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Multi-objective programming problem often contains numerous efficient solutions, which confuses the decision-maker. To assist in selecting the most desirable solution, optimizing a function over the efficient set becomes crucial. In this paper, we present a novel method for optimizing a general quadratic function over the efficient set of a multi-objective integer linear programming problem. To solve this problem, a ranking approach and efficiency test is utilized. The proposed methodology obtains a globally optimal solution by systematically scanning ranked solutions of an integer quadratic programming problem until the efficiency condition is satisfied. For generating ranked solutions, we construct a related integer linear programming problem. Then, ranked solutions of the integer linear programming problem are used for enumerating ranked solutions of the integer quadratic programming problem. The convergence of our algorithm is established theoretically, and its steps are illustrated using a numerical example. Aparticular case of the proposed method for optimizing a linear function over the efficient set of a multi-objective integer linear programming problem is also discussed. Further, extensive computational results demonstrate the effectiveness of our method for solving problems with large number of constraints, variables, and objective functions. Moreover, comparative analysis shows that the developed algorithm came out to be computationally more efficient as compared to the existing state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

The Barzilai–Borwein (BB) method is a popular and efficient gradient method for solving large-scale unconstrained optimization problems. In general, it converges much faster than the steepest descent (Cauchy) method. However, it may not converge, even when the objective function is strongly convex. To overcome this, by virtue of bounding the distance between sequential iterates, [Burdakov et al. (2019)] proposed a stabilized BB method. In this paper, by combining the stabilized BB method with a projection approach, we develop a stabilized BB projection (SBBP) method to solve nonlinear monotone equations with convex constraints. SBBP does not involve computing matrices proving the usefulness for solving large-scale problems. Its global convergence and Q-linear convergence rate are also established. We report numerical experiments on recovering sparse signals and restoring blurred images arising from compressive sensing, demonstrating the usefulness of SBBP. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce new iterative algorithms for approximating a solution to a class of monotone operator equations. More precisely, we study the split common solution problem with multiple output sets for monotone operator equations in Hilbert spaces. In order to solve this problem, we propose three new algorithms and establish strong convergence theorems for them. [ABSTRACT FROM AUTHOR]

Box-constrained optimization problems in the real m × n matrix space have been widely applied in big data mining. However, efficient solution of them is still a challenge. In this paper, a new nonmonotone line search rule is first proposed by extending the well-known ones and inheriting their advantages. Then, by analyzing and exploiting properties of this rule, a new nonmonotone spectral projected gradient algorithm is developed to solve the box-constrained optimization problems in the matrix space. Global convergence of the developed algorithm is also established. Numerical tests are conducted on a series of randomly generated test problems and those in the set of benchmark test problems. Compared with other existing nonmonotone line search rules, our rule shows its advantages in terms of the significantly reduced number of function evaluations and significantly reduced number of iterations. To further validate applicability of this research, we apply the studied optimization problem and the developed algorithm to solve the problems of image clustering. Numerical results demonstrate that the proposed method can generate better clustering results and is more robust than the similar ones available in the literature. [ABSTRACT FROM AUTHOR]

The rational fraction system is a special nonlinear system, the existence of the denominator polynomial leads to the difficulty of identifying rational fraction models. Inspired by the gradient search and the Newton method, the gradient-based iterative algorithm and Newton iterative algorithm are presented to estimate the parameters of rational fraction system models. Furthermore, in order to avoid a large amount of calculation and complex equations encountered in the process of solving partial derivatives, the model transformation-based gradient iterative algorithm and the model transformation-based Newton iterative algorithm are proposed for parameter identification. Two examples are carried out to show the effectiveness of the proposed algorithms. This paper focuses on solving the identification problem of rational fraction systems. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fast inertial self-adaptive projection based algorithm for solving large-scale monotone equations with the relaxation factor. The modified hyperplane projection technique accelerates the computational performance of the proposed algorithm, and the self-adaptive parameter enhances its stability. Under some appropriate assumptions, the global convergence of the proposed algorithm is established. Moreover, the proposed algorithm is suitable to solve large-scale problems because it does not need gradient information or Jacobian matrix. The numerical results indicate that the new algorithm is robust and efficient by comparing with other popular algorithms. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of finding poles of rational functions from function values on open curves in the complex plane. For this problem, Nara and Ando recently proposed an algorithm that reduces the problem to a system of linear equations through contour integration. The main aim of this paper is to analyze and improve this algorithm by giving a new interpretation to the algorithm in terms of orthogonal polynomials. It is demonstrated that the system of linear equations is not always uniquely solvable and that this difficulty can be remedied by doubling the number of the linear equations. Moreover, to cope with discretization errors caused by numerical integration, we introduce new polynomials similar, in spirit, to discrete orthogonal polynomials, which yield an algorithm free from discretization errors. [ABSTRACT FROM AUTHOR]

