Splitting Methods with Variable Metric for Kurdyka–Łojasiewicz Functions and General Convergence Rates

Abstract

We study the convergence of general descent methods applied to a lower semi-continuous and nonconvex function, which satisfies the Kurdyka-Łojasiewicz inequality in a Hilbert space. We prove that any precompact sequence converges to a critical point of the function, and obtain new convergence rates both for the values and the iterates. The analysis covers alternating versions of the forward-backward method with variable metric and relative errors. As an example, a nonsmooth and nonconvex version of the Levenberg-Marquardt algorithm is detailed.

Greedy approximate projection for magnetic resonance fingerprinting with partial volumes

“…We assume that we have a suitable estimate of K when the stopping criteria given in step 10 of Algorithm 3 is reached. The same process is adopted for the estimation of υ described (steps [11][12][13][14][15][16][17][18][19] and κ (steps 20-28). For these two estimates, we allow for a small tolerance (τ υ > 0 and τ κ > 0, respectively), for robustness purposes.…”

“…It consists in a projected gradient descent algorithm, where the projection is computed inexactly, through a memory efficient greedy approach. The proposed method is also generalised to compensate for phase errors in the model, due to timing or coil sensitivity errors, using an alternating minimisation approach [16][17][18][19][20]. Through simulations on a simulated PV phantom, we show that our approach outperforms state-of-the-art methods.…”

Backward–forward algorithms for structured monotone inclusions in Hilbert spaces

“…In [8,33], the method is coupled with approximation or penalization techniques. Based on Kurdyka-Lojasiewicz property, convergence of forward-backward algorithms has been recently obtained in a nonconvex, nonsmooth setting, for tame optimization and semi-algebraic problems [5,4,21,16]. For a recent account on these methods one can consult [6,13,15,19,40] and the bibliography therein.…”

A comparison of edge-preserving approaches for differential interference contrast microscopy

“…In addition, the convergence of Algorithm 1 can be asserted whenever the objective function J satisfies the KurdykaLojasiewicz (KL) property [31,32] at each point of its domain. More precisely, as shown in a number of recent papers [33,34,35], one can prove the convergence of a sequence {φ (n) } n∈N to a limit point (if any exists) which is stationary for J if the following three conditions are satisfied:…”