Abstract
Of key importance in convex analysis and optimization is the notion of duality, and in particular that of Fenchel duality. This work explores improvements to existing algorithms for the symbolic calculation of subdifferentials and Fenchel conjugates of convex functions defined on the real line. More importantly, these algorithms are extended to enable the symbolic calculation of Fenchel conjugates on a class of real-valued functions defined on \(\mathbb{R}^n\). These algorithms are realized in the form of the Maple package SCAT.
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References
Ahlfors L. (1966). Complex Analysis. McGraw-Hill, New York
Auslender A. and Teboulle M. (2003). Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, Berlin
Bauschke, H., Mohrenschildt, M.v.: Fenchel conjugates and subdifferentiation in Maple. In: Technical Report CORR 97-23, Department of Combinatorics and Optimization, University of Waterloo (1997)
Bauschke, H., Mohrenschildt, M.v.: Symbolic computation of Fenchel conjugates. ACM SIGSAM bulletin (2005) (to appear)
Borwein J. and Lewis A. (2000). Convex Analysis and Nonlinear Optimization CMS Books in Mathematics. Springer, New York
Borwein, J., Maréchal, P., Naugler, D.: A convex dual approach to the computation of NMR complex spectra. Math. Meth. Oper. Res. 51(1), 91–102 (2000). URL: http://locutus.cs.dal.ca:8088/archive/ 00000207/
Boyd S. and Vandenberghe L. (2004). Convex Optimization. Cambridge University Press, Cambridge
Hamilton, C.: Symbolic convex analysis. Master’s thesis, Department of Mathematics and Statistics, Simon Fraser University (2005)
Hoch J., Stern A., Donoho D. and Johnstone I. (1990). Maximum entropy reconstruction of complex (phase-sensitive) spectra. J. Magn. Reson. 86: 236–246
Lucet Y. (1996). A fast computational algorithm for the Legendre–Fenchel transform. Comput. Optim. Appl. 6(1): 27–57
Lucet Y. (1997). Faster than the fast Legendre transform, the linear-time Legendre transform. Numer. Algorithms 16: 171–185
Luenberger D. (1969). Optimization by Vector Space Methods Series in Decision and Control. Wiley, New York
Rockafellar R. (1970). Convex Analysis. Princeton University Press, Princeton
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In honour of Alfred Auslender.
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Borwein, J.M., Hamilton, C.H. Symbolic Fenchel Conjugation. Math. Program. 116, 17–35 (2009). https://doi.org/10.1007/s10107-007-0134-4
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DOI: https://doi.org/10.1007/s10107-007-0134-4