Optimization::Marquardt Function

Minimizes the function of several variables by using the Marquardt optimization algorithm.

What it computes. Minimizes a smooth real function of several
variables with the Levenberg-Marquardt method, solving a damped Newton step (H+lambda I) delta=-g and adapting the damping lambda (started from Lambda0) between gradient-descent and Newton behaviour. It needs both the gradient and the Hessian via GradHess. The standard test objective is

f(x_0,x_1) = 100 (x_1 - x_0^2)^2 + (1 - x_0)^2

with global minimum f=0 at $(1,1)$, gradient

(d f)/(d x_0) = -400 (x_1 - x_0^2) x_0 - 2 (1 - x_0), (d f)/(d x_1) = 200 (x_1 - x_0^2)

and Hessian

H(f) = [ 1200 x_0^2 - 400 x_1 + 2  -400 x_0 ]
       [ -400 x_0                  200      ]

Domain. Pars holds the n starting coordinates;
GradHess returns the gradient and (symmetric) Hessian; MaxIter > 0; Tol,GradTol ≥ 0; Lambda0 > 0.

Defined behaviour. Returns the iteration count; Pars
holds the minimizer, FMin the minimum and IHess the inverse Hessian. StopReason reports the cause - TOptStopReason::OptResConverged/TOptStopReason::OptResSmallGrad on success, TOptStopReason::optNANValue if the objective returned NaN, or TOptStopReason::OptResBigLambda/TOptStopReason::OptResSmallLambda/ TOptStopReason::OptResMaxIter otherwise.

Minimizes the function of several variables by using the Marquardt optimization method with no log.

Minimizes the function of several variables by using the Marquardt optimization method with no log.

Minimizes the function of several variables by using the Marquardt optimization method with no log.