Generalized Convex Functions (GCFs)

• Generalized convex functions are an umbrella family extending ordinary convexity by replacing classical interpolation rules or dual mappings with structures adapted to new geometries and models.
• They encompass variations such as s-convexity, GA-convexity, and X-convexity, each introducing modified inequalities and optimality conditions useful in both theoretical and applied contexts.
• These frameworks underlie practical advances in optimization, geometric function theory, and approximation, offering refined tools for global structure recovery in complex systems.

Generalized convex functions (GCFs) form an umbrella family of non-equivalent extensions of ordinary convexity rather than a single canonical class. In the works surveyed here, the label covers modified Jensen-type inequalities, generalized conjugacies, epigraphic and polyhedral structures, geodesic and connection-dependent convexity, and kernel-generated transforms. What unifies these constructions is that each replaces either the classical interpolation rule $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$ or the usual Fenchel pairing by a structure adapted to a different geometry, algebra, or optimization model (Mo et al., 2014, Ali et al., 2022, Alizadeh et al., 2024, Luan et al., 2017, Wang et al., 2024, Nehzati, 30 Aug 2025).

1. Scope and conceptual taxonomy

A first point of orientation is terminological. In the surveyed literature, “generalized convexity” may mean an interpolation inequality with altered weights, as in $s$-convex, $(\alpha,m)$-convex, GA-convex, or fractal-set convexity; a path-dependent notion generated by a map $g$, as in $X$-convexity; a conjugacy-based notion, as in Capra-convexity and $(e,y)$-conjugacy; a structural epigraph class, as in generalized polyhedral convexity; or a geometry-dependent notion, as in g-convexity on manifolds and convexity of order $\alpha$ in geometric function theory (Muddassar et al., 2013, İşcan et al., 2015, Chancelier et al., 2020, Páles, 2021, Li et al., 2016, Luan et al., 2023).

Ordinary convexity is recovered in several of these frameworks by specialization. In the fractal-set setting, taking $\alpha=1$ recovers standard convexity; in $s$-$(\alpha,m)$-convexity, $s$0, $s$1, $s$2 gives ordinary convexity; in $s$3-convexity, choosing $s$4 turns $s$5 into the usual affine combination $s$6; in e-convexity, $s$7 yields the classical convex inequality; and in the kernel-generated theory, $s$8 recovers Fenchel-Legendre convexity (Mo et al., 2014, Muddassar et al., 2013, Ali et al., 2022, Alizadeh et al., 2024, Nehzati, 30 Aug 2025).

2. Interpolation-based extensions

One major branch of GCF theory modifies the interpolation rule itself. On the fractal-number set $s$9, generalized convexity is defined by

$(\alpha,m)$0

with $(\alpha,m)$1. In that setting, the paper establishes a three-point slope characterization, equivalence with monotonicity of the local fractional derivative $(\alpha,m)$2, a second local fractional derivative criterion $(\alpha,m)$3, and generalized Jensen and Hermite–Hadamard inequalities (Mo et al., 2014).

A second family uses parametric perturbations of convex weights. The classes $(\alpha,m)$4 and $(\alpha,m)$5 are defined through $(\alpha,m)$6-$(\alpha,m)$7-convexity in the first and second senses, for example

$(\alpha,m)$8

and support weighted Hermite–Hadamard-type estimates with beta-function coefficients (Muddassar et al., 2013). The later notion of general $(\alpha,m)$9-convexity adds a perturbation map $g$0 and defines

$g$1

together with closure under positive combinations, maxima, an epigraph characterization via general $g$2-convex sets, and sufficient optimality conditions including a KKT-type theorem (Ali et al., 2023). In a different direction, $g$3-convexity replaces the affine segment by $g$4; this extends convexity to certain nonconvex sets, induces quasi-$g$5-convex, strictly quasi-$g$6-convex, and semi-strictly quasi-$g$7-convex variants, and preserves classical local-to-global minimality phenomena under an additional proximity condition (Ali et al., 2022).

