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1.
An inexact Newton algorithm for large sparse equality constrained non-linear programming problems is proposed. This algorithm is based on an indefinitely preconditioned smoothed conjugate gradient method applied to the linear KKT system and uses a simple augmented Lagrangian merit function for Armijo type stepsize selection. Most attention is devoted to the termination of the CG method, guaranteeing sufficient descent in every iteration and decreasing the number of required CG iterations, and especially, to the choice of a suitable preconditioner. We investigate four preconditioners, which have 2 × 2 block structure, and prove theoretically their good properties. The efficiency of the inexact Newton algorithm, together with a comparison of various preconditioners and strategies, is demonstrated by using a large collection of test problems. © 1998 John Wiley & Sons, Ltd. 相似文献
2.
We present a class of nested iteration schemes for solving large sparse systems of linear equations with a coefficient matrix with a dominant symmetric positive definite part. These new schemes are actually inner/outer iterations, which employ the classical conjugate gradient method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent and symmetric positive definite splitting of the coefficient matrix. Convergence properties of the new schemes are studied in depth, possible choices of the inner iteration steps are discussed in detail, and numerical examples from the finite-difference discretization of a second-order partial differential equation are used to further examine the effectiveness and robustness of the new schemes over GMRES and its preconditioned variant. Also, we show that the new schemes are, at least, comparable to the variable-step generalized conjugate gradient method and its preconditioned variant. 相似文献
3.
We present a nested splitting conjugate gradient iteration method for solving large sparse continuous Sylvester equation, in which both coefficient matrices are (non-Hermitian) positive semi-definite, and at least one of them is positive definite. This method is actually inner/outer iterations, which employs the Sylvester conjugate gradient method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent and Hermitian positive definite splitting of the coefficient matrices. Convergence conditions of this method are studied and numerical experiments show the efficiency of this method. In addition, we show that the quasi-Hermitian splitting can induce accurate, robust and effective preconditioned Krylov subspace methods. 相似文献
4.
D. G. Luenberger 《Journal of Optimization Theory and Applications》1974,14(5):477-495
A new programming algorithm for nonlinear constrained optimization problems is proposed. The method is based on the penalty function approach and thereby circumyents the necessity to maintain feasibility at each iteration, but it also behaves much like the gradient projection method. Although only first-order information is used, the algorithm converges asymptotically at a rate which is independent of the magnitude of the penalty term; hence, unlike the simple gradient method, the asymptotic rate of the proposed method is not affected by the ill-conditioning associated with the introduction of the penalty term. It is shown that the asymptotic rate of convergence of the proposed method is identical with that of the gradient projection method.Dedicated to Professor M. R. HestenesThis research was supported by the National Science Foundation, Grant No. GK-16125. 相似文献
5.
We propose the damped inexact Newton method, coupled with preconditioned inner iterations, to solve the finite element discretization of a class of nonlinear elliptic interface problems. The linearized equations are solved by a preconditioned conjugate gradient method. Both the inner and outer iterations exhibit mesh independent superlinear convergence. 相似文献
6.
This paper is concerned with a primal–dual interior point method for solving nonlinear semidefinite programming problems. The method consists of the outer iteration (SDPIP) that finds a KKT point and the inner iteration (SDPLS) that calculates an approximate barrier KKT point. Algorithm SDPLS uses a commutative class of Newton-like directions for the generation of line search directions. By combining the primal barrier penalty function and the primal–dual barrier function, a new primal–dual merit function is proposed. We prove the global convergence property of our method. Finally some numerical experiments are given. 相似文献
7.
We focus on efficient preconditioning techniques for sequences of Karush‐Kuhn‐Tucker (KKT) linear systems arising from the interior point (IP) solution of large convex quadratic programming problems. Constraint preconditioners (CPs), although very effective in accelerating Krylov methods in the solution of KKT systems, have a very high computational cost in some instances, because their factorization may be the most time‐consuming task at each IP iteration. We overcome this problem by computing the CP from scratch only at selected IP iterations and by updating the last computed CP at the remaining iterations, via suitable low‐rank modifications based on a BFGS‐like formula. This work extends the limited‐memory preconditioners (LMPs) for symmetric positive definite matrices proposed by Gratton, Sartenaer and Tshimanga in 2011, by exploiting specific features of KKT systems and CPs. We prove that the updated preconditioners still belong to the class of exact CPs, thus allowing the use of the conjugate gradient method. Furthermore, they have the property of increasing the number of unit eigenvalues of the preconditioned matrix as compared with the generally used CPs. Numerical experiments are reported, which show the effectiveness of our updating technique when the cost for the factorization of the CP is high. 相似文献
8.