In this paper, using the properties of graph convergence of H(.,.)-accretive operators, we construct a perturbed algorithm for solving systems of ill-posed variational inclusions involving H(.,.)-accretive operators. Under some suitable conditions, we prove that the sequence generated by the proposed method is strongly convergent to the unique solution of the problem. The main results presented in this paper generalize and improve some recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the error estimates of a decoupled algorithm for the fluid–fluid model. The system consists of two Navier–Stokes equations which are coupled by a set of linear interface conditions. We apply the partitioned time stepping method to decouple the system. The corresponding scheme is unconditionally stable and we prove that the error estimates for velocities in L 2 norm are optimal. Moreover, under a restriction on the time step scale, we prove that the convergent orders for the velocities in H 1 norm and for the pressures in L 2 norm are Δ t 7 8 + h and Δ t 3 4 + h , respectively. Two numerical examples are given to verify our theoretical results. Besides, by comparing the decoupled algorithm with the coupled one, numerical test shows the effectiveness of our decoupled method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, an effective method based upon the piecewise homotopy perturbation method (PHPM) is proposed for finding the numerical solutions of Fredholm integro-differential equations with weakly singular kernels. For these equations, the traditional homotopy perturbation method is divergent. So we present a modified homotopy perturbation method for solving this kind of equation. The advantages of the approach of this paper lie in the feature that this technique not only achieves an approximate solution with high accuracy, but also improves the calculation speed. Two examples are given to show the effectiveness and convenience of this method. [Copyright &y& Elsevier]

Abstract: The linear complementarity problem is to find a vector in satisfying , ,, where and are given. In this paper, we use the fact that solving is equivalent to solving the nonlinear equation where is a function from into itself defined by . We build a sequence of smooth functions which is uniformly convergent to the function . We show that, an approximation of the solution of the (when it exists) is obtained by solving for a parameter large enough. Then we give a globally convergent hybrid algorithm which is based on vector divisions and the secant method for solving . We close our paper with some numerical simulations to illustrate our theoretical results, and to show that this method can solve efficiently large-scale linear complementarity problems. [Copyright &y& Elsevier]

Abstract: In this paper, we study the following minimization problem where is a bounded linear operator, is some constant, is a potential function for , is the set of fixed points of nonexpansive mapping and is the solution set of an equilibrium problem. This paper introduces two new algorithms (one implicit and one explicit) that can be used to find the solution of the above minimization problem. [Copyright &y& Elsevier]

In this paper, we focus on the policy iteration algorithm (PIA) for the optimal dividend problem under the Cramér–Lundberg risk model. We conclude that the optimal value function is the minimum nonnegative solution of an optimization equation. Under any conditions, it can be approximated by iteration starting with the initial zero-valued policy, i.e., the policy of no dividend payment at all. An auxiliary optimization problem of maximizing dividend payment with terminal reward up to the first claim arrival time is introduced. This auxiliary optimization problem is solved completely in the sense that, no matter what the claim-size distribution is, the explicit formulae for both optimal strategy and value function are presented in general situation. These ensure that, at each iteration, we can get the explicit formulae of the improved strategy and its evaluation function. Finally, the PIA is illustrated with some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate the solution by the neural networks and, since the solution may change dramatically across the interface, we employ different neural networks for each sub-domain. By reformulating the interface problem as a least-squares problem, we discretize the objective function using mean squared error via sampling and solve the proposed deep least-squares method by standard training algorithms such as stochastic gradient descent. The discretized objective function utilizes only the point-wise information on the sampling points and thus no underlying mesh is required. Doing this circumvents the challenging meshing procedure as well as the numerical integration on the complex interfaces. To improve the computational efficiency for more challenging problems, we further design an adaptive sampling strategy based on the residual of the least-squares function and propose an adaptive algorithm. Finally, we present several numerical experiments in both 2D and 3D to show the flexibility, effectiveness, and accuracy of the proposed deep least-square method for solving interface problems. [ABSTRACT FROM AUTHOR]

In this paper, we study a special kind of matrices which are pseudo-Jacobi matrices. Four inverse eigenvalue problems are discussed, and the necessary and sufficient conditions are presented under which these problems are solvable. The corresponding algorithms and some examples are given. Numerical results demonstrate that these methods are of practical utility. [ABSTRACT FROM AUTHOR]

In this paper we survey all known (including own recent results) properties of the longest-edge n -section algorithms. These algorithms (in classical and recently designed conforming form) are nowadays used in many applications, including finite element simulations, computer graphics, etc. as a reliable tool for controllable mesh generation. In addition, we present a list of open problems arising in and around this topic. [ABSTRACT FROM AUTHOR]

