M. Al-Baali and R. Fletcher. Variational methods for non-linear least squares. J. Oper. Res. Soc., 36:405--421, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... TOMLAB [6] A prototype nonlinear least solver algorithm gn in the MATLAB toolbox NLPLIB TB [5] implements four different algorithms: ffl Gauss Newton with sub space minimization (GN) The idea of sub space minimization was discussed by Lindstrom [9] ffl The Al Baali Fletcher hybrid method (AF) [1]. ffl The Fletcher Xu hybrid method (FX) 4] ffl Huschens TSSM method (Hu) 7] The gn solver treats bound constraints, which is of vital importance for the convergence of the exponential fitting problem. In this paper we only present results using the Al BaaliFletcher hybrid method, Algorithm ....

M. Al-Baali and R. Fletcher. Variational methods for non-linear least squares. J. Oper. Res. Soc., 36:405--421, 1985.

.... the toolbox NLPLIB TB [7] A prototype nonlinear least solver algorithm gn in NLPLIB TBg implements four different algorithms: ffl Gauss Newton with subspace minimization (GN) The idea of subspace minimization was discussed by Lindstrom [12] ffl The Al Baali Fletcher hybrid method (AF) [2]. ffl The Fletcher Xu hybrid method (FX) 5] ffl Huschens TSSM method (Hu) 10] The gn solver treats bound constraints, which is of vital importance for the convergence of the exponential fitting problem. In this paper we only present results using the Al BaaliFletcher hybrid method, Algorithm ....

....if 1 p. Substitute z[1] into remaining equations. Pick out numerators=0 of remaining equations as new equations eq21 . Compute their collected forms for S[j] eq22) and z[j] eq23) Compute degrees in each equation for degmat3 each z[j] separately and degmat4 all z[j] together. Solve for z[2] and decompose so that (P p Q) Den and (P Gamma p Q) Den are solutions. Show that z2 sol includes the solution z1 sol. eq2: seq(subs(z[1] z1sol, eq13[i] i=2. p) eq21: numer(normal(eq2) eq22: collect(eq21, seq(z[j] j=2. p) eq23: collect(eq21, seq(S[j] j=1. 2 p) ....

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M. Al-Baali and R. Fletcher. Variational methods for non-linear least squares. J. Oper. Res. Soc., 36:405--421, 1985.

....in NLPLIB TB [37] and this works well for the exponential fitting problem. ffl Hybrid methods. As GN has good properties for (Z) and (S) and BFGS has good properties for general NLP, one can invent a criterion for switching between GN and BFGS. This strategy is used by Al Baali and Fletcher [6] and Fletcher and Xu [31] The critical point is when to switch between the two algorithms in a hybrid method. Fletcher simply suggests a GN step when s k Gamma s k 1 t Delta s k with e.g. t = 0:2 ( 32, page 117] ffl Structured secant methods: Another way is to invent a QN updated ....

M. Al-Baali and R. Fletcher. Variational methods for non-linear least squares. J. Oper. Res. Soc., 36:405--421, 1985.

....four methods to compute the search direction. The line search algorithm used is the same as for unconstrained problems. The prototype routine lsSolve includes four optimization methods for nonlinear least squares problems: ffl The Gauss Newton method. ffl The Al Baali Fletcher hybrid method [5]. ffl The Fletcher Xu hybrid method [19] ffl The Huschens TSSM method [31] To handle constrained nonlinear least squares problems a new prototype routine clsSolve based on lsSolve is developed. Currently clsSolve can treat linear equality and inequality TOMLAB A General Purpose, Open MATLAB ....

M. Al-Baali and R. Fletcher. Variational methods for non-linear least squares. J. Oper. Res. Soc., 36:405--421, 1985.

....nonlinear least solver algorithm lsSolve in the MATLAB toolbox NLPLIB TB [5] implements four different methods. The lsSolve solver treats bound constraints, which is of vital importance for the convergence of the exponential fitting problem. We use the Al Baali Fletcher hybrid method (AF) [2] as implemented in lsSolve for our tests. There are several ways to formulate the parameter estimation problem and we here present a non separated algorithm, which solve the following simply bounded weighted nonlinear least squares problem Problem NLLSw min a;b P n j=1 (w j Delta ( P p i=1 ....

