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31.
Serge Gratton Vincent Malmedy Philippe L. Toint 《Computational Optimization and Applications》2012,51(3):967-979
The properties of multilevel optimization problems defined on a hierarchy of discretization grids can be used to define approximate
secant equations, which describe the second-order behavior of the objective function. Following earlier work by Gratton and
Toint (2009) we introduce a quasi-Newton method (with a linesearch) and a nonlinear conjugate gradient method that both take
advantage of this new second-order information. We then present numerical experiments with these methods and formulate recommendations
for their practical use. 相似文献
32.
Exploiting sparsity is essential to improve the efficiency of solving large optimization problems. We present a method for
recognizing the underlying sparsity structure of a nonlinear partially separable problem, and show how the sparsity of the
Hessian matrices of the problem’s functions can be improved by performing a nonsingular linear transformation in the space
corresponding to the vector of variables. A combinatorial optimization problem is then formulated to increase the number of
zeros of the Hessian matrices in the resulting transformed space, and a heuristic greedy algorithm is applied to this formulation.
The resulting method can thus be viewed as a preprocessor for converting a problem with hidden sparsity into one in which
sparsity is explicit. When it is combined with the sparse semidefinite programming relaxation by Waki et al. for polynomial
optimization problems, the proposed method is shown to extend the performance and applicability of this relaxation technique.
Preliminary numerical results are presented to illustrate this claim.
S. Kim’s research was supported by Kosef R01-2005-000-10271-0.
M. Kojima’s research was supported by Grant-in-Aid for Scientific Research on Priority Areas 16016234. 相似文献
33.
Nicholas I.M. Gould Dominique Orban Annick Sartenaer Philippe L. Toint 《Mathematical Programming》2002,92(3):481-508
The asymptotic convergence of parameterized variants of Newton’s method for the solution of nonlinear systems of equations
is considered. The original system is perturbed by a term involving the variables and a scalar parameter which is driven to
zero as the iteration proceeds. The exact local solutions to the perturbed systems then form a differentiable path leading
to a solution of the original system, the scalar parameter determining the progress along the path. A path-following algorithm,
which involves an inner iteration in which the perturbed systems are approximately solved, is outlined. It is shown that asymptotically,
a single linear system is solved per update of the scalar parameter. It turns out that a componentwise Q-superlinear rate may be attained, both in the direct error and in the residuals, under standard assumptions, and that this
rate may be made arbitrarily close to quadratic. Numerical experiments illustrate the results and we discuss the relationships
that this method shares with interior methods in constrained optimization.
Received: September 8, 2000 / Accepted: September 17, 2001?Published online February 14, 2002 相似文献
34.
This note deals with the computational problem of determining the projection of a given symmetric matrix onto the subspace of symmetric matrices that have a fixed sparsity pattern. This projection is performed with respect to a weighted Frobenius norm involving a metric that is not diagonal. It is shown that the solution to this question is computationally feasible when the metric appearing in the norm is a low rank modification to the identity. Also, generalization to perturbations of higher rank is shown to be increasingly costly in terms of computation. 相似文献
35.
36.
An Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis,
Gould and Toint (Part I, Math Program, doi:, 2009), generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, University
of Cambridge), an algorithm by Nesterov and Polyak (Math Program 108(1):177–205, 2006) and a proposal by Weiser, Deuflhard
and Erdmann (Optim Methods Softw 22(3):413–431, 2007). In this companion paper, we further the analysis by providing worst-case
global iteration complexity bounds for ARC and a second-order variant to achieve approximate first-order, and for the latter
second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most O(e-3/2){\mathcal{O}(\epsilon^{-3/2})} iterations, or equivalently, function- and gradient-evaluations, to drive the norm of the gradient of the objective below
the desired accuracy e{\epsilon}, and O(e-3){\mathcal{O}(\epsilon^{-3})} iterations, to reach approximate nonnegative curvature in a subspace. The orders of these bounds match those proved for Algorithm
3.3 of Nesterov and Polyak which minimizes the cubic model globally on each iteration. Our approach is more general in that
it allows the cubic model to be solved only approximately and may employ approximate Hessians. 相似文献
37.
Andrew R. Conn Nicholas I. M. Gould Dominique Orban Philippe L. Toint 《Mathematical Programming》2000,87(2):215-249
A new primal-dual algorithm is proposed for the minimization of non-convex objective functions subject to general inequality
and linear equality constraints. The method uses a primal-dual trust-region model to ensure descent on a suitable merit function.
Convergence is proved to second-order critical points from arbitrary starting points. Numerical results are presented for
general quadratic programs.
Received: July 19, 1999 / Accepted: February 1, 2000?Published online March 15, 2000 相似文献
38.
This paper studies the speed of convergence of a general algorithm for function minimization without calculating derivatives. This algorithm contains Powell's 1964 algorithm as well as Zangwill's second modification of this procedure. The main results are Theorems 3.1 and 4.1 which show that, if the algorithm behaves well, then asymptotically almost conjugate directions are built; therefore, the algorithm has an every-iteration superlinear speed of convergence. The paper hinges on ideas of McCormick and Ritter and Powell.The authors wish to thank the Namur Department of Mathematics, especially its optimization group, for many discussions and encouragements. The authors also thank the reviewer for many helpful suggestions. 相似文献
39.
This paper considers the inverse shortest paths problem where arc costs are subject to correlation constraints. The motivation for this research arises from applications in traffic modelling and seismic tomography. A new method is proposed for solving this class of problems. It is constructed as a generalization of the algorithm presented in Burton and Toint (Mathematical Programming 53, 1992) for uncorrelated inverse shortest paths. Preliminary numerical experience with the new method is presented and discussed. 相似文献
40.
Ph. L. Toint 《Mathematical Programming》1986,36(3):290-306
Global convergence is proved for a partitioned BFGS algorithm, when applied on a partially separable problem with a convex
decomposition. This case convers a known practical optimization method for large dimensional unconstrained problems. Inexact
solution of the linear system defining the search direction and variants of the steplength rule are also shown to be acceptable
without affecting the global convergence properties. 相似文献