Search

In this paper, we propose two momentum-preserving finite difference schemes for solving one-dimensional Camassa–Holm equation with periodic boundary conditions. A two-level nonlinear difference scheme and a three-level linearized difference scheme are constructed by using the method of order reduction. For nonlinear scheme, we combine mid-point rule and a specific difference operator, which ensures that our obtained scheme is of second-order convergence in both temporal and spatial directions. For linearized scheme, we apply a linear implicit Crank–Nicolson scheme in the temporal direction, then unique solvability and momentum conservation are analysed in detail. Numerical experiments are provided for Camassa–Holm equation admitting different types of solutions, which demonstrate the convergence order and accuracy of the proposed methods coincide with theoretical analysis. Moreover, numerical results show that the nonlinear scheme exhibits better accuracy for mass conservation, while the linearized scheme is more time-saving in computation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a single timescale stochastic quasi-Newton method for solving the stochastic optimization problems. The objective function of the problem is a composition of two smooth functions and their derivatives are not available. The algorithm sets to approximate sequences to estimate the gradient of the composite objective function and the inner function. The matrix correction parameters are given in BFGS update form for avoiding the assumption that Hessian matrix of objective is positive definite. We show the global convergence of the algorithm. The algorithm achieves the complexity O (ϵ − 1) to find an ϵ − approximate stationary point and ensure that the expectation of the squared norm of the gradient is smaller than the given accuracy tolerance ϵ. The numerical results of nonconvex binary classification problem using the support vector machine and a multicall classification problem using neural networks are reported to show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we present two nonmonotone versions of adaptive cubic regularized (ARC) method for unconstrained optimization problems. The proposed methods are a combination of the ARC algorithm with the nonmonotone line search methods introduced by Zhang and Hager [A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14 (2004), pp. 1043–1056] and Ahookhosh et al. [A nonmonotone trust-region line search method for large-scale unconstrained optimization, Appl. Math. Model. 36 (2012), pp. 478–487]. The global convergence analysis for these iterative algorithms is established under suitable conditions. Several numerical examples are given to illustrate the efficiency and robustness of the newly suggested methods. The obtained results show the satisfactory performance of the proposed algorithms when compared to the basic ARC algorithm. [ABSTRACT FROM AUTHOR]

This paper proposes an affine scaling interior trust-region method in association with nonmonotone line search filter technique for solving nonlinear optimization problems subject to linear inequality constraints. Based on a Newton step which is derived from the complementarity conditions of linear inequality constrained optimization, a trust-region subproblem subject only to an ellipsoidal constraint is defined by minimizing a quadratic model with an appropriate quadratic function and scaling matrix. The nonmonotone schemes combining with trust-region strategy and line search filter technique can bring about speeding up the convergence progress in the case of high nonlinear. A new backtracking relevance condition is given which assures global convergence without using the switching condition used in the traditional line search filter technique. The fast local convergence rate of the proposed algorithm is achieved which is not depending on any external restoration procedure. The preliminary numerical experiments are reported to show effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We propose an infeasible active set QP-free algorithm for general constrained optimization in this paper. It starts from an arbitrary initial point. At each iteration, only two or three reduced linear equations with the same coefficients are solved to obtain the search direction. To determine the working set, the method makes use of multipliers from the last iteration, eliminating the need to compute a new estimate, and no additional linear systems are solved to select linear independent constraint gradients. The infeasibility measure and the objective function value are controlled separately by the filter technique. Without the positive definiteness assumption on the Hessian estimate, the sequence generated by the algorithm still globally converges to a Karush-Kuhn-Tucker point. And the algorithm obtains superlinear convergence without the strict complementarity. At last, preliminary numerical results are reported. [ABSTRACT FROM PUBLISHER]

Based on the Moreau–Yosida regularization and a modified line search technique, this paper presents an implementable memory gradient method for solving a possibly non-differentiable convex minimization problem by converting the original objective function to a once continuously differentiable function. A main feature of this proposed method is that at each iteration, it sufficiently uses the previous multi-step iterative information and avoids the storage and computation of some matrices. Moreover, the proposed method makes use of approximate function and gradient values of the Moreau–Yosida regularization instead of the corresponding exact values. Under reasonable conditions, the convergence properties of the proposed algorithm are analysed. Preliminary numerical results show that the proposed method is efficient and can be applied to solve large-scale non-smooth optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, an affine-scaling derivative-free trust-region method with interior backtracking line search technique is considered for solving nonlinear systems subject to linear inequality constraints. The proposed algorithm is designed to take advantage of the problem structured by building polynomial interpolation models for each function in the nonlinear system functionF. The proposed approach is developed by forming a quadratic model with an appropriate quadratic function and scaling matrix: there is no need to handle the constraints explicitly. By using both trust-region strategy and interior backing line search technique, each iteration switches to backtracking step generated by the trust-region subproblem and satisfies strict interior point feasibility by line search backtracking technique. Under reasonable conditions, the global convergence and fast local convergence rate of the proposed algorithm are established. The results of numerical experiments are reported to show the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive relaxation method and a discontinuity treatment of edges are proposed to improve the digital image denoising process by using the fourth-order partial differential equation (known as the YK model) first proposed by You and Kaveh. Since the YK model would generate some speckles into the denoised image, a relaxation method is incorporated into the model to reduce the formation of isolated speckles. An additional improvement is employed to handle the discontinuity on the edges of the image. In order to stop the iteration automatically, a control of the iteration is integrated into the denoising process. Numerical results demonstrate that such modifications not only make the denoised image look more natural, but also achieve a higher value of peak signal noise ratio (PSNR). [ABSTRACT FROM AUTHOR]

