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**CALL NLPLM(***rc, xr, "fun", x0, opt, blc, tc, par, "ptit", "jac">***);**

See "Nonlinear Optimization and Related Subroutines" for a listing of all NLP subroutines. See Chapter 11, "Nonlinear Optimization Examples," for a description of the inputs to and outputs of all NLP subroutines.

The NLPLM subroutine uses the Levenberg-Marquardt method, which is an efficient modification of the trust-region method for nonlinear least-squares problems and is implemented as in Mor (1978). This is the recommended algorithm for small to medium least-squares problems. Large least-squares problems can often be processed more efficiently with other subroutines, such as the NLPCG and NLPQN methods. In each iteration, the NLPLM subroutine solves a quadratically-constrained quadratic minimization problem that restricts the step to the boundary or interior of an

The

**Note:** In least-squares subroutines, you must set the first
element of the *opt* vector to *m*, the number of functions.

In addition to the standard iteration history, the NLPLM subroutine also prints the following information:

- Under the heading
*Iter*, an asterisk (*) printed after the iteration number indicates that the computed Hessian approximation was singular and had to be ridged with a positive value. - The heading
*lambda*represents the Lagrange multiplier, . This has a value of zero when the optimum of the quadratic function approximation is inside the trust region, in which case a trust-region-scaled Newton step is performed. It is greater than zero when the optimum is at the boundary of the trust region, in which case the scaled Newton step is too long to fit in the trust region and a quadratically-constrained optimization is done. Large values indicate optimization difficulties, and as in Gay (1983), a negative value indicates the special case of an indefinite Hessian matrix. - The heading
*rho*refers to , the ratio between the achieved and predicted difference in function values. Values that are much smaller than one indicate optimization difficulties. Values close to or larger than one indicate that the trust region radius can be increased.

Figure 17.5 shows the iteration history for the solution of the unconstrained Rosenbrock problem. See the section, "Unconstrained Rosenbrock Function", for the statements that generate this output.

Optimization Start Parameter Estimates Gradient Objective N Parameter Estimate Function 1 X1 -1.200000 -107.799999 2 X2 1.000000 -44.000000 Value of Objective Function = 12.1 Levenberg-Marquardt Optimization Scaling Update of More (1978) Gradient Computed by Finite Differences CRP Jacobian Computed by Finite Differences Parameter Estimates 2 Functions (Observations) 2 Optimization Start Active Constraints 0 Objective Function 12.1 Max Abs Gradient Element 107.7999987 Radius 2626.5613171 Actual Function Active Objective Iter Restarts Calls Constraints Function 1 0 4 0 2.18185 2 0 6 0 1.59370 3 0 7 0 1.32848 4 0 8 0 1.03891 5 0 9 0 0.78943 6 0 10 0 0.58838 7 0 11 0 0.34224 8 0 12 0 0.19630 9 0 13 0 0.11626 10 0 14 0 0.0000396 11 0 15 0 2.4652E-30 Ratio Between Actual Objective Max Abs and Function Gradient Predicted Iter Change Element Lambda Change 1 9.9181 17.4704 0.00804 0.964 2 0.5881 3.7015 0.0190 0.988 3 0.2652 7.0843 0.00830 0.678 4 0.2896 6.3092 0.00753 0.593 5 0.2495 7.2617 0.00634 0.486 6 0.2011 7.8837 0.00462 0.393 7 0.2461 6.6815 0.00307 0.524 8 0.1459 8.3857 0.00147 0.469 9 0.0800 9.3086 0.00016 0.409 10 0.1162 0.1781 0 1.000 11 0.000040 4.44E-14 0 1.000 Optimization Results Iterations 11 Function Calls 16 Jacobian Calls 12 Active Constraints 0 Objective Function 2.46519E-30 Max Abs Gradient Element 4.440892E-14 Lambda 0 Actual Over Pred Change 1 Radius 0.0178062912 ABSGCONV convergence criterion satisfied. Optimization Results Parameter Estimates Gradient Objective N Parameter Estimate Function 1 X1 1.000000 -4.44089E-14 2 X2 1.000000 2.220446E-14 Value of Objective Function = 2.46519E-30

**Figure 17.5:** Iteration History for the NLPLM Subroutine

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