M. Arioli, T.F. Chan, I. S. Duff, N.I.M. Gould and J. K. Reid (1993). Computing a search direction for large-scale linearly constrained nonlinear optimization calculations, Technical Report TR/PA/93/94, CERFACS Toulouse, France.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....I is always of good quality. An alternative [46] is to use the Lanczos iteration and continue past the point where negativecurvature is first detected. It is also possible to alter the CG iteration, after encountering negative curvature, so that it can 11 continue exploring other subspaces [3] I do not know if significant improvements in performance can be realized with these proposals, but this question is important and deserves careful investigation. Perhaps more importantthanany of these issues is the choice of preconditioner. The Hessian matrix B k can change drastically during ....

M. Arioli, T.F. Chan, I. S. Duff, N.I.M. Gould and J. K. Reid (1993). Computing a search direction for large-scale linearly constrained nonlinear optimization calculations, Technical Report TR/PA/93/94, CERFACS Toulouse, France.

....I is always of good quality. An alternative [46] is to use the Lanczos iteration and continue past the point where negative curvature is first detected. It is also possible to alter the CG iteration, after encountering negative curvature, so that it can continue exploring other subspaces [3]. I do not know if significant improvements in performance can be realized with these proposals, but this question is important and deserves careful investigation. Perhaps more important than any of these issues is the choice of preconditioner. The Hessian matrix B k can change drastically during ....

M. Arioli, T.F. Chan, I. S. Duff, N.I.M. Gould and J. K. Reid (1993). Computing a search direction for large-scale linearly constrained nonlinear optimization calculations, Technical Report TR/PA/93/94, CERFACS Toulouse, France.

....information on the curvature of the objective function during the iterative processes that computes the Newton type direction. We believe that some promising approaches to compute both a Newton type direction and a negative curvature direction are ffl the use of the conjugate gradient method [1, 5, 24], ffl the use of Lanczos decomposition [18, 20] ffl and several new modifications of Cholesky factorization [9, 10, 11, 21, 22] Acknowledgement We thank the referees for their constructive comments and remarks that helped improve the paper. ....

M. Arioli, T. F. Chan, I. S. Duff, N. I. M. Gould, and J. K. Reid. Computing a search direction for large-scale linearly-constrained nonlinear optimization calculations. Technical Report TR/PA/93/34, CERFACS, 1993.

....approximately the Newton equation H(x k )s k = Gammag(x k ) 3.1) The most popular approach used to find an approximate solution of (3. 1) which satisfies (or can be easily modified so as to satisfy) Condition 1 and Condition 2, is the use of the conjugate gradient method (see, for example, [1, 5, 6, 12, 22, 25]) An alternative approach to solve approximately (3.1) is the use of Lanczos algorithm; in fact, in [17, 18] Nash showed that a truncated scheme based on the Lanczos algorithm allows us to obtain an effective Newton type direction. As regards the direction d k , it should help the algorithm to ....

M. Arioli, T.F. Chan, I.S. Duff, N.I.M. Gould, and J.K.Reid. Computing a search direction for large-scale linearly-constrained nonlinear optimization calculations. Technical Report TR/PA/93/34, CERFACS, 1993.

....kH(x k ) p k g k k 2 j k where j k = min(0:1; kg(x k )k 1=2 2 ) kg(x k )k 2 ; by solving the system H(x k ) p k = Gammag(x k ) using the conjugate gradient method as supplied in PAREBE. If H(x k ) is found not to be positive definite then use a modification of the conjugate gradient (see Arioli, Chan, Duff, Gould and Reid, 1993). 3. x k 1 = x k ff k p k where ff k is the largest scalar of the form 2 Gammal ; l = 0; 1; 2; such that f(x k 2 Gammal p k ) f(x k ) fi k 2 Gammal p T k g(x k ) and fi k = 10 Gamma4 . 4. k k 1, goto 1 Figure 1: Algorithm to find arg min f(x) An interface between ....

Arioli, M., Chan, T. F., Duff, I. S., Gould, N. I. M. and Reid, J. K. (1993), Computing a search direction for large-scale linearly-constrained nonlinear optimization calculations, Technical Report TR/PA/93/34, CERFACS, Toulouse, France.

....of optimization problems, in which (1.4) is approximately minimized within the region defined by the linear constraints, is attempted. This proposal has the advantage that it can fully exploit a number of effective techniques specifically designed to handle linear constraints directly (see Arioli et al. 1993), 0 This research was supported in part by the Advanced Research Projects Agency of the Departement of Defense and was monitored by the Air Force Office of Scientific Research under Contract No F49620 91 C 0079. The United States Government is authorized to reproduce and distribute reprints for ....

M. Arioli, T.F. Chan, I.S. Duff, Nick Gould, and J.K. Reid. Computing a search direction for large-scale linearly constrained nonlinear optimization calculations. Technical Report (in preparation), CERFACS, Toulouse, France, 1993.

....that of (preconditioned) conjugate gradients. Thus we need to solve (2.8) where B is a (possibly perturbed) approximation to the Hessian matrix r xx f . The perturbation may be obtained as the conjugate gradient algorithm proceeds in what we think is an elegant way that preserves conjugacy, see Arioli et al. 1993). 3 Some Existing Methods Let us first consider the most venerable and best known nonlinear optimization algorithm that was designed with large scale problems in mind. The origins of MINOS (Murtagh and Saunders, 1987) come from Robinson (1972) and Rosen and Kreuser (1972) The method can be ....

