P. E. Gill, W. Murray, D. B. Ponceleon, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization. SIAM J. Matrix Anal. Appl., 13:292--311, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....largest part of the CPU time of an algorithm, it is the goal to utilize the special structure of the linear system in the linear system solver. This has been considered by [5] where several preconditioners were used and compared numerically and theoretically. Further discussions can be found in [10], 22] 26] 27] 18] 17] and [9] In [16] the authors use a multilevel technique on the necessary optimality conditions in connection with Newton s method under box constraints on the control. The structure of the Newton system for optimal control problems is exploited in [15] to design ....

P. E. GILL,W.MURRAY,D.B.PONCELE ON, AND M. A. SAUNDERS, Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl, 13 (1992), pp. 292--311.

....the diagonal entries of D are chosen as pivots [20] stability considerations apart. A similar result regarding preconditioners for iterative methods will be shown later. Implementations using the Bunch Parlett factorization proved to be more stable but they are slower than solving (2. 7) see [7, 10, 20]) A multifrontal approach applied to the augmented system has been investigated in [5] The conjugate gradient method is not well defined for indefinite systems. Thus, it is not used for solving (2.6) In [10] SYMMLQ is used to solve the augmented system for a few small problems. We are not ....

....proved to be more stable but they are slower than solving (2.7) see [7, 10, 20] A multifrontal approach applied to the augmented system has been investigated in [5] The conjugate gradient method is not well defined for indefinite systems. Thus, it is not used for solving (2. 6) In [10], SYMMLQ is used to solve the augmented system for a few small problems. We are not aware of any successful implementation of an iterative method for solving the indefinite system arising in interior point methods for large scale linear programming problems. However, this is an active research ....

[Article contains additional citation context not shown here]

P. E. Gill, W. Murray, D. B. Poncele' on and M. A. Saunders, Preconditioners for Indefinite Systems Arising in Optimization, SIAM J. Matrix Anal. and Applications, 13 (1992), pp. 292--311.

....directly with the linear system (1.2) This system is symmetric but not positive definite, and so the traditional conjugate gradient method cannot be applied. It is also a larger system than (1.1) having n m variables instead of n Gamma m. Preconditioning strategies for (1. 2) are discussed in [5, 14]. We concentrate on the solution of (1.1) In exact arithmetic, the number of iterations required by the conjugate gradient method is bounded by the number of distinct eigenvalues of Z T GZ. In addition, from iteration to iteration the method displays a linear rate of convergence with rate ....

P.E. Gill, W. Murray, D.B. Poncele'on, and M.A. Saunders, Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Analysis and Applications, 13 (1992), pp. 292--311.

....largest part of the CPU time of an algorithm, it is the goal to utilize the special structure of the linear system in the linear system solver. This has been considered by [5] where several preconditioners were used and compared numerically and theoretically. Further discussions can be found in [10], 21] 25] 26] 18] 17] and [9] In [16] the authors use a multilevel technique on the necessary optimality conditions in connection with Newton s method under box constraints on the control. The structure of the Newton system for optimal control problems is exploited in [15] to design ....

P. E. GILL, W. MURRAY, D. B. PONCELE ON, AND M. A. SAUNDERS,Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl, 13 (1992), pp. 292--311. Approximating the Newton Step by Defect Correction 29

....descent directions are always generated this is particularily important for nonlinear problems. Another iterative approach to (1) is to consider the (symmetric indefinite) system of equations defining the optimality conditions and to apply symmetric indefinite iterative techniques, e.g. [8]. We have not considered such methods here. ....

P. Gill, W. Murray, D. Ponceleon, and M. Saunders, Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 292 --311. 10

....approaches are due to Gill, Murray, and Wright [18, section 4.4.2.2] and Schnabel and Eskow [40] Recent work in this area includes Forsgren, Gill, and Murray [16] Cheng and Higham [8] and Neumaier [33] These approaches can be extended to sparse problems (see, for example, 10, section 3.3. 8] [17], 39] and [8] but only if all the elements in the factorization are retained. Thus, these approaches lose the advantage of having predictable storage requirements. Section 4 presents the results of our computational experiments. We test an implementation icfm of the proposed incomplete ....

