Projection onto cones generated by epigraphs of perspective functions

TL;DR

This paper introduces an efficient method for projecting onto cones generated by epigraphs of perspective functions, enabling computations for various cones including exponential, power, and hyperbolic cones, with demonstrated numerical efficiency.

Contribution

It provides a novel formula requiring only two scalar equations for projections onto these cones, extending computational capabilities to new conic types.

Findings

Efficient projection formula involving two scalar equations.

Application to exponential, power, and hyperbolic cones.

Numerical comparison shows improved efficiency over existing methods.

Abstract

In this paper we provide an efficient computation of the projection onto the cone generated by the epigraph of the perspective of any convex lower semicontinuous function. Our formula requires solving only two scalar equations involving the proximity operator of the function. This enables the computation of projections, for instance, onto exponential and power cones, and extends to previously unexplored conic projections, such as the projection onto the hyperbolic cone. We compare numerically the efficiency of the proposed approach in the case of exponential cones with an open source available method in the literature, illustrating its efficiency.