for analysis of convergence of suggested algorithms.

Nonlinear Noetherian Boundary-Value Problem in the Case of Parametric Resonance

Necessary and sufficient conditions are established for the existence of solutions of the nonlinear Noetherian boundary-value problem for a system of differential equations in the case of parametric resonance. We propose a convergent iterative algorithm for finding approximate solutions of the nonlinear Noetherian boundary-value problem for a system of ordinary differential equations in the case of parametric resonance. As an example, we apply the constructed iterative algorithm to find approximate solutions of the periodic boundary-value problem for the Duffing-type equation with parametric perturbation. The accuracy of the obtained approximate solutions of the periodic boundary-value problem for the Duffing-type equation is controlled by using deviations from the original equation.

On the Computational Efficiency of the Algorithm of the Numerical Solution of Optimal Control Problems for Models of Leontieff Type

The paper shows the efficiency of the numerical algorithm for the class of problems that is considered by the example of optimal control, hard control, start control and hard starting control for the Leontieff type models. There are presented actual results of computational experiment. As the initial condition is used Showalter - Sidorov condition. This eliminates the restrictions caused by the need to initial checking the data that existed when using Cauchy conditions. The introduction presents various problems of optimal control. Is given their economic interpretation. The first section presents a theorem an existence of a unique solution the problem of optimal control, kind of exact and approximate solutions, the main stages of the algorithm for finding approximate solutions, theorem on the convergence of the approximate solution to the exact one. The second section presents the results of a computational experiment of solving the problem of optimal control. The third section presents the results of a computational experiment of solving the problem of hard control. The fourth section contains the results of numerical experiments solving the problem of start control and the problem of hard starting control. The fifth section presents the results of computational experiments with different parameters of the algorithm as an example a model of Leontieff type. It is shown that the change of parameters leads to small computational error, indicating the computational efficiency.

A proximal point algorithm for a system of generalized mixed variational inequalities

This paper introduces and considers a new system of generalized mixed variational inequalities in a Hilbert space, which includes many new and known systems of variational inequalities and generalized variational inequalities as special cases. By using the two concepts of η-subdifferential and η-proximal mappings of a proper function, the authors try to demonstrate that the system of generalized mixed variational inequalities is equivalence with a fixed point problem. By applying the equivalence, a new and innovative η-proximal point algorithm for finding approximate solutions of the system of generalized mixed variational inequalities will be suggested and analyzed. The authors also study the convergence analysis of the new iterative method under much weaker conditions. The results can be viewed as a refinement and improvement of the previously known results for variational inequalities.

Aircraft Collision Avoidance Using Monte Carlo Real-Time Belief Space Search

The aircraft collision avoidance problem can be formulated using a decision-theoretic planning framework where the optimal behavior requires balancing the competing objectives of avoiding collision and adhering to a flight plan. Due to noise in the sensor measurements and the stochasticity of intruder state trajectories, a natural representation of the problem is as a partially-observable Markov decision process (POMDP), where the underlying state of the system is Markovian and the observations depend probabilistically on the state. Many algorithms for finding approximate solutions to POMDPs exist in the literature, but they typically require discretization of the state and observation spaces. This paper investigates the introduction of a sample-based representation of state uncertainty to an existing algorithm called Real-Time Belief Space Search (RTBSS), which leverages branch-and-bound pruning to make searching the belief space for the optimal action more efficient. The resulting algorithm, called Monte Carlo Real-Time Belief Space Search (MC-RTBSS), is demonstrated on encounter scenarios in simulation using a beacon-based surveillance system and a probabilistic intruder model derived from recorded radar data.

A SYSTEM OF NONLINEAR VARIATIONAL INCLUSIONS WITH GENERAL H-MONOTONE OPERATORS IN BANACH SPACES

A system of nonlinear variational inclusions involving general H-monotone operators in Banach spaces is introduced. Using the resolvent operator technique, we suggest an iterative algorithm for finding approximate solutions to the system of nonlinear variational inclusions, and establish the existence of solutions and convergence of the iterative algorithm for the system of nonlinear variational inclusions.