A harmonic restarted Arnoldi algorithm for calculating eigenvalues and determining multiplicity

A restarted Arnoldi algorithm is given that computes eigenvalues and eigenvectors. It is related to implicitly restarted Arnoldi, but has a simpler restarting approach. Harmonic and regular Rayleigh-Ritz versions are possible.For multiple eigenvalues, an approach is proposed that first computes eigenvalues with the new harmonic restarted Arnoldi algorithm, then uses random restarts to determine multiplicity. This avoids the need for a block method or for relying on roundoff error to produce the multiple copies.     

An efficient combined DCA and B&;B using DC/SDP relaxation for globally solving binary quadratic programs

This paper addresses a new continuous approach based on the DC (Difference of Convex functions) programming and DC algorithms (DCA) to Binary quadratic programs (BQP) which play a key role in combinatorial optimization. DCA is completely different from other avalaible methods and featured by generating a convergent finite sequence of feasible binary solutions (obtained by solving linear programs with the same constraint set) with decreasing objective values. DCA is quite simple and inexpensive to handle large-scale problems. In particular DCA is explicit, requiring only matrix-vector products for Unconstrained Binary quadratic programs (UBQP), and can then exploit sparsity in the large-scale setting. To check globality of solutions computed by DCA, we introduce its combination with a customized Branch-and-Bound scheme using DC/SDP relaxation. The combined algorithm allows checking globality of solutions computed by DCA and restarting it if necessary and consequently accelerates the B&B approach. Numerical results on several series test problems provided in OR-Library (Beasley in J Global Optim, 8:429–433, 1996), show the robustness and efficiency of our algorithm with respect to standard methods. In particular DCA provides ϵ-optimal solutions in almost all cases after only one restarting and the combined DCA-B&B-SDP always provides (ϵ−)optimal solutions.     

计算部分奇异值分解的隐式重新启动的双对角化Lanczos方法和精化的双对角化Lanczos方法

The singular value decomposition problem is mathematically equivalent to the eigenproblem of an argumented matrix. Golub et al. give a bidiagonalization Lanczos method for computing a number of largest or smallest singular values and corresponding singular vertors, but the method may encounter some convergence problems. In this paper we analyse the convergence of the method and show why it may fail to converge. To correct this possible nonconvergence, we propose a refined bidiagonalization Lanczos method and apply the implicitly restarting technique to it, and we then present an implicitly restarted bidiagonalization Lanczos algorithm(IRBL) and an implicitly restarted refined bidiagonalization Lanczos algorithm (IRRBL). A new implicitly restarting scheme and a reliable and efficient algorithm for computing refined shifts are developed for this special structure eigenproblem.Theoretical analysis and numerical experiments show that IRRBL performs much better than IRBL.     

A new framework for implicit restarting of the Krylov–Schur algorithm

This paper introduces a new framework for implicit restarting of the Krylov–Schur algorithm. It is shown that restarting with arbitrary polynomial filter is possible by reassigning some of the eigenvalues of the Rayleigh quotient through a rank‐one correction, implemented using only the elementary transformations (translation and similarity) of the Krylov decomposition. This framework includes the implicitly restarted Arnoldi (IRA) algorithm and the Krylov–Schur algorithm with implicit harmonic restart as special cases. Further, it reveals that the IRA algorithm can be turned into an eigenvalue assignment method. Copyright © 2014 John Wiley & Sons, Ltd.     

New GMRES(k)-Type Algorithms with Explicit Restarts and the Analysis of Their Convergence Properties Based on Matrix Relations in QR Form

The paper considers the problem of constructing a basic iterative scheme for solving systems of linear algebraic equations with unsymmetric and indefinite coefficient matrices. A new GMRES-type algorithm with explicit restarts is suggested. When restarting, this algorithm takes into account the spectral/singular data transferred using orthogonal matrix relations in the so-called QR form, which arise when performing inner iterations of Arnoldi type. The main idea of the algorithm developed is to organize inner iterations and the filtering of directions before restarting in such a way that, from one restart to another, matrix relations effectively accumulate information concerning the current approximate solution and, simultaneously, spectral/singular data, which allow the algorithm to converge with a rate comparable with that of the GMRES algorithm without restarts. Convergence theory is provided for the case of nonsingular, unsymmetric, and indefinite matrices. A bound for the rate of decrease of the residual in the course of inner Arnoldi-type iterations is obtained. This bound depends on the spectral/singular characterization of the subspace spanned by the directions retained upon filtering and is used in developing efficient filtering procedures. Numerical results are provided. Bibliography: 9 titles.     

Computation of matrix functions with deflated restarting

A deflated restarting Krylov subspace method for approximating a function of a matrix times a vector is proposed. In contrast to other Krylov subspace methods, the performance of the method in this paper is better. We further show that the deflating algorithm inherits the superlinear convergence property of its unrestarted counterpart for the entire function and present the results of numerical experiments.     

The Bound-Constrained Conjugate Gradient Method for Non-negative Matrices

Existing conjugate gradient (CG)-based methods for convex quadratic programs with bound constraints require many iterations for solving elastic contact problems. These algorithms are too cautious in expanding the active set and are hampered by frequent restarting of the CG iteration. We propose a new algorithm called the Bound-Constrained Conjugate Gradient method (BCCG). It combines the CG method with an active-set strategy, which truncates variables crossing their bounds and continues (using the Polak–Ribière formula) instead of restarting CG. We provide a case with n=3 that demonstrates that this method may fail on general cases, but we conjecture that it always works if the system matrix A is non-negative. Numerical results demonstrate the effectiveness of the method for large-scale elastic contact problems.     

