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This paper reviews recent studies on the Particle Swarm Optimization (PSO) algorithm. The review has been focused on high impact recent articles that have analyzed and/or modified PSO algorithms. This paper also presents some potential areas for future study.

This paper studies the convergence of the proximal point method for quasiconvex functions in finite dimensional complete Riemannian manifolds. We prove initially that, in the general case, when the objective function is proper and lower semicontinuous, each accumulation point of the sequence generated by the method, if it exists, is a limiting critical point of the function. Then, under the assumptions that the sectional curvature of the manifold is bounded above by some non negative constant and the objective function is quasiconvex we analyze two cases. When the constant is zero, the global convergence of the algorithm to a limiting critical point is assured and if it is positive, we prove the local convergence for a class of quasiconvex functions, which includes Lipschitz functions. [ABSTRACT FROM AUTHOR]

A numerical algorithm for the solution of an inverse coefficient problem for nonstationary, nonlinear production-destruction type model is proposed and tested on an example of the Lorenz'63 system. With an ensemble of adjoint problem solutions, the inverse problem is transformed into a quasi-linear matrix problem and solved with Newton-type algorithm. Two different ways of the adjoint ensemble construction are compared. In the first one, a trigonometric basis is used. In the second one in situ measurements are taken into account. Local convergence properties of the algorithm are studied numerically to find out when the use of more data can lead to the degradation of the reconstruction results. [ABSTRACT FROM AUTHOR]

We study the convergence issue for the gradient algorithm (employing general step sizes) for optimization problems on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity (resp., weak sharp minima), local/global convergence (resp., linear convergence) results are established. As an application, the linear convergence properties of the gradient algorithm employing the constant step sizes and the Armijo step sizes for finding the Riemannian Lp (p ∈ [1, + ∞)) centers of mass are explored, respectively, which in particular extend and/or improve the corresponding results in [B. Afsari, R. Tron, and R. Vidal, SIAM J. Control Optim., 51 (2013), pp. 2230-2260; G. C. Bento et al., J. Optim. Theory Appl., 183 (2019), pp. 977-992]. [ABSTRACT FROM AUTHOR]

We propose a gradient-based Jacobi algorithm for a class of maximization problems on the unitary group, with a focus on approximate diagonalization of complex matrices and tensors by unitary transformations. We provide weak convergence results, and prove local linear convergence of this algorithm. The convergence results also apply to the case of real-valued tensors. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present an inverse iteration algorithm, to find the H-spectral radius and the associated positive eigenvector of a nonnegative weakly irreducible tensor, which always preserve the positivity of approximate eigenvectors. The local quadratical convergence of the proposed algorithm is established based on a basic result of the Newton's method for solving nonlinear equations. Some numerical examples illustrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a class of structured set-valued operators, which we call union averaged nonexpansive. At each point in their domain, the value of such an operator can be expressed as a finite union of single-valued averaged nonexpansive operators. We investigate various structural properties of the class and show, in particular, that is closed under taking unions, convex combinations, and compositions, and that their fixed point iterations are locally convergent around strong fixed points. We then systematically apply our results to analyze proximal algorithms in situations, where union averaged nonexpansive operators naturally arise. In particular, we consider the problem of minimizing the sum two functions, where the first is convex and the second can be expressed as the minimum of finitely many convex functions. [ABSTRACT FROM AUTHOR]

In this paper, we present an interior point method for nonlinear programming that avoids the use of penalty function or filter. We use an adaptively perturbed primal dual interior point framework to computer trial steps and a central path technique is used to keep the iterate bounded away from 0 and not to deviate too much from the central path. A trust-funnel-like strategy is adopted to drive convergence. We also use second-order correction (SOC) steps to achieve fast local convergence by avoiding Maratos effect. Furthermore, the presented algorithm can avoid the blocking effect. It also does not suffer the blocking of productive steps that other trust-funnel-like algorithm may suffer. We show that, under second-order sufficient conditions and strict complementarity, the full Newton step (combined with an SOC step) will be accepted by the algorithm near the solution, and hence the algorithm is superlinearly local convergent. Numerical experiments results, which are encouraging, are reported. [ABSTRACT FROM AUTHOR]

