Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Heinz H. Bauschke

Dept. of Mathematics, University of British Columbia, Kelowna, Canada

heinz.bauschke@ubc.ca

Xianfu Wang

Dept. of Mathematics, University of British Columbia, Kelowna, Canada

shawn.wang@ubc.ca

Keywords

  • Convex quartic
  • cubic equation
  • cubic polynomial
  • Fenchel conjugate
  • projection
  • proximal mapping
  • root

Mathematics Subject Classification

90C25; 12D10, 26C10, 90C26

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Suggested citation

H. H. Bauschke, M. K. Lal, X. Wang. “Real Roots of Real Cubics and Optimization.” Journal of Convex Analysis 32 (2025), No. 1, 119–144. https://doi.org/10.68381/jca32009

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.