Management of wastewater has become a critical environmental problem during past decades, specially for the particular case of urban areas. In this work we use numerical simulation, and combine optimal control theory of partial differential equations with multi-objective optimization techniques to formulate and solve this type of problems from a new original hierarchical viewpoint with a leader and a follower (Stackelberg strategies). Specifically, the main aim of this paper deals with applying these Stackelberg techniques to the environmental problem related to determining the optimal location and the optimal purification profile when building a new treatment plant in a wastewater depuration system. We analyze the control problem, introduce a numerical algorithm, and show the possibilities of our approach by solving a realistic problem posed in the Estuary of Vigo (NW Spain). [ABSTRACT FROM AUTHOR]

In many phase-change problems of practical interest, it is important to know when a phase is depleted, a quantity referred to as the extinction time; however, there are no numerical schemes that are able to compute this with any degree of rigour or formal accuracy. In this paper, we develop such a scheme for the one-dimensional time-dependent problem of an evaporating spherical droplet. The Keller box finite-difference scheme is used, in tandem with the so-called boundary immobilization method. An important component of the work is the careful use of variable transformations that must be built into the numerical algorithm in order to preserve second-order accuracy in both time and space, in particular as regards resolving a square-root singularity in the droplet radius as the extinction time is approached. [ABSTRACT FROM AUTHOR]

In this paper we propose a variation of the Ehrlich-Aberth method for the simultaneous refinement of the zeros of H-palindromic polynomials. [ABSTRACT FROM AUTHOR]

Abstract: The time-dependent Vlasov–Maxwell equations are one of the most complete mathematical equations that model charged particle beams or plasma physics problems. However, the numerical solution of this system often requires a large computational effort. It is worthwhile, whenever possible, to take into account the geometrical or physical particularities of the problem to derive asymptotic simpler approximate models, leading to cheaper simulations. In this paper, we consider the case of high energy short beams, as for example the transport of a bunch of highly relativistic charged particles in the interior of a perfectly conducting hollow tube. We then derive and analyze a new paraxial asymptotic model, that approximates the Vlasov–Maxwell equations and is fourth order accurate with respect to a small parameter which reflects the physical characteristics of the problem. This approach promises to be very powerful in its ability to get an accurate and fast algorithm, easy to be developed. [Copyright &y& Elsevier]

Abstract: Let and be nontrivial involutions, i.e., and . A matrix is called -symmetric if . This paper presents a -symmetric matrix solution to the inverse eigenproblem with a leading principal submatrix constraint. The solvability condition of the constrained inverse eigenproblem is also derived. The existence, the uniqueness and the expression of the -symmetric matrix solution to the best approximation problem of the constrained inverse eigenproblem are achieved, respectively. An algorithm is presented to compute the -symmetric matrix solution to the best approximation problem. Two numerical examples are given to illustrate the effectiveness of our results. [Copyright &y& Elsevier]

Abstract: This paper shows the application of the generalized finite difference method (GFDM) to the problem of dynamic analysis of beams and plates. The stability conditions for a fully explicit algorithm are given for beams and plates. Measures of the irregularity of the clouds of points for beams and plates are given. Various cases of vibrations of beams and plates have been solved and the results show the accuracy of the method for irregular clouds of nodes. [Copyright &y& Elsevier]

Abstract: In this paper, an algorithm is presented for solving second-order nonlinear multi-point boundary value problems (BVPs). The method is based on an iterative technique and the reproducing kernel method (RKM). Two numerical examples are provided to show the reliability and efficiency of the present method. [Copyright &y& Elsevier]

Abstract: In this paper, we analyze a FEM and two-grid FEM decoupling algorithms for elliptic problems on disjoint domains. First, we study the rate of convergence of the FEM and, in particular, we obtain a superconvergence result. Then with proposed algorithms, the solution of the multi-component domain problem (simple example — two disjoint rectangles) on a fine grid is reduced to the solution of the original problem on a much coarser grid together with solution of several problems (each on a single-component domain) on fine meshes. The advantage is the computational cost although the resulting solution still achieves asymptotically optimal accuracy. Numerical experiments demonstrate the efficiency of the algorithms. [Copyright &y& Elsevier]

