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23
Nonmonotone spectral projected gradient methods on convex sets
 SIAM Journal on Optimization
, 2000
"... Abstract. Nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the Grippo–Lampariello–Lucidi nonmonotone lin ..."
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Cited by 205 (29 self)
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Abstract. Nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the Grippo–Lampariello–Lucidi nonmonotone line search. In particular, the nonmonotone strategy is combined with the spectral gradient choice of steplength to accelerate the convergence process. In addition to the classical projected gradient nonlinear path, the feasible spectral projected gradient is used as a search direction to avoid additional trial projections during the onedimensional search process. Convergence properties and extensive numerical results are presented.
A semidefinite framework for trust region subproblems with applications to large scale minimization
, 2002
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A New MatrixFree Algorithm for the LargeScale TrustRegion Subproblem
, 1995
"... The trustregion subproblem arises frequently in linear algebra and optimization applications. Recently, matrixfree methods have been introduced to solve large scale trustregion subproblems. These methods only require a matrixvector product and do not rely on matrix factorizations [4, 7]. The ..."
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Cited by 62 (9 self)
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The trustregion subproblem arises frequently in linear algebra and optimization applications. Recently, matrixfree methods have been introduced to solve large scale trustregion subproblems. These methods only require a matrixvector product and do not rely on matrix factorizations [4, 7]. These approaches recast the trust region subproblem in terms of a parameterized eigenvalue problem and then adjust the parameter to find the optimal solution from the eigenvector corresponding to the smallest eigenvalue of the parameterized eigenvalue problem. This paper presents a new matrixfree algorithm for the largescale trustregion subproblem. The new algorithm improves upon the previous algorithms by introducing a unified iteration that naturally includes the so called hard case. The new iteration is shown to be superlinearly convergent in all cases. Computational results are presented to illustrate convergence properties and robustness of the method.
A Subspace, Interior, and Conjugate Gradient Method for LargeScale BoundConstrained Minimization Problems
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 1999
"... A subspace adaptation of the ColemanLi trust region and interior method is proposed for solving largescale boundconstrained minimization problems. This method can be implemented with either sparse Cholesky factorization or conjugate gradient computation. Under reasonable conditions the convergenc ..."
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Cited by 61 (1 self)
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A subspace adaptation of the ColemanLi trust region and interior method is proposed for solving largescale boundconstrained minimization problems. This method can be implemented with either sparse Cholesky factorization or conjugate gradient computation. Under reasonable conditions the convergence properties of this subspace trust region method are as strong as those of its fullspace version. Computational
Incomplete Cholesky Factorizations With Limited Memory
 SIAM J. SCI. COMPUT
, 1999
"... We propose an incomplete Cholesky factorization for the solution of largescale trust region subproblems and positive definite systems of linear equations. This factorization depends on a parameter p that specifies the amount of additional memory (in multiples of n, the dimension of the problem) tha ..."
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Cited by 40 (6 self)
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We propose an incomplete Cholesky factorization for the solution of largescale trust region subproblems and positive definite systems of linear equations. This factorization depends on a parameter p that specifies the amount of additional memory (in multiples of n, the dimension of the problem) that is available; there is no need to specify a drop tolerance. Our numerical results show that the number of conjugate gradient iterations and the computing time are reduced dramatically for small values of p. We also show that in contrast with drop tolerance strategies, the new approach is more stable in terms of number of iterations and memory requirements.
Analysis of Inexact TrustRegion SQP Algorithms
 RICE UNIVERSITY, DEPARTMENT OF
, 2000
"... In this paper we extend the design of a class of compositestep trustregion SQP methods and their global convergence analysis to allow inexact problem information. The inexact problem information can result from iterative linear systems solves within the trustregion SQP method or from approximatio ..."
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Cited by 26 (2 self)
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In this paper we extend the design of a class of compositestep trustregion SQP methods and their global convergence analysis to allow inexact problem information. The inexact problem information can result from iterative linear systems solves within the trustregion SQP method or from approximations of firstorder derivatives. Accuracy requirements in our trustregion SQP methods are adjusted based on feasibility and optimality of the iterates. Our accuracy requirements are stated in general terms, but we show how they can be enforced using information that is already available in matrixfree implementations of SQP methods. In the absence of inexactness our global convergence theory is equal to that of Dennis, ElAlem, Maciel (SIAM J. Optim., 7 (1997), pp. 177207). If all iterates are feasible, i.e., if all iterates satisfy the equality constraints, then our results are related to the known convergence analyses for trustregion methods with inexact gradient information fo...