P. E. Gill, W. Murray, D. B. Poncel eon, and M. A. Saunders, Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 292--311.

....approaches are due to Gill, Murray, and Wright [18, Section 4.4.2.2] and Schnabel and Eskow [40] Recent work in this area includes Forsgren, Gill, and Murray [16] Cheng and Higham [8] and Neumaier [33] These approaches can be extended to sparse problems (see, for example, 10, Section 3.3. 8] [17], 39] and [8] but only if all the elements in the factorization are retained. Thus, these approaches lose the advantage of having predictable storage requirements. Section 4 presents the results of our computational experiments. We test an implementation icfm of the proposed incomplete ....

P. E. Gill, W. Murray, D. B. Poncel' eon, and M. A. Saunders, Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 292--311.

.... By contrast, the sparse Cholesky factorization primarily tries to order the rows and columns of B whilst maintaining reasonable stability by including the possibility of adding appropriate quantities to the diagonals of B, if necessary, Chapter 3 of Conn et al. 1992b, Gill and Murray, 1974, Gill et al. 1992, Schlick, 1993 and Schnabel and Eskow, 1991) For example, Schnabel and Eskow use Gerschgorin bounds to determine the amount to add to the diagonal. They choose diagonal pivots and change the diagonal as little as is 5 reasonable in order to maintain sufficient positive definiteness. All the ....

P. E. Gill, W. Murray, D. B. Poncel'eon, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization. SIAM Journal on Matrix Analysis and Applications, 13:292--311, 1992.

....current iteration. This Newton step d n is the modified Newton step (r 2 f(x c ) I) Gamma1 rf(x c ) where = 0 if r 2 f(x c ) is safely positive definite, and 0 otherwise. To obtain the perturbation , we use a modification of MA27 advocated by Gill, Murray, Ponceleon, and Saunders in [13]. In this method we first compute the LDL T of the Hessian matrix using the MA27 package, then change the block diagonal matrix D to D E. The modified matrix is block diagonal positive definite. This guarantees that the decomposition L(D E)L T is positive definite as well. Note that the ....

P. E. Gill, W. Murray, D. B. Ponceleon, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization and nonlinear least squares problems. Technical Report SOL 90-8, Department of Operations Research, Stanford University, California, 1990.

....Equation (85) can be solved using a sparse Cholesky decomposition [LMS94] The second strategy is to solve the sparse system (67) directly. Several researchers have argued that this method has better numerical properties (See Fourer and Mehrotra [FM91] Gill, Murray, Ponceleon, and Saunders [GMPS92], and Vanderbei and Carpenter [VC93] Moreover, directly solving (67) avoids the loss of sparsity caused by squaring A. Neither of these techniques works for semidefinite programs unfortunately, because they lead to systems with large dense blocks, even if the matrices F i are sparse. A third ....

....engineering. One can also consider solving the symmetric systems (62) or (86) iteratively, using Paige and Saunders SYMMLQ method [PS75] or Freund and Nachtigal s symmetric QMR method [FN94] Working on (62) or (86) has the advantage of allowing more freedom in the selection of preconditioners [GMPS92]. In practice, i.e. with roundoff error, convergence of these methods can be slow and the number of iterations can be much higher than m 1. There are techniques to improve the practical performance, but the implementation is very problem specific, and falls outside the scope of this paper. 8 ....

P. E. Gill, W. Murray, D. B. Ponceleon, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization. SIAM J. on Matrix Analysis and Applications, 13:292--311, 1992.

.... backtracks Trad 45 110 43 50 J and O 49 223 46 146 LBF 32 45 29 19 Table 3: OBSTCLBM (n = 15625) optimal value = 7:2958 MA27 (Duff and Reid, 1982) from the Harwell Subroutine Library (1990) and, if necessary, modifying the factorization to ensure a convex model using the techniques described by Gill et al. 1990). Then, a step along this direction is found as the smallest non negative power of 0:5 which is both feasible for the shifted constraints c(x) s (k) 0 and satisfies the Armijo sufficient decrease condition (see, for example, Dennis and Schnabel, 1983 or Fletcher, 1987) We appreciate that ....