A third branch replaces arithmetic interpolation by interpolation according to means or by coordinate slicing. GA-convexity requires

$g$8

and leads to Hermite–Hadamard–Fejér type inequalities via Hadamard fractional integrals, especially when $g$9 or $X$0 is GA-convex (İşcan et al., 2015). Coordinate-wise convexity, by contrast, is weaker than joint convexity: on a rectangle $X$1, it means convexity in each variable separately, and supports composite Hermite–Hadamard inequalities indexed by the number of partition subintervals $X$2 (Nwaeze, 2017).

3. Conjugacy-based and kernel-based generalized convexity

Another major direction changes the duality pairing rather than the interpolation rule. For functions of the support $X$3, ordinary Fenchel conjugacy is structurally inadequate because such functions are $X$4-homogeneous. The Capra coupling

$X$5

is constant along primal rays, and Capra-convexity is defined by equality with the Capra biconjugate. Under orthant-strict monotonicity of the source norm and its dual norm, every nondecreasing finite-valued set-function $X$6 yields a Capra-convex support function $X$7. The same paper also proves a hidden-convexity representation $X$8 and an exact variational formulation involving generalized local-$X$9-support dual norms (Chancelier et al., 2020).

The e-convex framework perturbs convexity by an error function $(e,y)$0: $(e,y)$1 Its central dual object is the $(e,y)$2-conjugate

$(e,y)$3

which satisfies $(e,y)$4. The e-subdifferential is linked to this conjugacy by the generalized Fenchel equality

$(e,y)$5

and the framework yields generalized Fermat rules: global minimizers satisfy $(e,y)$6, while local minimizers satisfy $(e,y)$7. The class is strictly broader than ordinary convexity; for example, $(e,y)$8 is e-convex for $(e,y)$9 (Alizadeh et al., 2024).

A related but more global kernel-based theory takes a surplus $\alpha$0 and defines generalized convexity through $\alpha$1-transforms. A $\alpha$2-convex function on $\alpha$3 can be approximated by finitely $\alpha$4-convex functions of the form

$\alpha$5

where $\alpha$6 is finite. Under compactness of $\alpha$7 and $\alpha$8 and local Lipschitz continuity of $\alpha$9, this finite class satisfies a universal approximation property in the uniform norm; under semiconvexity of $\alpha=1$0, the corresponding gradients also satisfy a universal approximation property. The same representation is used to reduce structured bilevel problems in optimal transport and mechanism design to optimization over a finite-dimensional generalized-convex parameterization (Nehzati, 30 Aug 2025).

4. Differential and geometric incarnations

In geometric function theory, generalized convexity appears as convexity of order $\alpha=1$1. The class

$\alpha=1$2

contains the ordinary convex class at $\alpha=1$3, becomes more rigid for $\alpha=1$4, and for $\alpha=1$5 consists of functions convex in one direction. A key structural formula is

$\alpha=1$6

from which the coefficient representation

$\alpha=1$7

follows. This makes sharp extremal problems tractable, and the paper solves the generalized Zalcman coefficient problem on $\alpha=1$8 for all $\alpha=1$9, all $s$0, and all $s$1, thereby proving the generalized Zalcman conjecture for convex functions of order $s$2 (Li et al., 2016).

A different geometric meaning of generalized convexity arises on manifolds. There, a smooth function is g-convex with respect to a torsion-free affine connection $s$3 if its restriction to every geodesic is convex in the ordinary one-variable sense, equivalently if $s$4 at every point. The paper proves that if a smooth function has no critical points, then one can prescribe its Hessian arbitrarily by a suitable connection, hence make it g-convex. At the same time, strong sparseness results hold: on a compact manifold, the set of g-convex functions with respect to some connection is nowhere dense in $s$5; for generic polynomials on $s$6, g-convexity under a geodesically complete connection forces at most one critical point; and the density of g-convex univariate, quadratic, monomial, and additively separable polynomials decreases asymptotically to zero in the senses made precise in the paper (Wang et al., 2024).

