Finding zeros of Hölder metrically subregular mappings via globally convergent Levenberg–Marquardt methods

Abstract

We introduce LMLS and LMQR, two globally convergent Levenberg–Marquardt methods for finding zeros of Hölder 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 non-monotone quadratic regularization 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 (Formula presented.) function and gradient evaluations, for an accuracy parameter (Formula presented.). 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 Hölder 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.