The aim of this paper is to find new iterative Newton-like schemes inspired by the modified Newton iterative algorithm and prove that these iterations are faster than the existing ones in the literature. We further investigate their behavior and finally illustrate the results by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive trust region algorithm in which the trust region radius converges to zero is presented for solving large-residual nonsmooth least squares problems. This algorithm uses the smoothing technique of the approximation function, and it combines an adaptive trust region radius. Moreover, this algorithm differs from the existing methods for solving nonsmooth equations through use of the approximation function of second-order information, which improves the convergence rate for large-residual nonsmooth least squares problems. Under some suitable conditions, the global and local superlinear convergences of the proposed method are proven. The preliminary numerical results indicate that the proposed algorithm is effective and suitable for solving large-residual nonsmooth least squares problems. [ABSTRACT FROM AUTHOR]

Solving equations of the form H ( x ) = 0 is usually done by applying iterative methods. The main interest of this paper is to improve the domain of starting points for Steffensen’s method. In general, the accessibility of iterative methods that use divided differences in their algorithms is reduced, since there are difficulties in the choice of starting points to guarantee the convergence of the methods. In particular, by using a decomposition of the operator H and applying a special type of iterative methods, which combine two iterative schemes in the algorithms, we can improve the accessibility of Steffensen’s method. Moreover, we analyze the local convergence of the new iterative method proposed in two cases: when H is differentiable and H is non-differentiable. The dynamical properties show that the method also improves the region of accessibility of Steffensen’s method for non-differentiable operators. So, we present an alternative for the non-applicability of Newton’s method to non-differentiable operators that improves the accessibility of Steffensen’s method. The theoretical results are illustrated with numerical experiments. [ABSTRACT FROM AUTHOR]

The gradient sampling method is a recently developed tool for solving unconstrained nonsmooth optimization problems. Using just first-order information about the objective function, it generalizes the steepest descent method, one of the most classical methods for minimizing a smooth function. This study aims at determining under which circumstances one can expect the same local convergence result of the Cauchy method for the gradient sampling algorithm under the assumption that the problem is stated by a finite max-function around the optimal point. Additionally, at the end, we show how to practically accomplish the required hypotheses during the execution of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose an algorithm to estimate the radius of convergence for the Picard iteration in the setting of a real Hilbert space. Numerical experiments show that the proposed algorithm provides convergence balls close to or even identical to the best ones. As the algorithm does not require to evaluate the norm of derivatives, the computing effort is relatively low. [ABSTRACT FROM AUTHOR]

We propose a trust region algorithm for solving nonconvex smooth optimization problems. For any $$\overline{\epsilon }\in (0,\infty )$$ , the algorithm requires at most $$\mathcal{O}(\epsilon ^{-3/2})$$ iterations, function evaluations, and derivative evaluations to drive the norm of the gradient of the objective function below any $$\epsilon \in (0,\overline{\epsilon }]$$ . This improves upon the $$\mathcal{O}(\epsilon ^{-2})$$ bound known to hold for some other trust region algorithms and matches the $$\mathcal{O}(\epsilon ^{-3/2})$$ bound for the recently proposed Adaptive Regularisation framework using Cubics, also known as the arc algorithm. Our algorithm, entitled trace, follows a trust region framework, but employs modified step acceptance criteria and a novel trust region update mechanism that allow the algorithm to achieve such a worst-case global complexity bound. Importantly, we prove that our algorithm also attains global and fast local convergence guarantees under similar assumptions as for other trust region algorithms. We also prove a worst-case upper bound on the number of iterations, function evaluations, and derivative evaluations that the algorithm requires to obtain an approximate second-order stationary point. [ABSTRACT FROM AUTHOR]

A trust-region algorithm in association with line search filter technique for solving nonlinear equality constrained programming is proposed in this paper. In the current iteration, the trial step providing sufficient descent is generated by solving a corresponding trust-region subproblem. Then, the step size is decided by backtracking line search together with filter technique to obtain the next iteration point. The advantage of this method is that resolving trust-region subproblem many times to determine a new iteration point in traditional trust-region method can be avoided and hence the expensive computation can be lessened. And the difficult decisions in regard to the choice of penalty parameters in the merit functions can be avoided by using filter technique. Second order correction steps are introduced in the proposed algorithm to overcome Maratos effect. Convergence analysis shows that fast local convergence can be achieved under some mild assumptions. The preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