The purpose of this paper is to present optimal preconditioned iterative methods to solve indefinite linear systems of equations arising from symmetric coupling of finite elements and boundary elements. This is a block‐diagonal preconditioner together with a conjugate residual method and a preconditioned inner–outer iteration. We prove the efficiency of these methods by showing that the number of iterations to preserve a given accuracy is bounded independent of the number of unknowns. Numerical examples underline the efficiency of these methods. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
9.
We propose two line search primal-dual interior-point methods for nonlinear programming that approximately solve a sequence
of equality constrained barrier subproblems. To solve each subproblem, our methods apply a modified Newton method and use
an ℓ2-exact penalty function to attain feasibility. Our methods have strong global convergence properties under standard assumptions.
Specifically, if the penalty parameter remains bounded, any limit point of the iterate sequence is either a Karush-Kuhn-Tucker
(KKT) point of the barrier subproblem, or a Fritz-John (FJ) point of the original problem that fails to satisfy the Mangasarian-Fromovitz
constraint qualification (MFCQ); if the penalty parameter tends to infinity, there is a limit point that is either an infeasible
FJ point of the inequality constrained feasibility problem (an infeasible stationary point of the infeasibility measure if
slack variables are added) or a FJ point of the original problem at which the MFCQ fails to hold. Numerical results are given
that illustrate these outcomes.
Research supported by the Presidential Fellowship of Columbia University.
Research supported in part by NSF Grant DMS 01-04282, DOE Grant DE-FG02-92EQ25126 and DNR Grant N00014-03-0514. 相似文献
10.
Klaus Neymeyr 《Linear algebra and its applications》2009,430(4):1039-1056
This paper deals with the convergence analysis of various preconditioned iterations to compute the smallest eigenvalue of a discretized self-adjoint and elliptic partial differential operator. For these eigenproblems several preconditioned iterative solvers are known, but unfortunately, the convergence theory for some of these solvers is not very well understood.The aim is to show that preconditioned eigensolvers (like the preconditioned steepest descent iteration (PSD) and the locally optimal preconditioned conjugate gradient method (LOPCG)) can be interpreted as truncated approximate Krylov subspace iterations. In the limit of preconditioning with the exact inverse of the system matrix (such preconditioning can be approximated by multiple steps of a preconditioned linear solver) the iterations behave like Invert-Lanczos processes for which convergence estimates are derived. 相似文献
11.
Huibin Chang Danping Yang 《Journal of Computational and Applied Mathematics》2011,235(17):5078-5094
A domain decomposition method (DDM) is presented to solve the distributed optimal control problem. The optimal control problem essentially couples an elliptic partial differential equation with respect to the state variable and a variational inequality with respect to the constrained control variable. The proposed algorithm, called SA-GP algorithm, consists of two iterative stages. In the inner loops, the Schwarz alternating method (SA) is applied to solve the state and co-state variables, and in the outer loops the gradient projection algorithm (GP) is adopted to obtain the control variable. Convergence of iterations depends on both the outer and the inner loops, which are coupled and affected by each other. In the classical iteration algorithms, a given tolerance would be reached after sufficiently many iteration steps, but more iterations lead to huge computational cost. For solving constrained optimal control problems, most of the computational cost is used to solve PDEs. In this paper, a proposed iterative number independent of the tolerance is used in the inner loops so as to save a lot of computational cost. The convergence rate of L2-error of control variable is derived. Also the analysis on how to choose the proposed iteration number in the inner loops is given. Some numerical experiments are performed to verify the theoretical results. 相似文献
12.
Subhashree Mohapatra Sashikumaar Ganesan 《Numerical Functional Analysis & Optimization》2016,37(10):1295-1311
In this article, we propose a non-conforming exponentially accurate least-squares spectral element method for Oseen equations in primitive variable formulation that is applicable to both two- and three-dimensional domains. First-order reformulation is avoided, and the condition number is controlled by a suitable preconditioner for velocity components and pressure variable. A preconditioned conjugate gradient method is used to obtain the solution. Navier-Stokes equations in primitive variable formulation have been solved by solving a sequence of Oseen type iterations. For numerical test cases, similar order convergence has been achieved for all Reynolds number cases at the cost of higher iteration number for higher Reynolds number. 相似文献
13.
We present the convergence analysis of the inexact infeasible path-following (IIPF) interior-point algorithm. In this algorithm,
the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented
system is preconditioned by using a block triangular matrix.
