Accelerated Matrix Recovery via Random Projection Based on Inexact Augmented Lagrange Multiplier Method

In this paper, a unified matrix recovery model was proposed for diverse corrupted matrices. Resulting from the separable structure of the proposed model, the convex optimization problem can be solved efficiently by adopting an inexact augmented Lagrange multiplier (IALM) method. Additionally, a random projection accelerated technique (IALM+RP) was adopted to improve the success rate. From the preliminary numerical comparisons, it was indicated that for the standard robust principal component analysis (PCA) problem, IALM+RP was at least two to six times faster than IALM with an insignificant reduction in accuracy; and for the outlier pursuit (OP) problem, IALM+RP was at least 6.9 times faster, even up to 8.3 times faster when the size of matrix was 2 000×2 000.

GLOBAL CONVERGENCE OF A TRUST REGION ALGORITHM USING INEXACT GRADIENT FOR EQUALITY-CONSTRAINED OPTIMIZATION

A trust-region algorithm is presented for a nonlinear optimization problem of equality-constraints. The characterization of the algorithm is using inexact gradient information. Global convergence results are demonstrated where the gradient values are obeyed a simple relative error condition.

ON SOME RESULTS ABOUT INEXACT LINEAR PROGRAMMING

In this paper we point out that some theorems about the inexact programming in [2]are false and give the modified statement.

CONVERGENCE OF INEXACT CONIC NEWTON METHODS

A conic Newton method is attractive because it converges to a local minimizzer rapidly from any sufficiently good initial guess. However, it may be expensive to solve the conic Newton equation at each iterate. In this paper we consider an inexact conic Newton method, which solves the couic Newton equation oldy approximately and in sonm unspecified manner. Furthermore, we show that such method is locally convergent and characterizes the order of convergence in terms of the rate of convergence of the relative residuals.

Inexact Newton method via Lanczos decomposed technique for solving box-constrained nonlinear systems

This paper proposes an inexact Newton method via the Lanczos decomposed technique for solving the box-constrained nonlinear systems. An iterative direction is obtained by solving an affine scaling quadratic model with the Lanczos decomposed technique. By using the interior backtracking line search technique, an acceptable trial step length is found along this direction. The global convergence and the fast local convergence rate of the proposed algorithm are established under some reasonable conditions. Furthermore, the results of the numerical experiments show the effectiveness of the pro- posed algorithm.

LOCAL CONVERGENCE OF INEXACT NEWTON-LIKE METHOD UNDER WEAK LIPSCHITZ CONDITIONS

The paper develops the local convergence of Inexact Newton-Like Method(INLM)for approximating solutions of nonlinear equations in Banach space setting.We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis.The obtained results compare favorably with earlier ones such as[7,13,14,18,19].A numerical example is also provided.

An Interval Probability-based Inexact Two-stage Stochastic Model for Regional Electricity Supply and GHG Mitigation Management under Uncertainty

In this study, an interval probability-based inexact two-stage stochastic (IP-ITSP) model is developed for environmental pollutants control and greenhouse gas (GHG) emissions reduction management in regional energy system under uncertainties. In the IP-ITSP model, methods of interval probability, interval-parameter programming (IPP) and two-stage stochastic programming (TSP) are introduced into an integer programming framework;the developed model can tackle uncertainties described in terms of interval values and interval probability distributions. The developed model is applied to a case of planning GHG -emission mitigation in a regional electricity system, demonstrating that IP-ITSP is applicable to reflecting complexities of multi-uncertainty, and capable of addressing the problem of GHG-emission reduction. 4 scenarios corresponding to different GHG -emission mitigation levels are examined;the results indicates that the model could help decision makers identify desired GHG mitigation policies under various economic costs and environmental requirements.

An Inexact Restoration Package for Bilevel Programming Problems

Bilevel programming problems are a class of optimization problems with hierarchical structure where one of the con-straints is also an optimization problem. Inexact restoration methods were introduced for solving nonlinear programming problems a few years ago. They generate a sequence of, generally, infeasible iterates with intermediate iterations that consist of inexactly restored points. In this paper we present a software environment for solving bilevel program-ming problems using an inexact restoration technique without replacing the lower level problem by its KKT optimality conditions. With this strategy we maintain the minimization structure of the lower level problem and avoid spurious solutions. The environment is a user-friendly set of Fortran 90 modules which is easily and highly configurable. It is prepared to use two well-tested minimization solvers and different formulations in one of the minimization subproblems. We validate our implementation using a set of test problems from the literature, comparing different formulations and the use of the minimization solvers.

