Iranian Journal of Numerical Analysis and Optimization
Volume:12 Issue: 1, Winter and Spring 2022

This study presents an algorithm for solving optimal control problems with the objective function of the Lagrange-type and multiple delays on both the state and control variables of the constraints, with bounds on the control variable. The full discretization of the objective functional and the multiple delay constraints is carried out by using the Simpson numerical scheme. The discrete recurrence relations generated from the discretization of both the objective functional and constraints are used to develop the matrix operators, which satisfy the basic spectral properties. The primal-dual residuals of the algorithm are derived in order to ascertain the rate of convergence of the algorithm, which performs faster when relaxed with an accelerator variant in the sense of Nesterov. The direct numerical approach for handling the multi-delay control problem is observed to obtain an accurate result at a faster rate of convergence when over-relaxed with an accelerator variant. This research problem is limited to linear constraints and objective functional of the Lagrange-type and can address real-life models with multiple delays as applicable to quadratic optimization of intensity modulated radiation theory planning. The novelty of this research paper lies in the method of discretization and its adaptation to handle linearly and proximal bound-constrained program formulated from the multiple delay optimal control problems.

We propose an effcient multistage method for solving a system of linear and nonlinear Volterra integral equations of the second kind. This numerical method is based on the Gauss–Legendre quadrature rule that obtains several values of unknown function at each step, and it will be shown that the order of the convergence is O(M-4), where M is the number of the nodes in the time discretization. The method has also the advantages of simplicity of application, less computational time, and useful performance for large intervals. In order to show the effciency of the method, numerical results for the avian human influenza epidemic model is obtained that is comparable with the fourth-order Runge–Kutta method.

This article deals with an effcient approximation method named successive complementary expansion method (SCEM) for solving singularly perturbed differential-difference equations with mixed shifts. It is compared with the method of matched asymptotic expansion (MMAE) and the parameter uniform upwind finite difference scheme for solving such a model. The comparison shows, unlike the MMAE, the SCEM method requires no matching procedure. It requires less computation when compared to the upwind finite difference scheme on the Shishkin mesh. The error analysis is carried out to prove the robustness of the method. Some numerical experiments are provided, which show the effectiveness of the proposed method.

This article considers a nonlinear inverse problem of the Ostrovsky–Burgers equation by using noisy data. Two B-Splines with different levels, the quintic B-spline and septic B-spline, are used to study this problem. For both B-splines, the stability and convergence analysis are calculated, and results show that an excellent estimation of the unknown functions of thenonlinear inverse problem.

We propose the preconditioned global generalized product-type method based on the preconditioned global BiCG method to solve nonsymmetric saddle point problems with multiple right-hand sides. We apply an indefinite preconditioner to enhance the convergence rate of the method. We also present some theoretical analysis and discuss the convergence of the PGl-GPBiCG method. Some useful properties of the preconditioned matrix are established. Moreover, we present the bounds for the residual norm of the PGl-GPBiCG method according to the residual norm of the global GMRES method that guarantees convergence. Finally, some numerical examples are presented to show the effciency of the new method in comparison with the preconditioned global BiCGSTAB method, and a comparison with another preconditioner is also provided.

The conditions for the sequence of complex numbers (bn,k) are obtained, such that the interpolation problem g(k-1)(λn) = bn,k, k ∈ 1, s, n ∈ N, where |λk/λk+1| ≤ ∆ < 1, has a unique solution in some classes of entire functions g for which Mg(r) ≤ c1 exp ((s - 1)N(r) + N(ρ1r)), where N(r) is the counting function of the sequence (λn), ρ1 ∈ (∆; 1), and c1 > 0. Moreover, these results have been applied to the description of the solution of the differential equation f(s) +A0(z)f = 0 for which (λn) is zero-sequence and the coeffcient A0 is an entire function from the mentioned class.

We consider a class of singularly perturbed semilinear three-point boundary value problems. An accelerated uniformly convergent numerical method is constructed via the exponential fitted operator method using Richardson extrapolation techniques to solve the problem. To treat the semilinear term, we use quasi-linearization techniques. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and ε-uniformly convergent for h ≥ ε, where the classical numerical methods fail to give a good result. It also improves the results of the methods existing in the literature. The method is shown to be second-order convergent independent of perturbation parameter ε.

We consider a new extension of the bin packing problem in which a set of connectivity constraints should be satisfied. An undirected graph with a weight function on the nodes is given. The objective is to pack all the nodes in the minimum number of unit-capacity bins, such that the induced subgraph on the set of nodes packed in each bin is connected. After analyzing some structural properties of the problem, we present a linear time approximation algorithm for this problem when the underlying graph is traceable. We show that the approximation factor of this algorithm is 2 and this factor is tight. Finally, concerning the investigated structural properties, we extend the algorithm for more general graphs. This extended algorithm also has a tight approximation factor of 2.

As known, outliers and multicollinearity in the data set are among the important diffculties in regression models, which badly affect the leastsquares estimators. Under multicollinearity and outliers’ existence in the data set, the prediction performance of the least-squares regression method is decreased dramatically. Here, proposing an approximation for the condition number, we suggest a nonlinear mixed-integer programming model to simultaneously control inappropriate effects of the mentioned problems. The model can be effectively solved by popular metaheuristic algorithms. To shed light on importance of our optimization approach, we make some numerical experiments on a classic real data set as well as a simulated data set.

When we design the payoff matrix of a game on the basis of the available information, then rarely the information is free from impreciseness, and as a result, the payoffs of the payoff matrix have a certain amount of ambiguity associated with them. In this work, we have developed a heuristic technique to solve two persons m × n zero-sum games (m > 2, n > 2), with interval-valued payoffs and interval-valued objectives. Thus the game has been formulated by representing the impreciseness of the payoffs with interval numbers. To solve the game, a real coded genetic algorithm with interval fitness function, tournament selection, uniform crossover, and uniform mutation has been developed. Finally, our proposed technique hasbeen demonstrated with a few examples and sensitivity analyses with respect to the genetic algorithm parameters have been done graphically to study the stability of our algorithm.

We offer a method for solving the fractional optimal control problems of multi-dimensional. We obtain a fractional derivative and multiplication operational matrix for Mott polynomials (M-polynomials). In the proposed method, the Caputo sense of the fractional derivative is applied on dynamical system. The main feature of this method is to reduce the problem into a system of algebraic equations in order to simplify it. We also show that by increasing the approximation points, the responses converge to the real answer. When the degree of fractional derivative approaches to 1, then the obtained solution approaches to the classical solution as well.

In this paper, we use a novel technique to solve the nonlinear fractional Volterra-Fredholm integro-differential equations (FVFIDEs). To this end, the Legendre wavelets are used in conjunction with the quadrature rule for converting the problem into a linear or nonlinear system of algebraic equations, which can be easily solved by applying mathematical programming techniques. Only a small number of Legendre wavelets are needed to obtain a satisfactory result. Better accuracies are also achieved within the method by increasing the number of polynomials. Furthermore, the existence and uniqueness of the solution are proved by preparing some theorems and lemmas. Also, error estimation and convergence analyses are given for the considered problem and the method. Moreover, some examples are presented and their results are compared to the results of Chebyshev wavelet, Nystro¨m, and Newton–Kantorovitch methods to show the capability and validity of this scheme.