We study first-order optimality for constrained composite optimization problems whose objective is the sum of a locally Lipschitz function and a proper lower semicontinuous function. We focus on the case in which the limiting subdifferential of the latter function is empty at a point of interest, so that the usual KKT conditions are unavailable. We examine what optimality conditions remain in this setting. We show that any local minimizer must have an active constraint and satisfies a Fritz–John condition based on the horizon subdifferential, independently of any constraint qualification. We then establish an approximate KKT condition that retains information from the Lipschitzian part of the objective. Under differentiability away from the point of interest, a safeguarded augmented Lagrangian method generates a sequence of distinct points satisfying this condition and converging to the minimizer. If the function values of the lower semicontinuous term also converge, the sequence yields the horizon-based condition.