A D.C. optimization algorithm for solving the trustregion subproblem
 SIAM Journal on Optimization
, 1988
"... Abstract. This paper is devoted to difference of convex functions (d.c.) optimization: d.c. duality, local and global optimality conditions in d.c. programming, the d.c. algorithm (DCA), and its application to solving the trustregion problem. The DCA is an iterative method that is quite different f ..."
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Cited by 23 (0 self)
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Abstract. This paper is devoted to difference of convex functions (d.c.) optimization: d.c. duality, local and global optimality conditions in d.c. programming, the d.c. algorithm (DCA), and its application to solving the trustregion problem. The DCA is an iterative method that is quite different from wellknown related algorithms. Thanks to the particular structure of the trustregion problem, the DCA is very simple (requiring only matrixvector products) and, in practice, converges to the global solution. The inexpensive implicitly restarted Lanczos method of Sorensen is used to check the optimality of solutions provided by the DCA. When a nonglobal solution is found, a simple numerical procedure is introduced both to find a feasible point having a smaller objective value and to restart the DCA at this point. It is shown that in the nonconvex case, the DCA converges to the global solution of the trustregion problem, using only matrixvector products and requiring at most 2m+2 restarts, where m is the number of distinct negative eigenvalues of the coefficient matrix that defines the problem. Numerical simulations establish the robustness and efficiency of the DCA compared to standard related methods, especially for largescale problems.
A LargeScale TrustRegion Approach to the Regularization of Discrete IllPosed Problems
 RICE UNIVERSITY
, 1998
"... We consider the problem of computing the solution of largescale discrete illposed problems when there is noise in the data. These problems arise in important areas such as seismic inversion, medical imaging and signal processing. We pose the problem as a quadratically constrained least squares pro ..."
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Cited by 17 (6 self)
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We consider the problem of computing the solution of largescale discrete illposed problems when there is noise in the data. These problems arise in important areas such as seismic inversion, medical imaging and signal processing. We pose the problem as a quadratically constrained least squares problem and develop a method for the solution of such problem. Our method does not require factorization of the coefficient matrix, it has very low storage requirements and handles the high degree of singularities arising in discrete illposed problems. We present numerical results on test problems and an application of the method to a practical problem with real data.
SPG: Software for ConvexConstrained Optimization
, 2001
"... this paper we describe Fortran 77 software that implements the nonmonotone spectral projected gradient (SPG) algorithm. The SPG method applies to problems of the form min f(x) subject to x 2 ; where is a closed convex set in IR n . It is assumed that f is dened and has continuous partial deriva ..."
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Cited by 13 (4 self)
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this paper we describe Fortran 77 software that implements the nonmonotone spectral projected gradient (SPG) algorithm. The SPG method applies to problems of the form min f(x) subject to x 2 ; where is a closed convex set in IR n . It is assumed that f is dened and has continuous partial derivatives on an open set that contains Users of the software must supply subroutines to compute the function f(x), the gradient rf(x) and projections of an arbitrary point x onto Information about the Hessian matrix is not required and the storage requirements are minimal. Therefore, the algorithm is appropriate for largescale convexconstrained optimization problems with aordable projections onto the feasible set. Notice that the algorithm is also suitable for unconstrained optimization problems simply by setting = IR n
On Some Properties of Quadratic Programs With a Convex Quadratic Constraint
, 1996
"... In this paper we consider the problem of minimizing a (possibly nonconvex) quadratic function with a quadratic constraint. We point out some new properties of the problem. In particular, in the first part of the paper, we show that (i) given a KKT point that is not a global minimizer, it is easy to ..."
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Cited by 12 (1 self)
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In this paper we consider the problem of minimizing a (possibly nonconvex) quadratic function with a quadratic constraint. We point out some new properties of the problem. In particular, in the first part of the paper, we show that (i) given a KKT point that is not a global minimizer, it is easy to find a "better" feasible point; (ii) strict complementarity holds at the localnonglobal minimizer. In the second part, we show that the original constrained problem is equivalent to the unconstrained minimization of a piecewise quartic merit function. Using the unconstrained formulation we give, in the nonconvex case, a new second order necessary condition for global minimizers. In the third part, algorithmic applications of the preceding results are briefly outlined and some preliminary numerical experiments are reported.