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In this paper, iterative algorithm for strong vector equilibrium problem (SVEP) is studied. Firstly, an auxiliary problem for SVEP is introduced and the relationships between these two problems are discussed. Then, based on the auxiliary problem, a projection iterative algorithm for SVEP is proposed. Moreover, analysis of convergence of this iterative algorithm is investigated under suitable conditions of continuity and convexity. The main result obtained in this paper generalizes and improves the corresponding ones of Iusem and Sosa [Iusem AN, Sosa W. Iterative algorithms for equilibrium problems. Optimization. 2003;52(3):301–316.] and Cheng and Liu [Cheng B, Liu SY. An iterative algorithm for vector equilibrium problems. J. Lanzhou Univ. (Nat. Sci.). 2009;45(5):105–109.]. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear iterative methods, which are different from the classical algorithms, to solve inverse problems are presented. Our methods by denoting some parameters and some properties of the algorithm in both noise and noiseless cases are studied. Finally, the convergence of the sequence generated by the algorithm without noise is discussed. [ABSTRACT FROM AUTHOR]

Bregman iterative algorithms have been extensively used forand total variation regularization problems, allowing to obtain simple, fast and effective algorithms. In this paper, three already-available algorithms for geosounding inversion are modified by including them in a Bregman iterative procedure. The resulting algorithms are easy to implement and do not require any optimization package. Modelling results are presented for synthetic and field data, observing better convergence properties than the original versions, avoiding the need of any continuation descent procedure. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following minimization problem:where,,,andare given. An efficient inequality relaxation technique is presented to relax the matrix inequality constraint so that there is an optimal solution which is(R,S)-symmetric that minimize, and also satisfies the corrected matrix inequality constraint. A hybrid algorithm with convergence analysis is given to solve this problem. Numerical examples show that the algorithm requires less CPU times when compared with some other methods. [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]

We propose a new family of subgradient- and gradient-based methods which converges with optimal complexity for convex optimization problems whose feasible region is simple enough. This includes cases where the objective function is non-smooth, smooth, have composite/saddle structure, or are given by an inexact oracle model. We unified the way of constructing the subproblems which are necessary to be solved at each iteration of these methods. This permitted us to analyse the convergence of these methods in a unified way compared to previous results which required different approaches for each method/algorithm. Our contribution rely on two well-known methods in non-smooth convex optimization: the mirror-descent method (MDM) by Nemirovski-Yudin and the dual-averaging method by Nesterov. Therefore, our family of methods includes them and many other methods as particular cases. For instance, the proposed family of classical gradient methods and its accelerations generalize Devolder et al.'s, Nesterov's primal/dual gradient methods, and Tseng's accelerated proximal gradient methods. Also our family of methods can partially become special cases of other universal methods, too. As an additional contribution, the novel extended MDM removes the compactness assumption of the feasible region and the fixation of the total number of iterations which is required by the original MDM in order to attain the optimal complexity. [ABSTRACT FROM AUTHOR]

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]

The study was conducted with the aim to find out random variational-like inclusion involving relaxed monotone, random operators in Hilbert space and construct a new iterative algorithm. The study also proves the existence of random solution of random variational-like inclusion problem and convergence of random iterative sequences generated by random iterative algorithm. The results in this paper unify, improve and extend some well-known results from the literature. [ABSTRACT FROM AUTHOR]

In this article, we deal with an iteration method for approximating a fixed point of a contraction mapping using Mann’s algorithm under functional random errors. We first show its almost complete convergence to the fixed point by mean of an exponential inequality and then we specify the induced rate of convergence. We finally build a confidence domain for the fixed point. Moreover, some numerical examples are considered. [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]

Motivated by the commonly encountered problem in which tracking is only required at selected intermediate points within the time interval, a general optimisation-based iterative learning control (ILC) algorithm is derived that ensures convergence of tracking errors to zero whilst simultaneously minimising a specified quadratic objective function of the input signals and chosen auxiliary (state) variables. In practice, the proposed solutions enable a repeated tracking task to be accurately completed whilst simultaneously reducing undesirable effects such as payload spillage, vibration tendencies and actuator wear. The theory is developed using the well-known norm optimal ILC (NOILC) framework, using general linear, functional operators between real Hilbert spaces. Solutions are derived using feedforward action, convergence is proved and robustness bounds are presented using both norm bounds and positivity conditions. Algorithms are specified for both continuous and discrete-time state-space representations, with the latter including application to multi-rate sampled systems. Experimental results using a robotic manipulator confirm the practical utility of the algorithms and the closeness with which observed results match theoretical predictions. [ABSTRACT FROM AUTHOR]