In this paper, we propose a nonmonotone sequential quadratic programming-filter method for solving nonlinear equality constrained optimization. This new method has more flexibility for the acceptance of the trial step and requires less computational costs compared with the monotone methods. Under reasonable conditions, we give the global convergence properties. Further, the second-order correction step and nonmonotone reduction conditions are used to overcome Maratos effect so that quadratic local convergence is achieved. The numerical experiments are reported to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Sequential quadratic programming (SQP) methods have been extensively studied to handle nonlinear programming problems. In this paper, a new SQP approach is employed to tackle nonlinear complementarity problems (NCPs). At each iterate, NCP conditions are divided into two parts. The inequalities and equations in NCP conditions, which are violated in the current iterate, are treated as the objective function, and the others act as constraints, which avoids finding a feasible initial point and feasible iterate points. NCP conditions are consequently transformed into a feasible nonlinear programming subproblem at each step. New SQP techniques are therefore successful in handling NCPs. [ABSTRACT FROM AUTHOR]

In this paper we present a new class of memory gradient methods for unconstrained optimization problems and develop some useful global convergence properties under some mild conditions. In the new algorithms, trust region approach is used to guarantee the global convergence. Numerical results show that some memory gradient methods are stable and efficient in practical computation. In particular, some memory gradient methods can be reduced to the BB method in some special cases. [ABSTRACT FROM AUTHOR]

This paper presents a new hybrid algorithm for unconstrained optimization problems, which combines the idea of the IMPBOT algorithm with the nonmonotone line search technique. A feature of the proposed method is that at each iteration, a system of linear equations is solved only once to obtain a trial step, via a modified limited-memory BFGS two loop recursion that requires only matrix–vector products, thus reducing the computations and storage. Furthermore, when the trial step is not accepted, the proposed method performs a line search along it using a modified nonmonotone scheme, thus a larger stepsize can be yielded in each line search procedure. Under some reasonable assumptions, the convergence properties of the proposed algorithm are analysed. Numerical results are also reported to show the efficiency of this proposed method. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a trust-region algorithm in association with line search filter technique for solving nonlinear equality constrained programming. At current iteration, a trial step is formed as the sum of a normal step and a tangential step which is generated by trust-region subproblem and the step size is decided by interior backtracking line search together with filter methods. Then, the next iteration is determined. This is different from general trust-region methods in which the next iteration is determined by the ratio of the actual reduction to the predicted reduction. The global convergence analysis for this algorithm is presented under some reasonable assumptions and the preliminary numerical results are reported. [ABSTRACT FROM PUBLISHER]

Conjugate gradient method is a root-finding algorithm to non-linear equations. In this paper, we suggest extending this method for a polynomial to the complex plane. Through the experimental and theoretical mathematics method, we drew the following conclusions: (1) the conjugate gradient is a dynamical system with two complex parameters; (2) locally conditions for convergence to any roots of complex functions is given; (3) the conjugate gradient method may fail to converge to all roots for cubic with three simple roots; (4) the boundary of conjugate gradient basins are fractals in some cases, and depends on the parameters; (5) the algorithm is then improved by introducing a method to determine the optimal parameters. [ABSTRACT FROM AUTHOR]

A model with six mutually exclusive compartments related to dengue is studied. Three vector control tools are considered: insecticides (larvicide and adulticide) and mechanical control. The basic reproduction number associated to the model is presented. The problem is studied using an optimal control approach. The human data are based on the dengue outbreak that occurred in Cape Verde. Control measures are simulated in different scenarios and their consequences analysed. [ABSTRACT FROM PUBLISHER]

We propose a trust region multidimensional filter SQP algorithm. The multidimensional filter technique proposed by Gould et al. [SIAM J. Optim., 15 (2005), pp. 17-38] is extended to solve constrained optimization problems. The constraints are partitioned into p parts. The entry of our filter consists of these different parts. Not only the criteria for accepting a trial step would be relaxed, but also the individual behaviour of each part of the constraints is considered. The filter's entries and the acceptance criteria are different from other filter-related algorithms in the literature. It should be noted that the undesirable link between the objective function and the constraint violation function in the filter acceptance criteria disappears. Our algorithm is also combined with the non-monotone technique for accepting a trial step, which leads to a more flexible acceptance criteria. Under mild conditions, global convergence is proved. Numerical results show the robustness and efficiency of our algorithm. [ABSTRACT FROM AUTHOR]