P. E. Gill, W. Murray, D. B. Poncel'eon, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization. SOL 90-8 Technical Report, Department of Operations Research, Stanford University, California, USA, 1990.

....that might be applied to (1.9) MINRES (see, Paige and Saunders, 1975) is probably the method of choice in this case, but the conflict between the form of permissible preconditioners (symmetric and positive definite) and the form of (1. 10) symmetric and indefinite) is unfortunate (see, however, Gill, Murray, Poncel eon and Saunders, 1992, for some possibilities) We will not explore this possibility further in this paper and shall concentrate on iterative methods for (1.1) In Section 2, we shall consider conjugate gradient methods for the problem. We start by reviewing the traditional preconditioned conjugate gradient method, ....

P. E. Gill, W. Murray, D. B. Poncel'eon, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization. SIAM Journal on Matrix Analysis and Applications, 13(1), 292--311, 1992.

....descent directions are always generated this is particularily important for nonlinear problems. Another iterative approach to (1) is to consider the (symmetric indefinite) system of equations defining the optimality conditions and to apply symmetric indefinite iterative techniques, e.g. [8]. We have not considered such methods here. ....

P. Gill, W. Murray, D. Ponceleon, and M. Saunders, Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 292 --311.

....pairs of authors are very careful to ensure that the modifications made are bounded and that no modification ensues when the Hessian is sufficiently positive definite. Extensions to large scale unconstrained and bound constrained optimization, using sparse factorizations, have been proposed by Gill et al. 1992), Conn et al. 1991) and Schlick (1992) We should also mention the philosophically different but mechanically similar class of 2 trust region methods (see, e.g. Hebden, 1973, Mor e, 1978, Sorensen, 1980 and Gay, 1981) Here perturbations of the form E (k) k) I may be made to H(x (k) ....

P. E. Gill, W. Murray, D. B. Poncel'eon, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization. SIAM Journal on Matrix Analysis and Applications, 13:292--311, 1992. REFERENCES 26

....rows, then both of these forms will suffer unnecessarily large amounts of fill in and one would prefer, in that case, to work with the larger system to find a pivot order that does not generate so much fill in. This idea was first suggested by Turner [11] and has been adopted by Saunders et.al. [6, 5, 4] and Mehrotra [2] All three use a Bunch Parlet factorization of the indefinite system. Solving the larger system is also the approach adopted in loqo, but loqo does not employ a Bunch Parlet factorization. Instead, loqo uses a modified Cholesky factorization code that has been altered to solve ....

P.E. Gill, W. Murray, D.B. Poncele'on, and M.A. Saunders. Preconditioners for indefinite systems arising in optimization. SIAM J. Matrix Anal. Appl., 13(1):292--311, 1992.

....matrix compared to the size of the largest entry of the original matrix. The choice of the diagonal blocks usually involves a symmetric permutation of rows and columns. Implementations using the Bunch Parlett factorization proved to be more stable but they are slower than solving (2. 6) see [19, 24, 55]) A multifrontal approach applied to the augmented system has been investigated in [16] The conjugate gradient method is not well defined for indefinite systems. Thus, it is not used for solving (2.5) MINRES and SYMMLQ [43] are two iterative methods for solving symmetric indefinite systems. ....

....systems. Thus, it is not used for solving (2.5) MINRES and SYMMLQ [43] are two iterative methods for solving symmetric indefinite systems. Extensions of the conjugate gradient method are presented in [32, 33] but the numerical properties of these methods are inferior to MINRES and SYMMLQ. In [24], SYMMLQ is used to solve the augmented system for a few small problems. Some of the computational results presented in this work use MINRES for solving the augmented system. We are not aware of any successful implementation of an iterative method for solving the indefinite system arising in ....

[Article contains additional citation context not shown here]

GILL, P. E., MURRAY, W., PONCELE ' ON, D. B. and SAUNDERS, M. A. Preconditioners for Indefinite Systems Arising in Optimization. SIAM Journal on Matrix Analysis and Applications, Vol. 13 pp.292-311, 1992.