The KKT system is solved approximately. Therefore, it becomes necessary to study the convergence of the interior-point method
for this specific inexact case. We present the convergence analysis of the inexact infeasible path-following (IIPF) algorithm,
prove the global convergence of this method and provide complexity analysis.
Communicated by Y. Zhang. 相似文献
14.
We propose to compute the search direction at each interior-point iteration for a linear program via a reduced augmented system
that typically has a much smaller dimension than the original augmented system. This reduced system is potentially less susceptible
to the ill-conditioning effect of the elements in the (1,1) block of the augmented matrix. A preconditioner is then designed
by approximating the block structure of the inverse of the transformed matrix to further improve the spectral properties of
the transformed system. The resulting preconditioned system is likely to become better conditioned toward the end of the interior-point
algorithm. Capitalizing on the special spectral properties of the transformed matrix, we further proposed a two-phase iterative
algorithm that starts by solving the normal equations with PCG in each IPM iteration, and then switches to solve the preconditioned
reduced augmented system with symmetric quasi-minimal residual (SQMR) method when it is advantageous to do so. The experimental
results have demonstrated that our proposed method is competitive with direct methods in solving large-scale LP problems and
a set of highly degenerate LP problems.
Research supported in parts by NUS Research Grant R146-000-076-112 and SMA IUP Research Grant. 相似文献
15.
W. W. Hager 《Journal of Optimization Theory and Applications》1987,55(1):37-71
Algorithms to solve constrained optimization problems are derived. These schemes combine an unconstrained minimization scheme like the conjugate gradient method, an augmented Lagrangian, and multiplier updates to obtain global quadratic convergence. Since an augmented Lagrangian can be ill conditioned, a preconditioning strategy is developed to eliminate the instabilities associated with the penalty term. A criterion for deciding when to increase the penalty is presented.This work was supported by the National Science Foundation, Grant Nos. MCS-81-01892, DMS-84-01758, and DMS-85-20926, and by the Air Force Office of Scientific Research, Grant No. AFOSR-ISSA-860091. 相似文献
16.
17.
带等式约束的光滑优化问题的一类新的精确罚函数 总被引:1,自引:0,他引:1
罚函数方法是将约束优化问题转化为无约束优化问题的主要方法之一. 不包含目标函数和约束函数梯度信息的罚函数, 称为简单罚函数. 对传统精确罚函数而言, 如果它是简单的就一定是非光滑的; 如果它是光滑的, 就一定不是简单的. 针对等式约束优化问题, 提出一类新的简单罚函数, 该罚函数通过增加一个新的变量来控制罚项. 证明了此罚函数的光滑性和精确性, 并给出了一种解决等式约束优化问题的罚函数算法. 数值结果表明, 该算法对于求解等式约束优化问题是可行的. 相似文献
18.
Sergey I. Solov’ëv 《Linear algebra and its applications》2006,415(1):210-229
This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to nonlinear eigenvalue problems with very large sparse ill-conditioned matrices monotonically depending on the spectral parameter. To compute the smallest eigenvalue of large-scale matrix nonlinear eigenvalue problems, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors, and inner products of vectors. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem. 相似文献
19.
Globally and Superlinearly Convergent QP-Free Algorithm for Nonlinear Constrained Optimization 总被引:2,自引:0,他引:2
A new, infeasible QP-free algorithm for nonlinear constrained optimization problems is proposed. The algorithm is based on a continuously differentiable exact penalty function and on active-set strategy. After a finite number of iterations, the algorithm requires only the solution of two linear systems at each iteration. We prove that the algorithm is globally convergent toward the KKT points and that, if the second-order sufficiency condition and the strict complementarity condition hold, then the rate of convergence is superlinear or even quadratic. Moreover, we incorporate two automatic adjustment rules for the choice of the penalty parameter and make use of an approximated direction as derivative of the merit function so that only first-order derivatives of the objective and constraint functions are used. 相似文献
20.
Vincent Darrigrand Andrei Dumitrasc Carola Kruse Ulrich Rüde 《Numerical Linear Algebra with Applications》2023,30(5):e2484
We study an inexact inner–outer generalized Golub–Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, an inner system has to be solved which in theory has to be done exactly. Whenever the system is getting large, an inner exact solver is, however, no longer efficient or even feasible and iterative methods must be used. We focus this article on a numerical study showing the influence of the accuracy of an inner iterative solution on the accuracy of the solution of the block system. Emphasis is further given on reducing the computational cost, which is defined as the total number of inner iterations. We develop relaxation techniques intended to dynamically change the inner tolerance for each outer iteration to further minimize the total number of inner iterations. We illustrate our findings on a Stokes problem and validate them on a mixed formulation of the Poisson problem. 相似文献