From Perspective Maps to Epigraphical Projections

Michael P. Friedlander (), Ariel Goodwin () and Tim Hoheisel ()
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Michael P. Friedlander: Department of Computer Science/Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada
Ariel Goodwin: Department of Mathematics and Statistics, McGill University, Montréal, Québec H3A 0B9, Canada
Tim Hoheisel: Department of Mathematics and Statistics, McGill University, Montréal, Québec H3A 0B9, Canada

Mathematics of Operations Research, 2023, vol. 48, issue 3, 1711-1740

Abstract: The projection onto the epigraph or a level set of a closed proper convex function can be achieved by finding a root of a scalar equation that involves the proximal operator as a function of the proximal parameter. This paper develops the variational analysis of this scalar equation. The approach is based on a study of the variational-analytic properties of general convex optimization problems that are (partial) infimal projections of the sum of the function in question and the perspective map of a convex kernel. When the kernel is the Euclidean norm squared, the solution map corresponds to the proximal map, and thus, the variational properties derived for the general case apply to the proximal case. Properties of the value function and the corresponding solution map—including local Lipschitz continuity, directional differentiability, and semismoothness—are derived. An SC 1 optimization framework for computing epigraphical and level-set projections is, thus, established. Numerical experiments on one-norm projection illustrate the effectiveness of the approach as compared with specialized algorithms.

Keywords: Primary: 49N15; 65K10; 90C25; 90C46; proximal map; Moreau envelope; subdifferential; Fenchel conjugate; perspective map; epigraph; infimal projection; infimal convolution; set-valued map; coderivative; graphical derivative; semismoothness*; SC 1 optimization (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:48:y:2023:i:3:p:1711-1740

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