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In this paper, we introduce a new concept of approximate optimal stepsize for gradient method, use it to interpret the Barzilai-Borwein (BB) method, and present an efficient gradient method with approximate optimal stepsize for large unconstrained optimization. If the objective function f is not close to a quadratic on a line segment between the current iterate xk and the latest iterate xk−1, we construct a conic model to generate the approximate optimal stepsize for gradient method if the conic model is suitable to be used. Otherwise, we construct a new quadratic model or two other new approximation models to generate the approximate optimal stepsize for gradient method. We analyze the convergence of the proposed method under some suitable conditions. Numerical results show the proposed method is very promising. [ABSTRACT FROM AUTHOR]

The matrix completion problem is easy to state: let A be a given data matrix in which some entries are unknown. Then, it is needed to assign 'appropriate values' to these entries. A common way to solve this problem is to compute a rank- k matrix, B , that approximates A in a least squares sense. Then, the unknown entries in A attain the values of the corresponding entries in B . This raises the question of how to determine a suitable matrix rank. The method proposed in this paper attempts to answer this question. It builds a finite sequence of matrices $B_{k}, k = 1, 2, \dots $ , where B is a rank- k matrix that approximates A in a least squares sense. The computational effort is reduced by using B as starting point in the computation of B . The ability of B to serve as substitute for A is measured with two objective functions: a 'training' function that measures the distance between the known part of A and the corresponding part of B , and a 'probe' function that assesses the quality of the imputed entries. Watching the changes in these functions as k increases enables us to find an optimal matrix rank. Numerical experiments illustrate the usefulness of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, we present primal-dual interior-point methods for convex quadratic optimization based on a finite barrier, which has been investigated earlier for the case of linear optimization by Bai et al. (SIAM J Optim 13(3):766-782, ). By means of the feature of the finite kernel function, we study the complexity analysis of primal-dual interior-point methods based on the finite barrier and derive the iteration bounds that match the currently best known iteration bounds for large- and small-update methods, namely, $O(\sqrt{n}\log{n}\log{\frac{n}{\varepsilon}})$ and $O(\sqrt{n}\log{\frac{n}{\varepsilon}})$, respectively, which are as good as the linear optimization analogue. Numerical tests demonstrate the behavior of the algorithms with different parameters. [ABSTRACT FROM AUTHOR]

Multiobjective optimization has a large number of real-life applications. Under this motivation, in this paper, we present a new method for solving multiobjective optimization problems with both linear constraints and bound constraints on the variables. This method extends, to the multiobjective setting, the classical reduced gradient method for scalar-valued optimization. The proposed algorithm generates a feasible descent direction by solving an appropriate quadratic subproblem, without the use of any scalarization approaches. We prove that the sequence generated by the algorithm converges to Pareto-critical points of the problem. We also present some numerical results to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a preconditioned general modulus-based matrix splitting iteration method for solving modulus equations arising from linear complementarity problems. Its convergence theory is proved when the system matrix is an H+-matrix, from which some new convergence conditions can be derived for the (general) modulus-based matrix splitting iteration methods. Numerical results further show that the proposed methods are superior to the existing methods. [ABSTRACT FROM AUTHOR]

This paper presents a modified ODE-based algorithm for unconstrained optimization problems. It combines the idea of IMPBOT algorithm with nonmonotone and subspace techniques. The main feature of this method is that at each iteration, a lower dimensional system of linear equations is solved to obtain a trial step. Under some standard assumptions, the method is proven to be globally convergent. Numerical results show the efficiency of this proposed method in practical computation. [ABSTRACT FROM AUTHOR]

Kernel functions play an important role in the design and analysis of primal-dual interior-point algorithms. They are not only used for determining the search directions but also for measuring the distance between the given iterate and the μ-center for the algorithms. In this paper we present a unified kernel function approach to primal-dual interior-point algorithms for convex quadratic semidefinite optimization based on the Nesterov and Todd symmetrization scheme. The iteration bounds for large- and small-update methods obtained are analogous to the linear optimization case. Moreover, this unifies the analysis for linear, convex quadratic and semidefinite optimizations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a full Nesterov-Todd (NT) step infeasible interior-point algorithm for convex quadratic symmetric cone optimization based on Euclidean Jordan algebra. The algorithm uses only one feasibility step in each main iteration. The complexity result coincides with the best-known iteration bound for infeasible interior-point methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical method for solving the stationary points of mathematical programs constrained by parameterized quasi-variational inequalities. The necessary optimality conditions (stationary conditions in the sense of Mordukhovich) for the optimization problem are reformulated as a system of nonsmooth equations without the strict complementarity condition and an inexact Newton method is constructed to find its solutions. The local convergence of the inexact Newton method is guaranteed under second order sufficient conditions and linear independence constraint qualification. Several illustrative examples are provided. [ABSTRACT FROM AUTHOR]

Abstract  Recently, Roos (SIAM J Optim 16(4):1110–1136, 2006) presented a primal-dual infeasible interior-point algorithm that uses full-Newton steps and whose iteration bound coincides with the best known bound for infeasible interior-point algorithms. In the current paper we use a different feasibility step such that the definition of the feasibility step in Mansouri and Roos (Optim Methods Softw 22(3):519–530, 2007) is a special case of our definition, and show that the same result on the order of iteration complexity can be obtained. [ABSTRACT FROM AUTHOR]

In this work, we present a new hybrid conjugate gradient method based on the approach of the convex hybridization of the conjugate gradient update parameters of DY and HS+, adapting a quasi-Newton philosophy. The computation of the hybrization parameter is obtained by minimizing the distance between the hybrid conjugate gradient direction and the self-scaling memoryless BFGS direction. Furthermore, a significant property of our proposed method is that it ensures sufficient descent independent of the accuracy of the line search. The global convergence of the proposed method is established provided that the line search satisfies the Wolfe conditions. Our numerical experiments on a set of unconstrained optimization test problems from the CUTEr collection indicate that our proposed method is preferable and in general superior to classic conjugate gradient methods in terms of efficiency and robustness. [ABSTRACT FROM AUTHOR]

We present a gradient descent algorithm with a line search procedure for solving unconstrained optimization problems which is defined as a result of applying Picard-Mann hybrid iterative process on accelerated gradient descent SM method described in Stanimirović and Miladinović (Numer. Algor. 54, 503-520, 2010). Using merged features of both analyzed models, we show that new accelerated gradient descent model converges linearly and faster then the starting SM method which is confirmed trough displayed numerical test results. Three main properties are tested: number of iterations, CPU time and number of function evaluations. The efficiency of the proposed iteration is examined for the several values of the correction parameter introduced in Khan (2013). [ABSTRACT FROM AUTHOR]

We present a primal-dual predictor-corrector interior-point method for symmetric cone optimization. The proposed algorithm is based on the Nesterov-Todd search directions and a wide neighborhood, which is an even wider neighborhood than a given negative infinity neighborhood. At each iteration, the method computes two corrector directions in addition to the Ai and Zhang directions (SIAM J. Optim. 16, 400-417, 2005), in order to improve performance. Moreover, we derive the complexity bound of the wide neighborhood predictor-corrector interior-point method for symmetric cone optimization that coincides with the currently best known theoretical complexity bounds for the short step algorithm. Finally, some numerical experiments are provided to reveal the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]