Abstract: This paper presents an augmented Lagrangian methodology with a stochastic population based algorithm for solving nonlinear constrained global optimization problems. The method approximately solves a sequence of simple bound global optimization subproblems using a fish swarm intelligent algorithm. A stochastic convergence analysis of the fish swarm iterative process is included. Numerical results with a benchmark set of problems are shown, including a comparison with other stochastic-type algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, we propose an interval version of the generalized accelerated overrelaxation methods, which we refer to as IGAOR, for solving the linear complementarity problems, LCP (M, q), and develop a class of multisplitting IGAOR methods which can be easily implemented in parallel. In addition, in regards to the H-matrix with positive diagonal elements, we prove the convergence of these algorithms and illustrate their efficiency through our numerical results. [Copyright &y& Elsevier]

Abstract: In this paper, we first investigate the invertibility of a class of matrices. Based on the obtained results, we then discuss the solvability of Newton equations appearing in the smoothing-type algorithm for solving the second-order cone complementarity problem (SOCCP). A condition ensuring the solvability of such a system of Newton equations is given. In addition, our results also show that the assumption that the Jacobian matrix of the function involved in the SOCCP is a -matrix is not enough for ensuring the solvability of such a system of Newton equations, which is different from the one of smoothing-type algorithms for solving many traditional optimization problems in . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove strong convergence theorems by the hybrid method for a family of hemi-relatively nonexpansive mappings in a Banach space. Our results improve and extend the corresponding results given by Qin et al. [Xiaolong Qin, Yeol Je Cho, Shin Min Kang, Haiyun Zhou, Convergence of a modified Halpern-type iteration algorithm for quasi--nonexpansive mappings, Appl. Math. Lett. 22 (2009) 1051–1055], and at the same time, our iteration algorithm is different from the Kimura and Takahashi algorithm, which is a modified Mann-type iteration algorithm [Yasunori Kimura, Wataru Takahashi, On a hybrid method for a family of relatively nonexpansive mappings in Banach space, J. Math. Anal. Appl. 357 (2009) 356–363]. In addition, we succeed in applying our algorithm to systems of equilibrium problems which contain a family of equilibrium problems. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with a new composite iteration approximating to common fixed points for a finite family of nonexpansive mappings in Banach spaces which have a uniformly Gâteaux differentiable norm. Utilizing the iterative algorithm, we obtain the strong convergence theorems for a finite family of nonexpansive mappings. Furthermore, the problem of image recovery is considered in the above result. Our results extend and improve the corresponding results. [Copyright &y& Elsevier]

Abstract: In this paper, we propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of asymptotically -strict pseudo-contractions in the setting of real Hilbert spaces. By using our proposed scheme, we get a weak convergence theorem for a finite family of asymptotically -strict pseudo-contractions and then we modify these algorithm to have strong convergence theorem by using the two hybrid methods in the mathematical programming. Our results improve and extend the recent ones announced by Ceng, et al.’s result [L.C. Ceng, Al-Homidan, Q.H. Ansari and J.C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Comput. Appl. Math. 223 (2009) 967–974] Qin, Cho, Kang, and Shang, [X. Qin, Y. J. Cho, S. M. Kang, and M. Shang, A hybrid iterative scheme for asymptotically -strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 70 (2009) 1902–1911] and other authors. [Copyright &y& Elsevier]

The multi-block linearly constrained separable convex optimization is frequently applied in numerous applications, including image/signal processing, statistical learning, and data mining, where the objective function is the sum of multiple individual convex functions, and the key constraints are linear. A classical approach to solving such optimization problem could be the alternating direction method of multipliers (ADMM). It decomposes the subproblem into a series of small-scale ones such that its per-iteration cost may be meager. ADMM, however, is designed initially for the two-block model, and its convergence cannot be guaranteed for a general multi-block model without additional assumptions. Dai et al. (2017) proposed the algorithm SUSLM (for Sequential Updating Scheme of the Lagrange Multiplier) for separable convex programming problems. The Lagrange multipliers are updated several times at each iteration, and a correction step is imposed at the end of each iteration. In order to derive its convergence property, a correction step is imposed at the end of each iteration. In this paper, we improve the SUSLM algorithm by introducing two controlled parameters in the updating expressions for decision variables and Lagrange multipliers. The condition of step sizes is then relaxed. We show experimentally that our SUSLM algorithm converges faster than SUSLM. Moreover, result comparisons on robust principal component analysis (RPCA) show better performances than other ADMM-based algorithms. [ABSTRACT FROM AUTHOR]

In this paper, based on the iteration framework (Tian et al., 2019) and relaxed two-step splitting (RTSS) iteration method (Xie and Ma, 2018), we present two relaxed iteration methods for solving the PageRank problem, which are the relaxed generalized inner–outer (RGIO) and relaxed generalized two-step splitting (RGTSS) iteration methods, respectively. Next, their overall convergence properties are analyzed in detail, and choices of the parameters in these algorithms are also discussed. Finally, several numerical examples are given to illustrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]