Finding Zeros of H"older Metrically Subregular Mappings via Globally Convergent Levenberg-Marquardt Methods

We present two globally convergent Levenberg-Marquardt methods for finding zeros of H"older metrically subregular mappings that may have non-isolated zeros. The first method unifies the Levenberg- Marquardt direction and an Armijo-type line search, while the second incorporates this direction with a nonmonotone trust-region technique. For both methods, we prove the global convergence to a first-order stationary point of the associated merit function. Furthermore, the worst-case global complexity of these methods are provided, indicating that an approximate stationary point can be computed in at most O(varepsilon-2) function and gradient evaluations, for an accuracy parameter varepsilon>0. We also study the conditions for the proposed methods to converge to a zero of the associated mappings. Computing a moiety conserved steady state for biochemical reaction networks can be cast as the problem of finding a zero of a H"older metrically subregular mapping. We report encouraging numerical results for finding a zero of such mappings derived from real-world biological data, which supports our theoretical foundations.