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In this paper, we present a feasible predictor-corrector interior-point method for symmetric cone optimization problem in the large neighbourhood of the central path. The method is generalization of Ai-Zhang's predictor-corrector algorithm to the symmetric cone optimization problem. Starting with a feasible point in given large neighbourhood of the central path, the algorithm still terminates in at most iterations. This matches the best known iteration bound that is usually achieved by short-step methods, thereby, closing the complexity gap between long- and short-step interior-point methods for symmetric cone optimization. The preliminary numerical results on a selected set of NETLIB problems show advantage of the method in comparison with the version of the algorithm that is not based on the predictor-corrector scheme. [ABSTRACT FROM AUTHOR]

In this paper, we propose a non-monotone line search method for solving optimization problems on Stiefel manifold. The main novelty of our approach is that our method uses a search direction based on a linear combination of descent directions and a Barzilai-Borwein line search. The feasibility is guaranteed by projecting each iterate on the Stiefel manifold through SVD (singular value decomposition) factorizations. Some theoretical results for analysing the algorithm are presented. Finally, we provide numerical experiments for comparing our algorithm with other state-of-the-art procedures. The code is available online. The experimental results show that the proposed algorithm is competitive with other approaches and for particular problems, the computational performance is better than the state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

This paper deals with generalized pattern search (GPS) algorithms for linearly constrained optimization. At each iteration, the GPS algorithm generates a set of directions that conforms to the geometry of any nearby linear constraints. This set is then used to construct trial points to be evaluated during the iteration. In a previous work, Lewis and Torczon developed a scheme for computing the conforming directions, however, the issue of degeneracy merits further investigation. The contribution of this paper is to provide a detailed algorithm for constructing the set of directions whether or not the constraints are degenerate. One difficulty in the degenerate case is the classification of constraints as redundant or nonredundant. We give a short survey of the main definitions and methods for treating redundancy and propose an approach to identify nonredundant ε-active constraints. We also introduce a new approach for handling nonredundant linearly dependent constraints, which maintains GPS convergence properties without significantly increasing computational cost. Some simple numerical tests illustrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

Most algorithms for solving linear program require a phase-1 procedure to find a feasible solution. Recently, a dual–primal algorithm for linear optimization has been proposed by Li [Dual–primal algorithm for linear optimization, Optimiz. Methods Softw. 28 (2013), pp. 327–338]. In the process of implementing the dual–primal algorithms, we found an interesting phenomenon that the phase-1 algorithm developed in [Li (2013)] always terminates in one iteration. This fact does not come by chance. A rigorous proof is given in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the properties of a simple locally kernel function. As an application, we present a full-step interior-point algorithm for second-order cone optimization (SOCO). The algorithm uses the simple locally kernel function to determine the search direction and define the neighbourhood of central path. The full step used in the algorithm has local quadratic convergence property according to the proximity function which is constructed by the simple locally kernel function. We derive the iteration complexity for the algorithm and obtain the best-known iteration bound for SOCO. [ABSTRACT FROM AUTHOR]

We propose a quasi-Newton line-search method that uses negative curvature directions for solving unconstrained optimization problems. In this method, the symmetric rank-one (SR1) rule is used to update the Hessian approximation. The SR1 update rule is known to have a good numerical performance; however, it does not guarantee positive definiteness of the updated matrix. We first discuss the details of the proposed algorithm and then concentrate on its practical behaviour. Our extensive computational study shows the potential of the proposed method from different angles, such as its performance compared with some other existing packages, the profile of its computations, and its large-scale adaptation. We then conclude the paper with the convergence analysis of the proposed method. [ABSTRACT FROM AUTHOR]

Sequential quadratic programming (SQP) algorithms have proven to be efficient for solving nonlinearly constrained optimization problems under some conditions. In analysis of global convergence for SQP algorithms, it is usually assumed that the quasi-Newton matrices, approximating the Hessian of Lagrangian function, are uniformly positive definite. However, it is not known whether this condition is satisfied for quasi-Newton matrices. In this paper, we present a new update criterion for the quasi-Newton matrix and a new update method for the penalty parameter. We establish the global convergence result of a modifying SQP algorithm for equality constrained optimization problems without any assumption on quasi-Newton matrix sequence. Moreover, if the second-order sufficient condition holds at an optimal solution, the new update criterion reduces to the ordinary BFGS update. The superlinear rate of convergence of the algorithm can be obtained if an additional step is performed. [ABSTRACT FROM AUTHOR]