On the Convergence of Inexact Alternate Minimization in Problems with \(\ell _0\) Penalties

In this work, we consider unconstrained nonlinear optimization problems where the objective function presents a penalty term on the cardinality of a subset of the variables vector; specifically, we prove that an alternate minimization scheme has global asymptotic convergence guarantees towards points satisfying first-order optimality conditions, even when the optimization step with respect to one of the blocks of variables is inexact and without introducing proximal terms. This result, supported by numerical evidence, justifies the use of pure alternate minimization in applications, even in absence of convexity assumptions.