....Now solve for z[1] z1sol: solve(eqz12[1] z[1] Continue if 1 p. Substitute z[1] into remaining equations. Pick out numerators=0 of remaining equations as new equations eq22 . Compute degrees in each equation for degmat3 each z[j] separately and degmat4 all z[j] together. Solve for z[2] and decompose so that (P p Q) Den and (P Gamma p Q) Den are solutions. Show that z2 sol includes the solution z1 sol. eqz21: normal(subs(z[1] z1sol[1] eqz12[2. p] eqz22: factor(numer(eqz21[1. p 1] degmat3: matrix(p 1,p 1, i,j) degree(eqz22[i] z[j 1] ....

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M. Al-Baali and R. Fletcher. Variational methods for non-linear least squares. J. Oper. Res. Soc., 36:405--421, 1985.

....[3] Mattias Bjorkman shows that clsSolve performs very well compared to other solvers. Constraints are handled using an active set strategy and four types of search step methods are possible to use: ffl Gauss Newton ffl The Fletcher Xu hybrid method [7] ffl The Al Baali Fletcher hybrid method [1]. ffl The Huschens method [17] The robust line search algorithm is a modified version of the algorithm in the text book by Fletcher [8] To handle ill conditioning clsSolve is using subspace minimization, based on ideas by Lindstrom and Wedin [20] 6 Conclusions This work is part of a new ....

M. Al-Baali and R. Fletcher. Variational methods for non-linear least squares. J. Oper. Res. Soc., 36:405--421, 1985.

....a structured BFGS secant method is used, for which superlinear convergence is proved by Dennis in [11] This result was extended by Engels and Martinez [15] to the whole convex Broyden class. Existing Gauss Newton Quasi Newton hybrid implementations include NL2SOL [10] and a package by Fletcher [1]. There are three important issues when designing quasi Newton methods for nonlinear least squares problems: First, we need a definition of y so that the secant equation Gs = y fulfills, where G now approximates S. Second, we need a technique for scaling in order to prevent G from getting too ....

M. Al-Baali and R. Fletcher. Variational methods for nonlinear least squares. Technical Report NA/71, Dept. of Math. Sci., University of Dundee, Dundee, Scotland, 1983.

....max Determine steplength ff k by a line search Update x k 1 = x k ff k p k Method update Gauss Newton Fletcher Xu Al Baali Fletcher Huschens TSSM 6 6 k = k 1 9 Psi q s z R AE ) q Figure 1. The NLPLIB TB prototype routine for nonlinear least squares 7 [1] and the Fletcher Xu [8] hybrid method, and the Huschens TSSM method [18] If rank problems occur the prototype algorithm is using subspace minimization. The line search algorithm used is the same as for unconstrained problems. The Constrained Nonlinear Least Squares Problem (cls) is defined as ....

M. Al-Baali and R. Fletcher. Variational methods for non-linear least squares. J. Oper. Res. Soc., 36:405--421, 1985.

....qpSolve. The prototype nonlinear least squares algorithm lsSolve treats problems with bound constraints in a similar way as the routine ucSolve. The prototype routine lsSolve includes four optimization methods for nonlinear least squares problems: the Gauss Newton method, the Al Baali Fletcher [3] and the Fletcher Xu [15] hybrid method, and the Huschens TSSM method [27] If rank problems occur the prototype algorithm is using subspace minimization. The constrained nonlinear least squares solver clsSolve is based on lsSolve and its search steps methods. Currently clsSolve treats linear ....

M. Al-Baali and R. Fletcher. Variational methods for non-linear least squares. J. Oper. Res. Soc., 36:405--421, 1985.

....return x = x t else Perform a standard backtracking line search [14] on d n to obtain x n return x = x n endif endif 6. 2 Test Results We have run the sparse tensor and Gauss Newton codes on versions of the nonlinear least squares problems described by Al Baali and Fletcher [1] and singular modifications of these problems. Both of these codes terminate successfully if the relative size of (x Gamma x c ) is less than macheps 2 3 , or jjF (x )jj 1 is less than macheps 2 3 , or the relative size of J(x ) T F (x ) is less than macheps 1 3 , and ....