A Hamiltonian Krylov-Schur-type method based on the symplectic Lanczos process

We discuss a Krylov-Schur like restarting technique applied within the symplectic Lanczos algorithm for the Hamiltonian eigenvalue problem. This allows us to easily implement a purging and locking strategy in order to improve the convergence properties of the symplectic Lanczos algorithm. The Krylov-Schur-like restarting is based on the SR algorithm. Some ingredients of the latter need to be adapted to the structure of the symplectic Lanczos recursion. We demonstrate the efficiency of the new method for several Hamiltonian eigenproblems.     

An Application of Little's Result to the G/G/1 (LCFS/P) Queue

This paper considers a quite general single-server queueing system, under a last-come-first-served queue discipline, with pre-emption and arbitrary restarting policy. Expressions are given for the queue-size limiting distribution when the system is considered at arrival (or departure) epochs and in continuous time, by using very simple arguments.     

A new technique for inconsistent QP problems in the SQP method

Successful treatment of inconsistent QP problems is of major importance in the SQP method, since such occur quite often even for well behaved nonlinear programming problems. This paper presents a new technique for regularizing inconsistent QP problems, which compromises in its properties between the simple technique of Pantoja and Mayne [36] and the highly successful, but expensive one of Tone [47]. Global convergence of a corresponding algorithm is shown under reasonable weak conditions. Numerical results are reported which show that this technique, combined with a special method for the case of regular subproblems, is quite competitive to highly appreciated established ones.     

Consistency method for measurements of the support function of a convex body in the metric of L<Subscript>∞</Subscript>

A new algorithm is proposed for estimation of convex body support function measurements in L _∞ metric, which allows us to obtain the solution in quadratic time (with respect to the number of measurements) not using linear programming. The rate of convergence is proved to be stable for quite weak conditions on input data. This fact makes the algorithm robust for a wider class of problems than it was previously. The implemented algorithm is stable and predictable unlike other existing support function estimation algorithms. Implementation details and testing results are presented.     

A low-rank Krylov squared Smith method for large-scale discrete-time Lyapunov equations

The squared Smith method is adapted to solve large-scale discrete-time Lyapunov matrix equations. The adaptation uses a Krylov subspace to generate the squared Smith iteration in a low-rank form. A restarting mechanism is employed to cope with the increase of memory storage of the Krylov basis. Theoretical aspects of the algorithm are presented. Several numerical illustrations are reported.     

Some observations on weighted GMRES

We investigate the convergence of the weighted GMRES method for solving linear systems. Two different weighting variants are compared with unweighted GMRES for three model problems, giving a phenomenological explanation of cases where weighting improves convergence, and a case where weighting has no effect on the convergence. We also present a new alternative implementation of the weighted Arnoldi algorithm which under known circumstances will be favourable in terms of computational complexity. These implementations of weighted GMRES are compared for a large number of examples. We find that weighted GMRES may outperform unweighted GMRES for some problems, but more often this method is not competitive with other Krylov subspace methods like GMRES with deflated restarting or BICGSTAB, in particular when a preconditioner is used.     

On combinatorial structures of line drawings of polyhedra

Combinatorial structures of line drawings of polyhedra were recently clarified by Sugihara in terms of the regularity of line drawings (references [6,8] in the paper). The definition of the regularity of drawings is quite similar to that mf matroids, and we here clarify both matroidal aspects and non-matroidal ones of the regularity of line drawings, especially concerning the problem of finding a maximum regular substructure of line drawing. A new class of regular line drawings and an algorithm for finding a maximal regular substructure are also discussed.     

Minimizing the sum of job completion times on capacitated two-parallel machines

This paper considers a scheduling problem with two identical parallel machines. One has unlimited capacity; the other can only run for a fixed time. A given set of jobs must be scheduled on the two machines with the goal of minimizing the sum of their completion times. The paper proposes an optimal branch and bound algorithm which employs three powerful elements, including an algorithm for computing the upper bound, a lower bound algorithm, and a fathoming condition. The branch and bound algorithm was tested on problems of various sizes and parameters. The results show that the algorithm is quite efficient to solve all the test problems. In particular, the total computation time for the hardest problem is less than 0.1 second for a set of 100 problem instances. An important finding of the tests is that the upper bound algorithm can actually find optimal solutions to a quite large number of problems.     

Minimizing the total completion time on-line on a single machine, using restarts

We give an algorithm to minimize the total completion time on-line on a single machine, using restarts, with a competitive ratio of 3/2. The optimal competitive ratio without using restarts is 2 for deterministic algorithms and e/(e−1)≈1.582 for randomized algorithms. This is the first restarting algorithm to minimize the total completion time that is proved to be better than an algorithm that does not restart.     

Stopping and restarting strategy for stochastic sequential search in global optimization

Two common questions when one uses a stochastic global optimization algorithm, e.g., simulated annealing, are when to stop a single run of the algorithm, and whether to restart with a new run or terminate the entire algorithm. In this paper, we develop a stopping and restarting strategy that considers tradeoffs between the computational effort and the probability of obtaining the global optimum. The analysis is based on a stochastic process called Hesitant Adaptive Search with Power-Law Improvement Distribution (HASPLID). HASPLID models the behavior of stochastic optimization algorithms, and motivates an implementable framework, Dynamic Multistart Sequential Search (DMSS). We demonstrate here the practicality of DMSS by using it to govern the application of a simple local search heuristic on three test problems from the global optimization literature.