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This paper presents a procedure to solve the classical location median problem where the distances are measured with ℓ-norms with p > 2. In order to do that we consider an approximated problem. The global convergence of the sequence generated by this iterative scheme is proved. Therefore, this paper closes the still open question of giving a modification of the Weiszfeld algorithm that converges to an optimal solution of the median problem with ℓ norms and $${p \in (2, \infty)}$$. The paper ends with a computational analysis of the different provided iterative schemes. [ABSTRACT FROM AUTHOR]

Under the condition that the values of mapping F are evaluated approximately, we propose a proximal analytic center cutting plane algorithm for solving variational inequalities. It can be considered as an approximation of the earlier cutting plane method, and the conditions we impose on the corresponding mappings are more relaxed. The convergence analysis for the proposed algorithm is also given at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, a three-term conjugate gradient algorithm is developed for solving large-scale unconstrained optimization problems. The search direction at each iteration of the algorithm is determined by rectifying the steepest descent direction with the difference between the current iterative points and that between the gradients. It is proved that such a direction satisfies the approximate secant condition as well as the conjugacy condition. The strategies of acceleration and restart are incorporated into designing the algorithm to improve its numerical performance. Global convergence of the proposed algorithm is established under two mild assumptions. By implementing the algorithm to solve 75 benchmark test problems available in the literature, the obtained results indicate that the algorithm developed in this paper outperforms the existent similar state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

Coordinate descent algorithms solve optimization problems by successively performing approximate minimization along coordinate directions or coordinate hyperplanes. They have been used in applications for many years, and their popularity continues to grow because of their usefulness in data analysis, machine learning, and other areas of current interest. This paper describes the fundamentals of the coordinate descent approach, together with variants and extensions and their convergence properties, mostly with reference to convex objectives. We pay particular attention to a certain problem structure that arises frequently in machine learning applications, showing that efficient implementations of accelerated coordinate descent algorithms are possible for problems of this type. We also present some parallel variants and discuss their convergence properties under several models of parallel execution. [ABSTRACT FROM AUTHOR]

A globally convergent optimization algorithm for solving large nonlinear optimal power flow (OPF) problems is presented. As power systems become heavily loaded, there is an increasing need for globally convergent OPF algorithms. By global convergence, one means the optimization algorithm being able to converge to an OPF solution, if at least one exists, for any choice of initial point. The globally convergent OPF presented is based on an infinity-norm trust region approach, using interior-point methods to solve the trust region subproblems. The performance of the proposed trust region interior-point OPF algorithm, when applied to the IEEE 30-, 57-, 118-, and 300-bus systems, and to an actual 1211-bus system, is compared with that of two widely used nonlinear interior-point methods, namely, a pure primal-dual and its predictor-corrector variant. [ABSTRACT FROM PUBLISHER]

We consider the best approximation problem (BAP) of projecting a point onto the intersection of a number of convex sets. It is known that Dykstra's algorithm is alternating minimization on the dual problem. We extend Dykstra's algorithm so that it can be enhanced by the supporting halfspace and quadratic programming (SHQP) strategy of using quadratic programming to project onto the intersection of supporting halfspaces generated by earlier projection operations. By looking at a structured alternating minimization problem, we show the convergence rate of Dykstra's algorithm when reasonable conditions are imposed to guarantee a dual minimizer. We also establish convergence of using a warmstart iterate for Dykstra's algorithm, show how all the results for the Dykstra's algorithm can be carried over to the simultaneous Dykstra's algorithm, and discuss a different way of incorporating the SHQP strategy. Last, we show that the dual of the best approximation problem can have an O(1=k²) accelerated algorithm that also incorporates the SHQP strategy. [ABSTRACT FROM AUTHOR]

In this work we focus on the problem of approximating multiple roots of nonlinear equations. Multiple roots appear in some applications such as the compression of band-limited signals and the multipactor effect in electronic devices. We present a new family of iterative methods for multiple roots whose multiplicity is known. The methods are optimal in Kung-Traub's sense (Kung and Traub in J Assoc Comput Mach 21:643-651, []), because only three functional values per iteration are computed. By adding just one more function evaluation we make this family derivative free while preserving the convergence order. To check the theoretical results, we codify the new algorithms and apply them to different numerical examples. [ABSTRACT FROM AUTHOR]

SUMMARY A squared Smith type algorithm for solving large-scale discrete-time Stein equations is developed. The algorithm uses restarted Krylov spaces to compute approximations of the squared Smith iterations in low-rank factored form. Fast convergence results when very few iterations of the alternating direction implicit method are applied to the Stein equation beforehand. The convergence of the algorithm is discussed and its performance is demonstrated by several test examples. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We consider the augmented Lagrangian method (ALM) as a solver for the fused lasso signal approximator (FLSA) problem. The ALM is a dual method in which squares of the constraint functions are added as penalties to the Lagrangian. In order to apply this method to FLSA, two types of auxiliary variables are introduced to transform the original unconstrained minimization problem into a linearly constrained minimization problem. Each updating in this iterative algorithm consists of just a simple one-dimensional convex programming problem, with closed form solution in many cases. While the existing literature mostly focused on the quadratic loss function, our algorithm can be easily implemented for general convex loss. We also provide some convergence analysis of the algorithm. Finally, the method is illustrated with some simulation datasets. [ABSTRACT FROM AUTHOR]

The performance of the generalized belief propagation algorithm to compute the noiseless capacity and mutual information rates of finite-size two-dimensional and three-dimensional run-length limited constraints is investigated. In both cases, the problem is reduced to estimating the partition function of graphical models with cycles. The partition function is then estimated using the region-based free energy approximation technique. For each constraint, a method is proposed to choose the basic regions and to construct the region graph which provides the graphical framework to run the generalized belief propagation algorithm. Simulation results for the noiseless capacity of different constraints as a function of the size of the channel are reported. In the cases that tight lower and upper bounds on the Shannon capacity exist, convergence to the Shannon capacity is discussed. For noisy constrained channels, simulation results are reported for mutual information rates as a function of signal-to-noise ratio. [ABSTRACT FROM AUTHOR]

The self-similar converging-diverging shock wave problem introduced by Guderley in 1942 has been the source of considerable mathematical and physical interest. We investigate a novel application of the Guderley solution as a unique and challenging code verification test problem for compressible flow algorithms; this effort requires a unified understanding of the problem's mathematical and computational subtleties. Hence, we review the simplifications and group invariance properties that reduce the compressible flow equations for a polytropic gas to two coupled nonlinear eigenvalue problems: the first for the similarity exponent in the converging regime, and the second for a trajectory multiplier in the diverging regime. The information we provide, together with previously published material, gives a complete description of the computational steps required to construct a semi-analytic Guderley solution. We employ the problem in a quantitative code verification analysis of a cell-centred, finite volume, Eulerian compressible flow algorithm. Lastly, in appended material, we introduce a new approximation for the similarity exponent, which may prove useful in the future construction of certain semi-analytic Guderley solutions. [ABSTRACT FROM PUBLISHER]

An equilibrium problem is investigated based on a hybrid projection iterative algorithm. Strong convergence theorems for solutions of the equilibrium problem are established in a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property. [ABSTRACT FROM AUTHOR]

The particle filter has become an important tool in solving nonlinear filtering problems for dynamic systems. This correspondence extends our recent work, where we proved that the particle filter converges for unbounded functions, using L^4-convergence. More specifically, the present contribution is that we prove that the particle filter converge for unbounded functions in the sense of L^p-convergence, for an arbitrary p\geq 2. [ABSTRACT FROM PUBLISHER]