....s B , where s B solves the block diagonal trust region subproblem minimize s B2R n hg B ; s B i 1 2 hs B ; Bs B i subject to hs B ; jBjs B i Delta 2 : 3.8) Once again, a single factorization suffices, but this time the factorization may be affordable even when n is large. Note that Gill, Murray, Poncel eon and Saunders (1992) proposed this modified factorization as a preconditioner for iterative methods, while Cheng and Higham (1996) suggest it as an alternative to the modified Cholesky factorizations of Gill and Murray (1974) Gill, Murray and Wright (1981) and Schnabel and Eskow (1991) within linesearch based ....

P. E. Gill, W. Murray, D. B. Poncel'eon, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization. SIAM Journal on Matrix Analysis and Applications, 13(1), 292--311, 1992.

.... Linear systems (1) with coefficient matrices of the form (3) often with C the zero matrix) also arise as subproblems in optimization algorithms, such as interior point methods for linear and nonlinear programs, or sequential programming methods for constrained nonlinear programs; see, e.g. [4,9]. Again, the matrix A is typically highly indefinite. The standard conjugate gradient type Krylov subspace algorithms for the iterative solution of symmetric indefinite linear systems are SYMMLQ and MINRES due to Paige and Saunders [13] Both SYMMLQ and MINRES are based on the Lanczos process for ....

....L is a unit lower triangular matrix and D is a nonsingular block diagonal matrix with 1 Theta 1 and 2 Theta 2 blocks. We ran the symmetric QMR Algorithm 1 with right preconditioner M , and we compared it to MINRES with a positive definite right preconditioner M . Following a proposal made in [9], we used the positive definite matrix M = L DL T that is obtained from (10) by constructing a block diagonal matrix D whose eigenvalues are just the absolute values of those of the indefinite matrix D. The resulting relative residual norms (9) for symmetric QMR (solid line) and MINRES ....

P.E. Gill, W. Murray, D.B. Poncele'on, and M.A. Saunders, "Preconditioners for indefinite systems arising in optimization", SIAM J. Matrix Anal. Appl. 13, pp. 292-- 311, 1992.

....6 (s T d) s T ffi) 3 fl 24 (s T ffi ) 4 : 2:1) Then we set the tensor step d t of the original tensor model (1.3) to ffi d. In step 4. 6, we obtain a perturbation such as r 2 f(x c ) I is safely positive definite by using the Gill, Murray, Ponceleon, and Saunders method [14]. After we compute the LDL T of the Hessian matrix using the MA27 package [13] we change the block diagonal matrix D to D E. The modified matrix is block diagonal positive definite. This guarantees that the decomposition L(D E)L T is sufficiently positive definite. Note that the Hessian ....

P. E. Gill, W. Murray, D. B. Ponceleon, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization and nonlinear least squares problems. Technical Report SOL 90-8, Department of Operations Research, Stanford University, California, 1990.

No context found.

P. E. Gill, W. Murray, D. B. Ponceleon, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization. SIAM J. Matrix Anal. Appl., 13:292--311, 1992.

No context found.

P. E. Gill, W. Murray, D. B. Poncele on, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization. SIAM J. Matrix Anal. Appl., 13:292--311, 1992.

No context found.

P.E. Gill, W. Murray, D.B. Poncele' on and M.A. Saunders, Preconditioners for indefinite systems arising in optimisation, SIAM J. Matrix Anal. Appl.13, pp. 292-311.

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P.E. Gill, W. Murray, D.B. Poncele' on and M.A. Saunders, Preconditioners for indefinite systems arising in optimisation, SIAM J. Matrix Anal. Appl.13, pp. 292-311.

No context found.

P. E. Gill, W. Murray, D. B. Ponceleon, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization. SIAM J. on Matrix Analysis and Applications, 13:292--311, 1992.

No context found.

P. E. Gill, W. Murray, D. B. Poncel'eon, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization. SOL 90-8 Technical Report, Department of Operations Research, Stanford University, California, USA, 1990.

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