TY - JOUR
T1 - Virtuous smoothing for global optimization
AU - Lee, Jon
AU - Skipper, Daphne
N1 - Funding Information:
Acknowledgements The authors gratefully acknowledge the anonymous referee who proposed the performance measure studied in Sect. 5. J. Lee gratefully acknowledges partial support from ONR Grant N00014-14-1-0315.
Publisher Copyright:
© 2017, Springer Science+Business Media New York.
PY - 2017/11/1
Y1 - 2017/11/1
N2 - In the context of global optimization and mixed-integer non-linear programming, generalizing a technique of D’Ambrosio, Fampa, Lee and Vigerske for handling the square-root function, we develop a virtuous smoothing method, using cubics, aimed at functions having some limited non-smoothness. Our results pertain to root functions (wp with 0 < p< 1) and their increasing concave relatives. We provide (i) a sufficient condition (which applies to functions more general than root functions) for our smoothing to be increasing and concave, (ii) a proof that when p= 1 / q for integers q≥ 2 , our smoothing lower bounds the root function, (iii) substantial progress (i.e., a proof for integers 2 ≤ q≤ 10 , 000) on the conjecture that our smoothing is a sharper bound on the root function than the natural and simpler “shifted root function”, and (iv) for all root functions, a quantification of the superiority (in an average sense) of our smoothing versus the shifted root function near 0.
AB - In the context of global optimization and mixed-integer non-linear programming, generalizing a technique of D’Ambrosio, Fampa, Lee and Vigerske for handling the square-root function, we develop a virtuous smoothing method, using cubics, aimed at functions having some limited non-smoothness. Our results pertain to root functions (wp with 0 < p< 1) and their increasing concave relatives. We provide (i) a sufficient condition (which applies to functions more general than root functions) for our smoothing to be increasing and concave, (ii) a proof that when p= 1 / q for integers q≥ 2 , our smoothing lower bounds the root function, (iii) substantial progress (i.e., a proof for integers 2 ≤ q≤ 10 , 000) on the conjecture that our smoothing is a sharper bound on the root function than the natural and simpler “shifted root function”, and (iv) for all root functions, a quantification of the superiority (in an average sense) of our smoothing versus the shifted root function near 0.
KW - Global optimization
KW - Non-differentiable
KW - Non-smooth
KW - Piece-wise functions
KW - Roots
KW - Smoothing
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U2 - 10.1007/s10898-017-0533-x
DO - 10.1007/s10898-017-0533-x
M3 - Article
AN - SCOPUS:85019708260
SN - 0925-5001
VL - 69
SP - 677
EP - 697
JO - Journal of Global Optimization
JF - Journal of Global Optimization
IS - 3
ER -