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Projection onto cones generated by epigraphs of perspective functions

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Pith's one-line read Projection onto the epigraph of the perspective of any convex lower semicontinuous function reduces to solving two scalar equations involving that function's proximity operator.

arxiv 2411.08000 v1 pith:D32KR7YJ submitted 2024-11-12 math.OC

Luis M. Briceño-Arias , Cristóbal Vivar-Vargas This is my paper · ORCID
classification math.OC
keywords ontoconesprojectioncomputationconeefficiencyexponentialfunction
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The reading

Many optimization problems ask for the closest point to a specified set. For sets defined by cones, this closest point is called the projection, and fast projection routines are the engine behind conic solvers. The paper studies cones of a special shape: they are built from a convex function f by taking its perspective, which scales the function as a new variable changes, and then taking the epigraph, the set of points lying above the function graph. The exponential cone, used in entropy and neural network problems, and the power cone are famous examples.

The authors prove that projecting onto any such cone only requires solving two scalar equations in one variable each. The equations involve the proximity operator of f, a standard tool in convex optimization that is already available or computable for many functions. The projection formula itself is a transfer of known epigraph projection theory to perspective functions, combined with a proximity result the authors published earlier. The paper then supplies explicit formulas for the exponential cone and a hyperbolic penalty cone, the latter previously unavailable, plus a bisection algorithm with error bounds.

Numerical tests compare the method with an open-source solver on the exponential cone. In one test region the new method is faster and more accurate; in another it is slower and less accurate. A high-dimensional radial test shows much better precision, and the hyperbolic test shows high precision, though no baseline exists for it. The authors do not ship code or data.

Extended reading notes

Core claim

Theorem 3.1 asserts that for any convex lower semicontinuous f in a real Hilbert space, the projection onto epi(tilde f) is either (P_dom(tilde f)(x,eta), delta) when tilde f(P_dom(tilde f)(x,eta)) <= delta, or (prox_mu(tilde f)(x,eta), delta+mu) otherwise, where mu is the unique positive solution of mu + delta - tilde f(prox_mu(tilde f)(x,eta)) = 0. If correct, this computes projections onto exponential, power, and hyperbolic cones by solving at most two scalar equations involving the proximity operator of f.

Load-bearing premise

The load-bearing premise is Proposition 2.1, which gives an explicit formula for prox_{gamma tilde f} in terms of the proximity operators of f and f^*, and is stated as a slight modification of the authors' own prior result [2, Theorem 3.1] without proof in this preprint. If that prox formula fails on boundary or non-supercoercive cases, Theorem 3.1 and all derived examples inherit the failure. The uniqueness and monotonicity of the root curve phi, imported from [3, Lemma 3.27] and [9, Lemma 3.3], is a second external assumption.

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Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. It relies on the prior prox-for-perspective formula [2] and on monotonicity and continuity lemmas from [3,9], neither of which is re-derived in this preprint.

assumptions (3)
  • domain assumption Proposition 2.1: the proximity operator of the perspective function has the stated two-case form involving P_dom f^*, prox of f, and a unique scalar root mu.
    Invoked in Theorem 3.1 and Proposition 3.1; stated as a slight modification of [2, Theorem 3.1] and not proved in this preprint. The main projection formula inherits any failure of this prox formula.
  • domain assumption The function phi(mu) = mu + delta - tilde f(prox_mu tilde f(x,eta)) is continuous, strictly increasing, has limit delta - tilde f(P_dom tilde f(x,eta)) at 0, and diverges to +infinity.
    Used in Algorithm 1 and Theorem 3.2 to guarantee a unique root; attributed to [3, Lemma 3.27] and [9, Lemma 3.3] rather than proved here.
  • standard math Lemma 2.2: the perspective function tilde f is in Gamma0, its conjugate is the indicator of {eta + f^*(x) <= 0}, and epi(tilde f) is a closed convex cone.
    Cites [16, Proposition 2.3] and [5]; this structural fact underlies the Moreau decomposition used in Theorem 3.1.

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Pith. "Pith review of Projection onto cones generated by epigraphs of perspective functions." pith.science (2026). https://pith.science/paper/D32KR7YJ

@misc{pith2026241108000,
  author       = {Pith},
  title        = {Pith review of: Projection onto cones generated by epigraphs of perspective functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D32KR7YJ}},
  note         = {Machine review of arXiv:2411.08000}
}
read the original 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.

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  1. Homogeneous Self-Dual Embedding via Perspective Functions

    math.OC 2026-07 conditional novelty 8.0 of 10

    A single perspective-function inequality encodes primal-dual solutions and infeasibility certificates for minimizing a sum of two convex functions, and Douglas-Rachford on that inequality recovers and extends SCS.

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