Fully lifted random duality theory

Abstract

We study a generic class of \emph{random optimization problems} (rops) and their typical behavior. The foundational aspects of the random duality theory (RDT), associated with rops, were discussed in \cite{StojnicRegRndDlt10}, where it was shown that one can often infer rops' behavior even without actually solving them. Moreover, \cite{StojnicRegRndDlt10} uncovered that various quantities relevant to rops (including, for example, their typical objective values) can be determined (in a large dimensional context) even completely analytically. The key observation was that the \emph{strong deterministic duality} implies the, so-called, \emph{strong random duality} and therefore the full exactness of the analytical RDT characterizations. Here, we attack precisely those scenarios where the strong deterministic duality is not necessarily present and connect them to the recent progress made in studying bilinearly indexed (bli) random processes in \cite{Stojnicnflgscompyx23,Stojnicsflgscompyx23}. In particular, utilizing a fully lifted (fl) interpolating comparison mechanism introduced in \cite{Stojnicnflgscompyx23}, we establish corresponding \emph{fully lifted} RDT (fl RDT). We then rely on a stationarized fl interpolation realization introduced in \cite{Stojnicsflgscompyx23} to obtain complete \emph{statitionarized} fl RDT (sfl RDT). A few well known problems are then discussed as illustrations of a wide range of practical applications implied by the generality of the considered rops.