Splitting methods for a class of non-potential mean field games

Abstract

<p style='text-indent:20px;'>We extend the methods from [<xref ref-type="bibr" rid="b39">39</xref>, <xref ref-type="bibr" rid="b37">37</xref>] to a class of <i>non-potential</i> mean-field game (MFG) systems with mixed couplings. Up to now, splitting methods have been applied to <i>potential</i> MFG systems that can be cast as convex-concave saddle-point problems. Here, we show that a class of non-potential MFG can be cast as primal-dual pairs of monotone inclusions and solved via extensions of convex optimization algorithms such as the primal-dual hybrid gradient (PDHG) algorithm. A critical feature of our approach is in considering dual variables of nonlocal couplings in <i>Fourier</i> or <i>feature spaces</i>.</p>