Journal of Convex Analysis 32 (2025), No. 1, 119--144
Copyright Heldermann Verlag 2025
Real Roots of Real Cubics and Optimization
Heinz H. Bauschke
Dept. of Mathematics, University of British Columbia, Kelowna, Canada
heinz.bauschke@ubc.ca
Manish Krishan Lal
Dept. of Mathematics, Technical University, Munich, Germany
manish.krishanlal@tum.de
Xianfu Wang
Dept. of Mathematics, University of British Columbia, Kelowna, Canada
shawn.wang@ubc.ca
The solution of the cubic equation has a century-long history; however, the usual presentation is geared towards applications in algebra and is somewhat inconvenient to use in optimization where frequently the main interest lies in real roots. In this paper, we first present the roots of the cubic in a form that makes them convenient to use and we also focus on information on the location of the real roots. Armed with this, we provide several applications in convex analysis and optimization where we compute Fenchel conjugates, proximal mappings and projections.
Keywords: Convex quartic, cubic equation, cubic polynomial, Fenchel conjugate, projection, proximal mapping, root.
MSC: 90C25; 12D10, 26C10, 90C26.
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