A Strongly Semismooth Integral Function and Its Application |
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Authors: | Liqun Qi Hongxia Yin |
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Institution: | (1) Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China;(2) Department of Mathematics, Graduate School, Chinese Academy of Sciences, P.O. Box 3908, Beijing, 100039, People's Republic of China |
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Abstract: | As shown by an example, the integral function f :
n
, defined by f(x) = a
bB(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
n
. 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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Keywords: | integral function strong semismoothness piecewise smoothness generalized Newton method quadratic convergence |
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