On Logarithmic Smoothing of the Maximum Function

F. Guerra Vazquez, H. Günzel and H.Th. Jongen

Annals of Operations Research, 2001, vol. 101, issue 1, 209-220

Abstract: 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 g ε of f itself, where ε>0 denotes the approximation parameter. The one-parametric family g ε 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 g ε 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 g ε , 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). Copyright Kluwer Academic Publishers 2001

Keywords: maximum function; logarithmic barrier function; interior approximation; stationary point; Morse index (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1023/A:1010924609000

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