We study the local convergence of Chebyshev-Halley-type methods of convergence order at least five to approximate a locally unique solution of a nonlinear equation. Earlier studies such as Behl (), Bruns and Bailey (Chem. Eng. Sci 32, 257-264, ), Candela and Marquina (Computing 44, 169-184, ), (Computing 45(4):355-367, ), Chicharro et al. (), Chun (Appl. Math. Comput, 190(2):1432-1437, ), Cordero et al. (Appl.Math. Lett. 26, 842-848, ), Cordero et al. (Appl. Math. Comput. 219, 8568-8583, ), Cordero and Torregrosa (Appl. Math. Comput. 190, 686-698, ), Ezquerro and Hernández (Appl. Math. Optim. 41(2):227-236, ), (BIT Numer. Math. 49, 325-342, ), (J. Math. Anal. Appl. 303, 591-601, ), Gutiérrez and Hernández (Comput. Math. Applic. 36(7):1-8, ), Ganesh and Joshi (IMA J. Numer. Anal. 11, 21-31, ), Hernández (Comput. Math. Applic. 41(3-4):433-455, ), Hernández and Salanova (Southwest J. Pure Appl. Math. 1, 29-40, ), Jarratt (Math. Comput. 20(95):434-437, ), Kou and Li (Appl. Math. Comput. 189, 1816-1821, ), Li (Appl. Math. Comput. 235, 221-225, ), Ren et al. (Numer. Algorithm. 52(4):585-603, ), Wang et al. (Numer. Algorithm. 57, 441-456, ), Kou et al. (Numer. Algorithm. 60, 369-390, ) show convergence under hypotheses on the third derivative or even higher. The convergence in this study is shown under hypotheses on the first derivative. Hence, the applicability of the method is expanded. The dynamical analyses of these methods are also studied. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply. [ABSTRACT FROM AUTHOR]

To better predict time series, in this paper the single multiplicative recurrent neuron (SMRN) is constructed by adding feedforward and feedback links at the nodes of the original single multiplicative neuron (SMN). Glowworm swarm optimization (GSO) algorithm as a method for training the parameters of various kinds of neural network models gets easily into locally optimal traps during optimization process and its movement stability is also poor because of no memory about search history. To overcome the aforementioned disadvantages, firstly the linearly declining inertia weight is incorporated into the location update formula of standard GSO (LWGSO). After that in order to further enhance the robustness capability, differential evolution (DE) algorithm is introduced into LWGSO forming LWGSODE. Standard unimodal and multi-modal static test functions in high dimensions have been used to test its properties. The statistically experimental results show that the proposed LWGSODE approach performs much better than basic GSO whatever in terms of solutions precision, robustness or convergence speed. Moreover, the function optimization results are also competitive when compared with other state-of-the-art methods in the literature. Finally, the LWGSODE algorithm is used to train SMRN for time series prediction, and approximation results have improved significantly. All the results obtained reveal the novel SMRN model combined with the proposed LWGSODE algorithm provides a promising means to approximate nonlinear series in the future. [ABSTRACT FROM AUTHOR]

Abstract: We present a class of trust region algorithms that do not use any penalty function or a filter for nonlinear equality constrained optimization. In each iteration, the infeasibility is controlled by a progressively decreasing upper limit and trial steps are computed by a Byrd–Omojokun-type trust region strategy. Measures of optimality and infeasibility are computed, whose relationship serves as a criterion on which the algorithm decides which one to focus on improving. As a result, the algorithm keeps a balance between the improvements on optimality and feasibility even if no restoration phase which is required by filter methods is used. The framework of the algorithm ensures the global convergence without assuming regularity or boundedness on the iterative sequence. By using a second order correction strategy, Marato’s effect is avoided and fast local convergence is shown. The preliminary numerical results are reported. [Copyright &y& Elsevier]

Abstract: In this paper, the multistability of a class of Amari’s -divergence based nonnegative matrix factorization learning algorithms is analyzed. The analysis results show that invariant sets for the update algorithms can be constructed. In these invariant sets, the non-convergence of the discussed algorithms can be guaranteed. Based on Lyapunov’s stability theorem, the local convergence of this class of learning algorithms is proved in the domain of their update rules. In the simulation, the analysis results are applied to image representation. Experiment results demonstrate that selecting suitable initial data for different applications of these nonnegative matrix factorization algorithms is very important. [Copyright &y& Elsevier]

