Algorithms for nonlinear M-estimation (Q1381515)

An algorithm of minimization of a sum \( \sum_{i=1}^m\rho(f_i(x)) \) by \(x\) is proposed, where \(\rho(t)\) is some robust (\(L_p\), Huber or Fair) cost function. The algorithm is of the trust region type, i.e. at each iteration the functions \(f_j\) are linearized in some neighborhood \(N_k\) of the current approximation \(x_k\) and a new approximation \[ x_{k+1}=\text{arg min}_{x_\in N_k}\sum_{j=1}^m \rho(l_j(x_k,x)) \] is considered, where \( l_j(x_k,x)=f_i(x_k)+f'_j(x_k)^T(x-x_k) \) is the linearization of \(f_j\). Special algorithms to derive \(N_k\) (the trust region) and the minimum of the ``linearized'' model are described. Results of simulations are presented.