5. Generalized polyhedral convexity and multifunctions

Generalized polyhedral convexity is a structural, epigraph-based notion rather than an axiomatic modification of Jensen’s inequality. In a locally convex Hausdorff topological vector space $s$7, a function is generalized polyhedral convex iff its epigraph is a generalized polyhedral convex set. Equivalently, a proper function $s$8 is generalized polyhedral convex iff $s$9 is a generalized polyhedral convex set and

$(\alpha,m)$0

The same paper proves that proper convex $(\alpha,m)$1 is generalized polyhedral convex iff it is generalized piecewise linear, that the conjugate of a proper generalized polyhedral convex function is again proper generalized polyhedral convex, that directional derivatives stay in the same class, and that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function (Luan et al., 2017).

The multifunction extension replaces epigraphs by graphs. A multifunction $(\alpha,m)$2 is generalized polyhedral convex if $(\alpha,m)$3 is a generalized polyhedral convex set, equivalently if

$(\alpha,m)$4

Within this class, each value $(\alpha,m)$5 is generalized polyhedral convex, domains and ranges are generalized polyhedral convex under a finite-codimensional closedness assumption on the relevant linear map, and compositions are again generalized polyhedral convex under a corresponding closedness assumption on $(\alpha,m)$6. The paper also studies direct and inverse images and optimal value functions $(\alpha,m)$7, extending generalized polyhedral convexity from sets and functions to graph-defined set-valued maps (Luan et al., 2023).

These structural results feed directly into nonconvex optimization with convex components. In generalized polyhedral DC optimization on locally convex Hausdorff spaces, the objective has the form $(\alpha,m)$8 with $(\alpha,m)$9, and generalized polyhedral convexity of one or both components yields much sharper conclusions than arbitrary convex data. If $s$00 is proper generalized polyhedral convex, then at interior points of $s$01 local optimality is equivalent to

$s$02

In the fully generalized polyhedral case, the local solution set is a finite union of semi-closed generalized polyhedral convex sets, the global solution set is a finite union of generalized polyhedral convex sets, and selection-based DCA becomes eventually periodic because the relevant subdifferentials take only finitely many values (Huong et al., 2024).

6. Optimization-theoretic consequences and recurrent themes

Some generalized-convex classes are designed almost entirely around optimization consequences. In the algebraic framework of a nonempty set $s$03 equipped with a binary operation $s$04, a function is $s$05-convex if

$s$06

This single inequality subsumes Jensen convexity and subadditivity as special cases. The paper proves a maximum theorem: if $s$07 are $s$08-convex and $s$09 for all $s$10, then there exists $s$11 such that $s$12 for all $s$13. From this it derives a generalized Karush–Kuhn–Tucker theorem for the constrained problem $s$14 subject to $s$15 (Páles, 2021).

Across the surveyed literature, optimization results repeatedly take the form “generalized convexity restores global structure.” In the $s$16-convex framework, local minima become global under the proximity condition used in the paper; in general $s$17-convexity, first-order conditions and KKT-type multipliers become sufficient for global optimality; in e-convexity, conjugacy and e-subdifferentials produce generalized Fermat rules; and in kernel-generated $s$18-convexity, optimal transport and mechanism design reduce to optimization over generalized-convex potentials and their gradients (Ali et al., 2022, Ali et al., 2023, Alizadeh et al., 2024, Nehzati, 30 Aug 2025).

A recurring source of confusion is that these theories are not interchangeable. Coordinate-wise convexity is weaker than joint convexity, with $s$19 on $s$20 serving as the standard example; $s$21-convexity can hold on nonconvex sets such as $s$22; e-convexity includes genuinely nonconvex functions such as $s$23; generalized polyhedral convexity is a finite-representability property of epigraphs and graphs; and manifold g-convexity depends on the chosen connection or metric rather than on an intrinsic Jensen inequality (Nwaeze, 2017, Ali et al., 2022, Alizadeh et al., 2024, Luan et al., 2017, Wang et al., 2024). The most accurate general characterization is therefore umbrella-like: generalized convex functions are ordinary convex functions recast through altered interpolation, altered duality, altered geometry, or altered epigraphic structure.