An Innovative Genetic Algorithms-Based Inexact Non-Linear Programming Problem Solving Method

In this paper, an innovative Genetic Algorithms (GA)-based inexact non-linear programming (GAINLP) problem solving approach has been proposed for solving non-linear programming optimization problems with inexact information (inexact non-linear operation programming). GAINLP was developed based on a GA-based inexact quadratic solving method. The Genetic Algorithm Solver of the Global Optimization Toolbox (GASGOT) developed by MATLABTM was adopted as the implementation environment of this study. GAINLP was applied to a municipality solid waste management case. The results from different scenarios indicated that the proposed GA-based heuristic optimization approach was able to generate a solution for a complicated nonlinear problem, which also involved uncertainty.

ADAPTIVE REGULARIZED QUASI-NEWTON METHOD USING INEXACT FIRST-ORDER INFORMATION

Classical quasi-Newton methods are widely used to solve nonlinear problems in which the first-order information is exact.In some practical problems,we can only obtain approximate values of the objective function and its gradient.It is necessary to design optimization algorithms that can utilize inexact first-order information.In this paper,we propose an adaptive regularized quasi-Newton method to solve such problems.Under some mild conditions,we prove the global convergence and establish the convergence rate of the adaptive regularized quasi-Newton method.Detailed implementations of our method,including the subspace technique to reduce the amount of computation,are presented.Encouraging numerical results demonstrate that the adaptive regularized quasi-Newton method is a promising method,which can utilize the inexact first-order information effectively.

AN INEXACT PROXIMAL DC ALGORITHM FOR THE LARGE-SCALE CARDINALITY CONSTRAINED MEAN-VARIANCE MODEL IN SPARSE PORTFOLIO SELECTION

Optimization problem of cardinality constrained mean-variance(CCMV)model for sparse portfolio selection is considered.To overcome the difficulties caused by cardinality constraint,an exact penalty approach is employed,then CCMV problem is transferred into a difference-of-convex-functions(DC)problem.By exploiting the DC structure of the gained problem and the superlinear convergence of semismooth Newton(ssN)method,an inexact proximal DC algorithm with sieving strategy based on a majorized ssN method(siPDCA-mssN)is proposed.For solving the inner problems of siPDCA-mssN from dual,the second-order information is wisely incorporated and an efficient mssN method is employed.The global convergence of the sequence generated by siPDCA-mssN is proved.To solve large-scale CCMV problem,a decomposed siPDCA-mssN(DsiPDCA-mssN)is introduced.To demonstrate the efficiency of proposed algorithms,siPDCA-mssN and DsiPDCA-mssN are compared with the penalty proximal alternating linearized minimization method and the CPLEX(12.9)solver by performing numerical experiments on realword market data and large-scale simulated data.The numerical results demonstrate that siPDCA-mssN and DsiPDCA-mssN outperform the other methods from computation time and optimal value.The out-of-sample experiments results display that the solutions of CCMV model are better than those of other portfolio selection models in terms of Sharp ratio and sparsity.

An Efficient Inexact Newton-CG Algorithm for the Smallest Enclosing Ball Problem of Large Dimensions

In this paper,we consider the problem of computing the smallest enclosing ball(SEB)of a set of m balls in Rn,where the product mn is large.We first approximate the non-differentiable SEB problem by its log-exponential aggregation function and then propose a computationally efficient inexact Newton-CG algorithm for the smoothing approximation problem by exploiting its special(approximate)sparsity structure.The key difference between the proposed inexact Newton-CG algorithm and the classical Newton-CG algorithm is that the gradient and the Hessian-vector product are inexactly computed in the proposed algorithm,which makes it capable of solving the large-scale SEB problem.We give an adaptive criterion of inexactly computing the gradient/Hessian and establish global convergence of the proposed algorithm.We illustrate the efficiency of the proposed algorithm by using the classical Newton-CG algorithm as well as the algorithm from Zhou et al.(Comput Optim Appl 30:147–160,2005)as benchmarks.

ON SEMILOCAL CONVERGENCE OF INEXACT NEWTON METHODS

Inexact Newton methods are constructed by combining Newton＇s method with another iterative method that is used to solve the Newton equations inexactly. In this paper, we establish two semilocal convergence theorems for the inexact Newton methods. When these two theorems are specified to Newton＇s method, we obtain a different Newton-Kantorovich theorem about Newton＇s method. When the iterative method for solving the Newton equations is specified to be the splitting method, we get two estimates about the iteration steps for the special inexact Newton methods.

