We consider the maximum function f resulting from a finite number of smooth functions. The logarithmic barrier function of the epigraph of f gives rise to a smooth approximation gϵ of f itself, where ϵ > 0 denotes the approximation parameter. The one-parametric family gϵ converges – relative to a compact subset – uniformly to the function f as ϵ tends to zero. Under nondegeneracy assumptions we show that the stationary points of gϵ and f correspond to each other, and that their respective Morse indices coincide. The latter correspondence is obtained by establishing smooth curves x(ϵ) of stationary points for gϵ, where each x(ϵ) converges to the corresponding stationary point of f as ϵ tends to zero. In case of a strongly unique local minimizer, we show that the nondegeneracy assumption may be relaxed in order to obtain a smooth curve x(ϵ).
