A very general algorithm for orthogonal rotation is identified. It is shown that when an algorithm parameter α is sufficiently large the algorithm converges monotonically to a stationary point of the rotation criterion from any starting value. Because a sufficiently largeα is in general hard to find, a modification that does not require it is introduced. Without this requirement the modified algorithm is not only very general, but also very simple. Its implementation involves little more than computing the gradient of the rotation criterion. While the modified algorithm converges monotonically from any starting value, it is not guaranteed to converge to a stationary point. It, however, does so in all of our examples. While motivated by the rotation problem in factor analysis, the algorithms discussed may be used to optimize almost any function of a not necessarily square column-wise orthonormal matrix. A number of these more general applications are considered. Empirical examples show that the modified algorithm can be reasonably fast, but its purpose is to save an investigator's effort rather than that of his or her computer. This makes it more appropriate as a research tool than as an algorithm for established methods.

Each of several speed-limited planar robots is driven by the acceleration, limited in magnitude. There is an unpredictable dynamic complex object, for example, a group of moving targets or an extended moving and deforming body. The robots should reach and then repeatedly trace a certain object-dependent moving and deforming curve that encircles the object and also achieve an effective self-deployment over it. This may be, for example, the locus of points at a desired mean distance or distance from a group of targets or a single extended object, respectively. Every robot has access to the nearest point of the curve and its own velocity and “sees” the objects within a finite sensing range. The robots have no communication facilities, cannot differentiate the peers, and are to be driven by a common law. Necessary conditions for the solvability of the mission are established. Under their slight and partly unavoidable enhancement, a new decentralized control strategy is proposed and shown to solve the mission, while excluding inter-robot collisions, and for the case of a steady curve, to evenly distribute the robots over the curve and to ensure a prespecified speed of their motion over it. These are justified via rigorous global convergence results and confirmed via computer simulations.

In this paper, a primal-dual interior point method is proposed for general constrained optimization, which incorporated a penalty function and a kind of new identification technique of the active set. At each iteration, the proposed algorithm only needs to solve two or three reduced systems of linear equations with the same coefficient matrix. The size of systems of linear equations can be decreased due to the introduction of the working set, which is an estimate of the active set. The penalty parameter is automatically updated and the uniformly positive definiteness condition on the Hessian approximation of the Lagrangian is relaxed. The proposed algorithm possesses global and superlinear convergence under some mild conditions. Finally, some preliminary numerical results are reported.

We propose a modified projected Polak–Ribière–Polyak (PRP) conjugate gradient method, where a modified conjugacy condition and a method which generates sufficient descent directions are incorporated into the construction of a suitable conjugacy parameter. It is shown that the proposed method is a modification of the PRP method and generates sufficient descent directions at each iteration. With an Armijo-type line search, the theory of global convergence is established under two weak assumptions. Numerical experiments are employed to test the efficiency of the algorithm in solving some benchmark test problems available in the literature. The numerical results obtained indicate that the algorithm outperforms an existing similar algorithm in requiring fewer function evaluations and fewer iterations to find optimal solutions with the same tolerance.

In this paper, we propose a regularized version of the generalized NCP-function proposed by Hu, Huang and Chen [J. Comput. Appl. Math., 230 (2009), pp. 69-82]. Based on this regularized function, we propose a semismooth Newton method for solving nonlinear complementarity problems, where a non-monotone line search scheme is used. In particular, we show that the proposed non-monotone method is globally and locally superlinearly convergent under suitable assumptions. We test the proposed method by solving the test problems from MCPLIB. Numerical experiments indicate that this algorithm has better numerical performance in the case of p = 5 and Θ ∈ [0.25,075] than other cases.

In this paper, we propose a new class of adaptive trust region methods for unconstrained optimization problems and develop some convergence properties. In the new algorithms, we use the current iterative information to define a suitable initial trust region radius at each iteration. The initial trust region radius is more reasonable in the sense that the trust region model and theobjective function are more consistent at the current iterate. The global convergence, super-linear and quadratic convergence rate are analyzed under some mild conditions. Numericalresults show that some special adaptive trust region methods are available and efficient in practical computation.