Nonlinear multidimensional optimization

Having evaluated the system performance for each setting of the six variables, the variables are optimized simultaneously in a multidimensional optimization, using for example SQP, to maximize or minimize an objective function evaluated at each setting of the variables. However, in practice, many models tend to be nonlinear and hence a stochastic method can be more effective. 

It is worth stressing that there are significant differences between the solution of nonlinear systems and multidimensional optimization. 

As far as multidimensional optimization problems are concerned, the matrix B can be a bad approximation of the Hessian (provided it is positive definite) yet still be able to guarantee a reduction in the merit function. Conversely, the matrix B involved in the solution of nonlinear systems should be a good estimate of the Jacobian. 

Unconstrained optimization (nonlinear programming), 2546-2553 classictil methods, 2546-2547 conjugate gradient methods, 2552-2553 golden section method, 2547-2549 line search techniques for, 2547 multidimensional search techniques for, 2549-2552... 

It is, therefore, reasonable to think that the problem of solving nonlinear systems can be tackled by a minimization program applied to a merit function of this kind. Unfortunately, multimodality issues arise if this is done and many additional minima, to which the real solution does not correspond, are introduced, transforming the original problem of the nonlinear system into a multidimensional multimodal optimization problem. 

The properties of the parameter estimates of linear models are no longer valid exactly for the estimates of nonlinear parameters, for the optimal estimates cannot be expressed as linear combinations of the observation outcomes. The estimating vector is not multidimensional normally distributed. Its distribution is not even known The true covariance matrix and the joint confidence region of the parameter estimates are approximated by the expression obtained in the case of linear model regression ... 

---