An Inexact Affine Scaling Levenberg-Marquardt Method Under Local Error Bound Conditions

We propose an inexact affine scaling Levenberg-Marquardt method for solving bound-constrained semismooth equations under the local error bound assumption which is much weaker than the standard nonsingularity condition. The affine scaling Levenberg-Marquardt equation is based on a minimization of the squared Euclidean norm of linearized model adding a quadratic affine scaling matrix to find a solution which belongs to the bounded constraints on variable. The global convergence and the superlinear convergence rate are proved.Numerical results show that the new algorithm is efficient.

A smoothing inexact Newton method for P0 nonlinear complementarity problem

We first propose a new class of smoothing functions for the non- linear complementarity function which contains the well-known Chen-Harker- Kanzow-Smale smoothing function and Huang-Han-Chen smoothing function as special cases, and then present a smoothing inexact Newton algorithm for the P0 nonlinear complementarity problem. The global convergence and local superlinear convergence are established. Preliminary numerical results indicate the feasibility and efficiency of the algorithm.

On convergence of the inexact Rayleigh quotient iteration with the Lanczos method used for solving linear systems

For the Hermitian inexact Rayleigh quotient iteration （RQI）, we consider the local convergence of the inexact RQI with the Lanczos method for the linear systems involved. Some attractive properties are derived for the residual, whose norm is ξk, of the linear system obtained by the Lanczos method at outer iteration k ＋ 1. Based on them, we make a refined analysis and establish new local convergence results. It is proved that （i） the inexact RQI with Lanezos converges quadratically provided that ξk ≤ξ with a constant ξ≥） 1 and （ii） the method converges linearly provided that ξk is bounded by some multiple of 1/‖τk‖ with ‖τk‖ the residual norm of the approximate eigenpair at outer iteration k. The results are fundamentally different from the existing ones that always require ξk 〈 1, and they have implications on effective implementations of the method. Based on the new theory, we can design practical criteria to control ξk to achieve quadratic convergence and implement the method more effectively than ever before. Numerical experiments confirm our theory and demonstrate that the inexact RQI with Lanczos is competitive to the inexact RQI with MINRES.

An inexact alternating proximal gradient algorithm for nonnegative CP tensor decomposition

Nonnegative tensor decomposition has become increasingly important for multiway data analysis in recent years. The alternating proximal gradient(APG) is a popular optimization method for nonnegative tensor decomposition in the block coordinate descent framework. In this study, we propose an inexact version of the APG algorithm for nonnegative CANDECOMP/PARAFAC decomposition, wherein each factor matrix is updated by only finite inner iterations. We also propose a parameter warm-start method that can avoid the frequent parameter resetting of conventional APG methods and improve convergence performance.By experimental tests, we find that when the number of inner iterations is limited to around 10 to 20, the convergence speed is accelerated significantly without losing its low relative error. We evaluate our method on both synthetic and real-world tensors.The results demonstrate that the proposed inexact APG algorithm exhibits outstanding performance on both convergence speed and computational precision compared with existing popular algorithms.

THE SCHWARZ CHAOTIC RELAXATION METHOD WITH INEXACT SOLVERS ON THE SUBDOMAINS

In this paper, a S-CR method with inexact solvers on the subdomains is presented, and then its convergence property is proved under very general conditions. This result is important because it guarantees the effectiveness of the Schwarz alternating method when executed on message-passing distributed memory multiprocessor system.

ROBUST INEXACT ALTERNATING OPTIMIZATION FOR MATRIX COMPLETION WITH OUTLIERS

We investigate the problem of robust matrix completion with a fraction of observation corrupted by sparsity outlier noise.We propose an algorithmic framework based on the ADMM algorithm for a non-convex optimization,whose objective function consists of an l1 norm data fidelity and a rank constraint.To reduce the computational cost per iteration,two inexact schemes are developed to replace the most time-consuming step in the generic ADMM algorithm.The resulting algorithms remarkably outperform the existing solvers for robust matrix completion with outlier noise.When the noise is severe and the underlying matrix is ill-conditioned,the proposed algorithms are faster and give more accurate solutions than state-of-the-art robust matrix completion approaches.