Invex Function¶
A differentiable function admitting a comparison map that bounds every value gap below by a gradient pairing at the base point.
Core Idea¶
A differentiable function \(f\) is invex if one direction map \(\eta(y,x)\) makes \(f(y)-f(x)\geq\eta(y,x)\cdot\nabla f(x)\) true for every pair \(x,y\) in its domain. For a convex differentiable function, \(\eta(y,x)=y-x\) works. A different map can make the inequality hold even when the function is nonconvex.[^ref-5b1fb6cc5621]
If the gradient at \(x\) is zero, the inequality says \(f(y)\geq f(x)\) for every \(y\). Every stationary point of an invex function is therefore a global minimum. This conclusion applies to a stationary point once reached; invexity alone does not guarantee that a numerical method reaches one.[^ref-5b1fb6cc5621]
Scope of Application¶
The concept belongs to generalized convexity and differentiable optimization. The convex quadratic \(f(t)=t^2\) is invex using the ordinary direction \(y-x\). Barik, Sra and Honorio also present the nonconvex \(f(t)=t^2+3\sin^2t\) as invex. It has only one stationary point, its global minimum at zero. Constrained invex programs use related ideas but require additional objective, constraint and theorem assumptions.[^ref-5b1fb6cc5621]
Clarity¶
The decisive question is whether a single map works for all comparison pairs. A bound near one candidate or successful solver run is insufficient. The live Quasiconvex Function tests convexity of sublevel sets instead; that is a different property. A nonglobal stationary point rules invexity out because its zero gradient cannot support the required inequality.
Manages Complexity¶
Once the all-pairs bound is proved, checking the gradient at a candidate stationary point settles its global optimality without separately comparing it to every other point. Finding a suitable \(\eta\) may itself be difficult, and a mathematically valid map need not be easy to use in an algorithm.[^ref-5b1fb6cc5621]
Abstract Reasoning¶
State the differentiable domain and proposed \(\eta\), then prove the inequality for every ordered pair. At a stationary point, set the gradient term to zero to derive global minimality. To reject invexity, exhibit one stationary point with a lower-valued competitor: no comparison map can overcome a zero gradient there.[^ref-5b1fb6cc5621]
Knowledge Transfer¶
The same inequality covers convex and some nonconvex objectives, and has a tangent-space formulation for suitable manifolds. Live Gradient supplies the local directional comparison presupposed by the definition. Outside differentiable analysis, the word “invex” is only analogy unless the gradient, map and universal inequality can be supplied.[^ref-5b1fb6cc5621]
[^ref-5b1fb6cc5621]: Adarsh Barik, Suvrit Sra and Jean Honorio, “Invex Programs: First Order Algorithms and Their Convergence”, 2023, Fig. 1©, §2 Definition 2.1, and §3.1–3.2.
Relationships to Other Abstractions¶
Current abstraction Invex Function Domain-specific
Parents (1) — more general patterns this builds on
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Invex Function presupposes Gradient Prime
Invexity compares each value gap with a direction paired against the gradient at a base point.
Hierarchy path (1) — routes to 1 parentless root
- Invex Function → Gradient
Neighborhood in Abstraction Space¶
Invex Function sits in a sparse region of the domain-specific corpus (82nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Jensen's Inequality — 0.84
- Epigraph — 0.83
- Multilinear form — 0.82
- Ridders' Method — 0.82
- Bundle metric — 0.81
Computed from structural-signature embeddings · 2026-10-08