In Zhang et al. (accepted by SIAM J. Optim., ), we developed a class of derivative-free algorithms, called DFLS, for least-squares minimization. Global convergence of the algorithm as well as its excellent numerical performance within a limited computational budget was established and discussed in the same paper. Here we would like to establish the local quadratic convergence of the algorithm for zero residual problems. Asymptotic convergence performance of the algorithm for both zero and nonzero problems is tested. Our numerical experiments indicate that the algorithm is also very promising for achieving high accuracy solutions compared with software packages that do not exploit the special structure of the least-squares problem or that use finite differences to approximate the gradients. [ABSTRACT FROM AUTHOR]

n Adaptive Regularisation algorithm using Cubics (ARC) is proposed for unconstrained optimization, generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, University of Cambridge), an algorithm by Nesterov and Polyak (Math Program 108(1):177-205, 2006) and a proposal by Weiser et al. (Optim Methods Softw 22(3):413-431, 2007). At each iteration of our approach, an approximate global minimizer of a local cubic regularisation of the objective function is determined, and this ensures a significant improvement in the objective so long as the Hessian of the objective is locally Lipschitz continuous. The new method uses an adaptive estimation of the local Lipschitz constant and approximations to the global model-minimizer which remain computationally-viable even for large-scale problems. We show that the excellent global and local convergence properties obtained by Nesterov and Polyak are retained, and sometimes extended to a wider class of problems, by our ARC approach. Preliminary numerical experiments with small-scale test problems from the CUTEr set show encouraging performance of the ARC algorithm when compared to a basic trust-region implementation. [ABSTRACT FROM AUTHOR]

In Gonzaga et al. [A globally convergent filter method for nonlinear programming, SIAM J. Optimiz. 14 (2003), pp. 646-669] we discuss general conditions to ensure global convergence of inexact restoration filter algorithms for non-linear programming. In this article we show how to avoid the Maratos effect by means of a second-order correction. The algorithms are based on feasibility and optimality phases, which can be either independent or not. The optimality phase differs from the original one only when a full Newton step for the tangential minimization of the Lagrangian is efficient but not acceptable by the filter method. In this case a second-order corrector step tries to produce an acceptable point keeping the efficiency of the rejected step. The resulting point is tested by trust region criteria. Under the usual hypotheses, the algorithm inherits the quadratic convergence properties of the feasibility and optimality phases. This article includes a comparison between classical Sequential Quadratic Programming (SQP) and Inexact Restoration (IR) iterations, showing that both methods share the same asymptotic convergence properties. [ABSTRACT FROM AUTHOR]

Abstract: Point set registration is important for calibration of multiple cameras, 3D reconstruction and recognition, etc. The iterative closest point (ICP) algorithm is accurate and fast for point set registration in a same scale, but it does not handle the case with different scales. This paper instead introduces a novel approach named the scaling iterative closest point (SICP) algorithm which integrates a scale matrix with boundaries into the original ICP algorithm for scaling registration. At each iterative step of this algorithm, we set up correspondence between two m–D point sets, and then use a simple and fast iterative algorithm with the singular value decomposition (SVD) method and the properties of parabola incorporated to compute scale, rotation and translation transformations. The SICP algorithm has been proved to converge monotonically to a local minimum from any given parameters. Hence, to reach desired global minimum, good initial parameters are required which are successfully estimated in this paper by analyzing covariance matrices of point sets. The SICP algorithm is independent of shape representation and feature extraction, and thereby it is general for scaling registration of m–D point sets. Experimental results demonstrate its efficiency and accuracy compared with the standard ICP algorithm. [Copyright &y& Elsevier]