....pseudo random numbers. The a ijk are uniformly distributed random integers in [0,3] and with likelihood p (p = min(100 Gamma[ 200 n ] 90) these values are randomly reset to zero. The parameters c ik and e i are in [ 100,100] and [ 10,10] respectively, the initial vector x 0 has elements in [1,2], and the index l = 8 is chosen. The solution of this type of problem is not known a priori, and indeed different local solutions may exist. The size of the residuals at the solution is typically large and is determined mainly by the bounds [ 10,10] on e i and can be changed by varying these ....

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M. Al-Baali and R. Fletcher. Variational methods for nonlinear least squares. Technical report NA/71, Department of Mathematical Sciences, University of Dundee, 1983.

....the prototype algorithm is using subspace minimization, see Lindstrom [24] The line search algorithm used is a modified version of the algorithm in Fletcher [12, chap. 2] The prototype algorithms includes the following search step methods: ffl Gauss Newton ffl Al Baali Fletcher hybrid method [4] ffl Fletcher Xu hybrid method [13] ffl Huschens method [23] As shown in Holmstrom [22] the gn routine performs very well on ill conditioned non linear least squares problems compared to other routines, like leastsq in the Optimization Toolbox. In Figure 3 we see the result of running the ....

M. Al-Baali and R. Fletcher. Variational methods for non-linear least squares. J. Oper. Res. Soc., 36:405--421, 1985.

....different update strategy by Engels and Martinez in [10] for the whole convex Broyden class. In Huschen [13] a totally structured update was proposed that further improved the convergence rate. Existing Gauss Newton quasi Newton hybrid implementations include NL2SOL [7] and a package by Fletcher [2]. It is interesting that the first papers in the subject did not involved the linear term in the update. Biggs [3] stressed that since S(x) very well can be indefinite at the solution, it is not necessary to use a BFGS update for the reason of positive definite approximations. The results ....

....iterations so we could not find any difference between the methods. We found one suitable, the trigonometric problem, which is constructed as f i (x) Gammae i n X j 1 (a ij sinx j b ij cosx j ) i = 1; m where a ij and b ij are random integers in [ Gamma100; 100] See Fletcher [2], for a complete description. In this test, T4) another global approach is chosed. Here we do not use the quasi Newton until necessary. That is, when the Gauss Newton method performs poorly a switch is made to the quasi Newton method. For these problems it is reasonable to switch when kPR fk ....

M. Al-Baali and R. Fletcher. Variational methods for nonlinear least squares. Technical Report NA/71, Dept. of Math. Sci., University of Dundee, Dundee, Scotland, 1983.

....their use as initial values when nonlinear least squares fitting (1.2) to data and make robustness tests of the optimal parameter values. For the refinement part of the fitting problem, nonlinear least squares algorithms is a standard tool. We have used the hybrid method of Al Baali and Fletcher [5]. and the TSSM of Huschens [20] In radiotherapy the goal is to sterilize tumour cells without causing injury to normal, healthy tissue cells. Ionizing radiation acts biologically by inducing damage to biomolecular structures, most importantly to the DNA strings. The rapidly reproducing tumour ....

.... Gamma J T J. The exponential terms are expected to give the derivatives strongly nonlinear properties and least squares algorithms especially designed for nonlinear problems are Hybrid methods and Structured Secant methods. One example of the former one is an algorithm by Al Baali and Fletcher [5] and Fletcher and Xu [11] Hybrid algorithms switch between Gauss Newton for small residuals and for example BFGS algorithm (Broyden Fletcher Goldfarb Shanno) for large residuals. The critical point is when to switch between the two algorithms in a hybrid method . Fletcher simply suggests a ....

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M. Al-Baali and R. Fletcher, Variational methods for non-linear least squares, J. Oper. Res. Soc., 36 (1985), pp. 405--421.

....subjective and we have compared some examples of V and U presented in next section. Details are discussed in [9] 4 Parameter estimation of exponential time series All tests have been done in MATLAB using the optimization environment TOMLAB [6, 5] We use the Al Baali Fletcher hybrid method (AF) [3] as implemented in lsSolve for our tests. There are several ways to formulate the parameter estimation problem and we here present a non separated algorithm, which solve the following simply bounded weighted nonlinear least squares problem Problem NLLSw min a;b P n j=1 (w j Delta ( P p i=1 a ....

M. Al-Baali and R. Fletcher. Variational methods for non-linear least squares. J. Oper. Res. Soc., 36:405--421, 1985.

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