Trust region algorithms for the nonlinear least distance problem (Q1893536)

This paper discusses the nonlinear least distance problem, which is a special case of equality constrained optimization. Let a curve or surface be given in implicit form via the equation \(f(x)= 0\), \(x\in \mathbb{R}^d\), and let \(z\in \mathbb{R}^d\) be a fixed data point. This paper discusses two algorithms for solving the following problem: Find a point \(x^*\) such that \(f(x^*)= 0\) and \(|z- x^*|_2\) is minimal among all such \(x\). The algorithms presented use the trust region approach in which, at each iteration, an approximation to the objective function or merit function is given in the neighborhood (the trust region) of the current iterate. Among other things, this allows one to prove global convergence of the algorithm.