....program is crucial. One has to repeatedly solve a linear system with the Karush Kuhn Tucker matrix B (k) A T A 0 ; where A = A I : 3:9) It is worth remarking that solving such systems has general applicability to problems with linear constraints (see, for example, Arioli et al. 1993 and Forsgren and Murray, 1993) Gill, Murray and Saunders use generalized TQ factorizations with A Q = 0 T ) and Q T HQ = R T R: 3:10) Now, the Hessian H required for the gradient of the quadratic program s objective function can be determined from H = Q 0T R T RQ 01 : 3:11) The ....

M. Arioli, T. F. Chan, I. S. Duff, N. I. M. Gould, and J. K. Reid. Computing a search direction for large-scale linearly constrained nonlinear optimization calculations. Technical Report TR/PA/93/34, CERFACS, Toulouse, France, 1993.

....of optimization problems, in which (1.4) is approximately minimized within the region defined by the linear constraints, is attempted. This proposal has the advantage that it can fully exploit a number of effective techniques specifically designed to handle linear constraints directly (see Arioli et al. 1993), Forsgren and Murray (1993) or Lustig et al. 1989) for instance) Such an approach is especially worthwhile for largescale problems. This strategy has been implemented and successfully applied within the LANCELOT package for large scale nonlinear optimization (see Conn et al. 1992) in the ....

M. Arioli, T.F. Chan, I.S. Duff, Nick Gould, and J.K. Reid. Computing a search direction for large-scale linearly constrained nonlinear optimization calculations. Technical Report (in preparation), CERFACS, Toulouse, France, 1993.

....suggested by the above example) and the number of inner iterations required to determine the truncated Newton direction. ffl The model is modified, if necessary, to ensure that it is strictly convex. The modification is carried out as the conjugate gradient iteration proceeds using the method of Arioli et al. 1993). ffl A BFGS linesearch method is used to solve the inner minimization problem. An Armijo backtracking linesearch is used, starting with a unit step and dividing the step by two until the Armijo sufficient decrease condition is satisfied. If a step of one proves acceptable, but the model has been ....

M. Arioli, T. F. Chan, I. S. Duff, N. I. M. Gould, and J. K. Reid (1993) Computing a search direction for large-scale linearly constrained nonlinear optimization calculations. Technical Report TR/PA/93/34, CERFACS, Toulouse, France.

.... applied within the LANCELOT package for large scale nonlinear optimization (see Conn et al. 1992) However, such a method may be inefficient when linear constraints are present as there are a number of effective techniques specifically designed to handle such constraints directly (see Arioli et al. 1993), Forsgren and Murray (1993) or Lustig et al. 1989) for instance) This is especially noticeable for large scale problems. The purpose of the present paper is therefore to define and analyze an algorithm where the constraints (1.3) are kept outside the augmented Lagrangian and handled at the ....

M. Arioli, T.F. Chan, I.S. Duff, Nick Gould, and J.K. Reid. Computing a search direction for large-scale linearly constrained nonlinear optimization calculations. Technical Report (in preparation), CERFACS, Toulouse, France, 1993.

....] Bjorck, 1991 ] We will consider this special case further in our discussions in Sections 5 8. Iterative methods are used widely in solving linear systems arising in optimization although they have not been used so often on system (1.1) because it is indefinite. Recent work, for example [Arioli et al. 1993], has investigated extensions of conjugate gradients for this case. 3.2 Computational fluid dynamics Although we concentrate in this section on applications from computational fluid dynamics, there are many other areas in differential equations giving rise to augmented systems. They occur in ....

M. Arioli, T. F. Chan, I. S. Duff, N. I. M. Gould, and J. K. Reid. Computing a search direction for large-scale linearly-constrained nonlinear optimization calculations. Technical Report RAL 93-066, Central Computing Department, Rutherford Appleton Laboratory, 1993.

.... within the LANCELOT package for large scale nonlinear optimization (see Conn et al. 6] However, such a method may be inefficient when linear constraints are present as there are a number of effective techniques specifically designed to handle such constraints directly (see Arioli et al. [1], Forsgren and Murray [14] Toint and Tuyttens [24] or Vanderbei and Carpenter [25] for instance) This is especially important for largescale problems. The purpose of the present paper is therefore to define and analyze an algorithm where the constraints (1.3) are kept outside the augmented ....

M. Arioli, T.F. Chan, I.S. Duff, N.I.M. Gould, and J.K. Reid. Computing a search direction for large-scale linearly constrained nonlinear optimization calculations. Technical Report TR/PA/93/34, CERFACS, Toulouse, France, 1993.

....suggested by the above example) and the number of inner iterations required to determine the truncated Newton direction. ffl The model is modified, if necessary, to ensure that it is strictly convex. The modification is carried out as the conjugate gradient iteration proceeds using the method of Arioli et al. 1993). ffl A BFGS linesearch method is used to solve the inner minimization problem. An Armijo backtracking linesearch is used, starting with a unit step and dividing the step by two until the Armijo sufficient decrease condition is satisfied. If a step of one proves acceptable, but the model has been ....

M. Arioli, T. F. Chan, I. S. Duff, N. I. M. Gould, and J. K. Reid. Computing a search direction for large-scale linearly constrained nonlinear optimization calculations. Technical Report TR/PA/93/34, CERFACS, Toulouse, France, 1993.

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