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Concave knapsack problems with integer variables have many applications in real life, and they are NP-hard. In this paper, an exact and efficient algorithm is presented for concave knapsack problems. The algorithm combines the contour cut with a special cut to improve the lower bound and reduce the duality gap gradually in the iterative process. The lower bound of the problem is obtained by solving a linear underestimation problem. A special cut is performed by exploiting the structures of the objective function and the feasible region of the primal problem. The optimal solution can be found in a finite number of iterations, and numerical experiments are also reported for two different types of concave objective functions. The computational results show the algorithm is efficient. [ABSTRACT FROM AUTHOR]

Recently, Salkuyeh and Fahim [A new iterative refinement of the solution of ill-conditioned linear system of equations, Int. Comput. Math. 88(5) (2011), pp. 950–956] have proposed a two-step iterative refinement of the solution of an ill-conditioned linear system of equations. In this paper, we first present a generalized two-step iterative refinement procedure to solve ill-conditioned linear system of equations and study its convergence properties. Afterward, it is shown that the idea of an orthogonal projection technique together with a basic stationary iterative method can be utilized to construct a new efficient and neat hybrid algorithm for solving the mentioned problem. The convergence of the offered hybrid approach is also established. Numerical examples are examined to demonstrate the feasibility of proposed algorithms and their superiority to some of existing approaches for solving ill-conditioned linear system of equations. [ABSTRACT FROM PUBLISHER]

In this paper,we propose a new predictor–corrector interior-point method for symmetric cone programming. This algorithm is based on a wide neighbourhood and the Nesterov–Todd direction. We prove that, besides the predictor steps, each corrector step also reduces the duality gap by a rate of, whereris the rank of the associated Euclidean Jordan algebras. In particular, the complexity bound is, whereis a given tolerance. To our knowledge, this is the best complexity result obtained so far for interior-point methods with a wide neighbourhood over symmetric cones. The numerical results show that the proposed algorithm is effective. [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]

This study presents a new trust-region algorithm to solve the -minimization problem with applications to compressed sensing (CS) and image deblurring that will be augmented with a shrinkage operation to produce a new iteration whenever an approximated solution of the trust-region subproblem lies within one and iterate is successful, simultaneously. Otherwise, a nonmonotone Armijo-type line search strategy incorporates with shrinkage technique, which includes a convex combination of the maximum function value of some preceding iterates and the current function value. Therefore, the proposed approach takes advantages of both the effective trust-region and nonmonotone Armijo-type line search with a shrinkage operation. It is believed that selecting an appropriate shrinkage parameter according to a new procedure can improve the efficiency of our algorithm. The global convergence and the R-linear convergence rate of the proposed approach are proved for which numerical results are also reported. [ABSTRACT FROM AUTHOR]

In this article, a recently introduced iterative method based on Green's functions and fixed-point iteration schemes is presented for the approximate solutions of delay and functional differential equations. The approach is especially suited for handling boundary value problems (BVPs). The algorithm is illustrated through a number of examples that confirm the high accuracy and efficiency of the strategy. The results of the test examples show excellent agreement with exact solutions and outperforms other existing numerical iterative schemes. [ABSTRACT FROM AUTHOR]

We present an adaptive trust-region algorithm to solve systems of nonlinear equations. Using the nonmonotone technique of Grippo, Lampariello and Lucidi, we introduce a new adaptive radius to decrease the total number of iterations and function evaluations. In contrast with the pervious methods, the new adaptive radius ensures that the size of radius is not too large or too small. We show that the sequence generated by the proposed adaptive radius is decreasing, so it prevents the production of too large radius as possible. Furthermore, it is shown that this sequence is reduced slowly, so it prevents the production of the intensely small radius. The global and quadratic convergence of the proposed approach are proved. Preliminary numerical results of our algorithm are also reported which indicate the promising behaviour of the new procedure to solve systems of nonlinear equations. [ABSTRACT FROM AUTHOR]