Convexoid: A Minimal Theory of Conjugate Convexity

TL;DR

This work introduces convexoids as a minimal, flexible framework to extend conjugate functions and duality beyond classical convex analysis. By defining a convexoid through a lattice codomain $( ext{Ω}, ext{≤}, )$, index sets $( ext{Δ}, ext{Λ})$, and a coupling function $ ext{φ}$, it derives generalized conjugates, subdifferentials, and a double-conjugate theorem, enabling broad function and structure approximations. It then develops two duality systems, Type-I and Type-II, with general weak duality guaranteed and specific sufficient conditions for strong duality, applicable to functional and structural (set- and graph-like) objects, as well as to topoi in extended settings. The framework supports diverse approximation schemes (e.g., symmetric-conic, bilinear, radial, piecewise-constant) and can model complex structures such as graphs, fuzzy sets, and toposes, yielding new bounds and structural verifications. Overall, convexoids unify and extend classical and radial dualities, offering a minimal yet powerful toolkit for primal–dual analysis across broad domains and problem spaces.

Abstract

A key idea in convex optimization theory is to use well-structured affine functions to approximate general functions, leading to impactful developments in conjugate functions and convex duality theory. This raises the question: what are the minimal requirements to establish these results? This paper aims to address this inquiry through a carefully crafted system called the convexoid. We demonstrate that fundamental constructs, such as conjugate functions and subdifferentials, along with their relationships, can be derived within this minimal system. Building on this, we define the associated duality systems and develop conditions for weak and strong duality, generalizing the classic results from conjugate duality and radial duality theories. Due to its flexibility, our framework supports various approximation schemes, including approximating general functions using symmetric-conic, bilinear, radial, or piecewise constant functions, and representing general structures such as graphs, set systems, fuzzy sets, or toposes using special membership functions. The associated duality results for these systems also open new opportunities for establishing bounds on objective values and verifying structural properties.

Figures (3)

• Figure 1: The product of $G_1$ and $G_2$ with the universal property.
• Figure 2: A morphism in $\mathcal{G}$ from a graph $G$ to the subobject classifier $\Omega$. Each edge of $G$ is mapped to the one in $\Omega$ with the same pattern.
• Figure 3: Definition of the conjugate $f^*$ in a topos concavoid.

Theorems & Definitions (35)

• Definition 1: Convexoid Codomain
• Lemma 1
• proof
• Definition 2: Convexoid
• Definition 3: Fenchel-Young Relationship & Conjugate & Subdifferential
• Definition 4: Affine & Convex Functions
• Theorem 1
• proof
• Proposition 1
• proof