We develop an affine-scaling algorithm for box-constrained optimization which has the property that each iterate is a scaled cyclic Barzilai–Borwein (CBB) gradient iterate that lies in the interior of the feasible set. Global convergence is established for a nonmonotone line search, while there is local R-linear convergence at a nondegenerate local minimizer where the second-order sufficient optimality conditions are satisfied. Numerical experiments show that the convergence speed is insensitive to problem conditioning. The algorithm is particularly well suited for image restoration problems which arise in positron emission tomography where the cost function can be infinite on the boundary of the feasible set. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a new modified Newton method for nonlinear equations. This method uses a part of elements of the Jacobian matrix to obtain the next iteration point and is refereed to as the incomplete Jacobian Newton (IJN) method. The IJN method may be fit for solving large scale nonlinear equations with dense Jacobian. The conditions of linear, superlinear and quadratic convergence of the IJN method are given and the local convergence results are analyzed and proved. Some special IJN algorithms are designed and numerical experiments are given. The results show that the IJN method is promising. [Copyright &y& Elsevier]

In this paper we analyze the rate of local convergence of the Newton primal-dual interior-point method when the iterates are kept strictly feasible with respect to the inequality constraints. It is shown under the classical conditions that the rate is q-quadratic when the functions associated to the binding inequality constraints are concave. In general, the q-quadratic rate is achieved provided the step in the primal variables does not become asymptotically orthogonal to any of the gradients of the binding inequality constraints. Some preliminary numerical experience showed that the feasible method can be implemented in a relatively efficient way, requiring a reduced number of function and derivative evaluations. Moreover, the feasible method is competitive with the classical infeasible primal-dual interior-point method in terms of number of iterations and robustness. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents the new convergence analysis for the Newton-like method proposed by [Xinyuan Wu, A new continuation Newton-like method and its deformation, Appl. Math. Comput. 112 (2000) 75–78.] Compared with the original version of the convergence analysis, the restriction imposed on f″(x) is removed thoroughly. In order to guarantee the quadratic convergence of the Newton-like method, it is suffices to suppose that f′(x ∗)≠0 and f′(x) is local Lipschitz near x ∗, where x ∗ is a solution of nonlinear equation f(x)=0. Moreover, some comments are given with examples for the Newton-like method, in comparison with Newton’s method. It can be concluded that the new algorithm is more feasible, effective than Newton’s method. In particularly, when it happens that, the derivative of the function f(x) at an iterate is singular or almost singular, the Newton-like method is vast superior to the classical Newton method. The numerical results of the paper strongly support the conclusion. [Copyright &y& Elsevier]

A general algorithmic scheme for solving inclusions in a Banach space is investigated in respect to its local convergence behavior. Particular emphasis is placed on applications to certain proximal-point-type algorithms in Hilbert spaces. The assumptions do not necessarily require that a solution be isolated. In this way, results existing for the case of a locally unique solution can be extended to cases with nonisolated solutions. Besides the convergence rates for the distance of the iterates to the solution set, strong convergence to a sole solution is shown as well. As one particular application of the framework, an improved convergence rate for an important case of the inexact proximal-point algorithm is derived. [ABSTRACT FROM AUTHOR]

Local convergence of an inexact-restoration method for nonlinear programming is proved. Numerical experiments are performed with the objective of evaluating the behavior of the purely local method against a globally convergent nonlinear programming algorithm. [ABSTRACT FROM AUTHOR]

A line search method is proposed for nonlinear programming using Fletcher and Leyffer's filter method, which replaces the traditional merit function. A simple modification of the method proposed in a companion paper [SIAM J. Optim., 16 (2005), pp. 1--31] introducing second order correction steps is presented. It is shown that the proposed method does not suffer from the Maratos effect, so that fast local convergence to second order sufficient local solutions is achieved. [ABSTRACT FROM AUTHOR]

The robustness of robot calibration with respect to sensor noise is sensitive to the manipulator poses used to collect measurement data. In this paper we propose an algorithm based on a constrained optimization method, which allows us to choose a set of measurement configurations. It works by selecting iteratively one pose after another inside the workspace. After a few steps, a set of configurations is obtained, which maximizes an index of observability associated with the identification Jacobian. This algorithm has been shown, in a former work, to be sensitive to local minima. This is why we propose here meta-heuristic methods to decrease this sensibility of our algorithm. Finally, a validation through the simulation of a calibration experience shows that using selected configurations significantly improve the kinematic parameter identification by dividing by 10-15 the noise associated with the results. Also, we present an application to the calibration of a parallel robot with a vision-based measurement device. [ABSTRACT FROM AUTHOR]

