Abstract
As shown by an example, the integral function f :ℝn → ℝ, defined by f(x) = ∫a b[B(x, t)]+ g(t) dt, may not be a strongly semismooth function, even if g(t) ≡ 1 and B is a quadratic polynomial with respect to t and infinitely many times smooth with respect to x. We show that f is a strongly semismooth function if g is continuous and B is affine with respect to t and strongly semismooth with respect to x, i.e., B(x, t) = u(x)t + v(x), where u and v are two strongly semismooth functions in ℝn. We also show that f is not a piecewise smooth function if u and v are two linearly independent linear functions, g is continuous and g ≢ 0 in [a, b], and n ≥ 2. We apply the first result to the edge convex minimum norm network interpolation problem, which is a two-dimensional interpolation problem.
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Qi, L., Yin, H. A Strongly Semismooth Integral Function and Its Application. Computational Optimization and Applications 25, 223–246 (2003). https://doi.org/10.1023/A:1022969507994
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DOI: https://doi.org/10.1023/A:1022969507994