Periodic signals can be estimated recursively by exploiting the fact that a sine wave passing through a static nonlinear function generates a spectrum of overtones. A real wave with unknown period in cascade with a piecewise linear function is therefore used as a parameterization for the estimated signal model. In this paper the driving periodic wave can be chosen depending on any prior knowledge. A recursive Gauss–Newton prediction error identification algorithm for joint estimation of the driving frequency and the parameters of the nonlinear output function is introduced. Local convergence properties as well as the Crame´r-Rao bound (CRB) are derived for the suggested algorithm. [Copyright &y& Elsevier]

This paper studies the convergence of the classical proximal point algorithm without assuming monotonicity of the underlying mapping. Practical conditions are given that guarantee the local convergence of the iterates to a solution of T(x) ∋ 0, where T is an arbitrary set-valued mapping from a Hilbert space to itself. In particular, when the problem is "nonsingular" in the sense that T[sup -1] has a Lipschitz localization around one of the solutions, local linear convergence is obtained. This kind of regularity property of variational inclusions has been extensively studied in the literature under the name of strong regularity, and it can be viewed as a natural generalization of classical constraint qualifications in nonlinear programming to more general problem classes. Combining the new convergence results with an abstract duality framework for variational inclusions, the author proves the local convergence of multiplier methods for a very general class of problems. This gives as special cases new convergence results for multiplier methods for nonmonotone variational inequalities and nonconvex nonlinear programming. [ABSTRACT FROM AUTHOR]

We present and analyze a class of evolutionary algorithms for unconstrained and bound constrained optimization on R[sup n]: evolutionary pattern search algorithms (EPSAs). EPSAs adaptively modify the step size of the mutation operator in response to the success of previous optimization steps. The design of EPSAs is inspired by recent analyses of pattern search methods. We show that EPSAs can be cast as stochastic pattern search methods, and we use this observation to prove that EPSAs have a probabilistic, weak stationary point convergence theory. This convergence theory is distinguished by the fact that the analysis does not approximate the stochastic process of EPSAs, and hence it exactly characterizes their convergence properties. [ABSTRACT FROM AUTHOR]

In this paper, we propose two complementary variants of the projected Levenberg–Marquardt (LM) algorithm for solving convex constrained nonlinear equations. Since the orthogonal projection onto the feasible set may be computationally expensive, we first propose a local LM algorithm in which inexact projections are allowed. The feasible inexact projections used in our algorithm can be easily obtained by means of iterative methods, such as conditional gradient. Local convergence of the proposed algorithm is established by using an error bound condition which is weaker than the standard full-rank assumption. We further present and analyze a global version of this algorithm by means of a nonmonotone line search technique. Numerical experiments are reported to showcase the effectiveness of the proposed algorithms, especially when the projection onto the feasible set is difficult to compute. [ABSTRACT FROM AUTHOR]

A variety of strategies are used to construct algorithms for solving equations. However, higher order derivatives are usually assumed to calculate the convergence order. More importantly, bounds on error and uniqueness regions for the solution are also not derived. Therefore, the benefits of these algorithms are limited. We simply use the first derivative to tackle all these issues and study the ball analysis for two sixth order algorithms under the same set of conditions. In addition, we present a calculable ball comparison between these algorithms. In this manner, we enhance the utility of these algorithms. Our idea is very general. That is why it can also be used to extend other algorithms as well in the same way. [ABSTRACT FROM AUTHOR]

In the Fermat-Weber problem, the location of a source point in RN is sought which minimizes the sum of weighted Euclidean distances lo a set of destinations. A classical iterative algorithm known as the Weiszfeld procedure is used to find the optimal location. Kuhn proves global convergence except for a denumerable set of starting points, while Katz provides local convergence results for this algorithm. In this paper, we consider a generalized version of the Fermat-Weber problem, where distances are measured by an lp norm and the parameter p takes on a value in the closed interval [1, 2]. This permits the choice of a continuum of distance measures from rectangular (p = 1) to Euclidean (p = 2). An extended version of the Weiszfeld procedure is presented and local convergence results obtained for the generalized problem. Linear asymptotic convergence rates are typically observed. However, in special cases where the optimal solution occurs at a singular point of the iteration functions, this rate can vary from sublinear to quadratic. It is also shown that for sufficiently large values of p exceeding 2, convergence of the Weiszfeld algorithm will not occur in general. [ABSTRACT FROM AUTHOR]