Pseudolinear Functions: Theory & Applications

• Pseudolinear functions are those that are both pseudoconvex and pseudoconcave, characterized by hyperplane level sets in differentiable and nonsmooth settings.
• They bridge multiple domains, appearing in optimization, machine learning, tropical algebra, discrete geometry, coding theory, and field theory with domain-specific nuances.
• Their structure simplifies complex problems by ensuring that KKT conditions guarantee global optimality and by reducing non-convex measures to tractable linear forms.

Pseudolinear functions belong, in the standard optimization sense, to generalized convexity: a function is pseudolinear when it is both pseudoconvex and pseudoconcave, and in differentiable finite-dimensional settings this is equivalent to a hyperplane structure of level sets. Across recent arXiv literature, the same adjective also appears in several non-equivalent domain-specific senses, including tropical objective functions, pseudo-linear performance measures for classification, and, by terminological extension, pseudolinear drawings, pseudolinear codes, and pseudolinear derivative terms in spin-2 field theory (Ivanov, 2 Sep 2025, Parambath et al., 2015, Parsons et al., 2020, Arroyo et al., 2015, Ruzomberka et al., 2023, Gao, 2014).

1. Core definition in generalized convexity

In the nonsmooth Banach-space formulation, let $f$ be a locally Lipschitz function on an open set $X$ in a Banach space $E$. The cited characterization defines $f$ as pseudoconvex when

$$f(y) < f(x) \implies \langle x^*, y-x \rangle < 0,\ \forall x^*\in \partial f(x),$$

and pseudolinear when it is both pseudoconvex and pseudoconcave, equivalently when $-f$ is pseudoconvex as well (Ivanov, 2 Sep 2025).

A complementary finite-dimensional description used in statistical learning treats pseudo-linear functions as differentiable functions $F: U \subseteq \mathbb{R}^d \to \mathbb{R}$ on an open convex set $U$ for which the level sets are hyperplanes: $$\forall t \in \mathrm{im}(F),~\exists ~a(t) \in \mathbb{R}^d,~b(t) \in \mathbb{R},$$

$F(e) \geq t \iff a(t)^\top e + b(t) \leq 0,$

$X$0

This hyperplane-level-set formulation is given as an equivalent characterization in the cited learning-theoretic treatment (Parambath et al., 2015).

A principal example is the linear-fractional function

$X$1

with $X$2. The source states that linear-fractional functions are pseudo-linear on open half-spaces where their denominator is positive (Parambath et al., 2015). This places pseudolinearity at the intersection of ratio-type objectives, monotonic first-order behavior, and exact geometric separability of upper and lower contour sets.

2. First-order, nonsmooth, and derivative-free characterizations

The 2025 characterization paper formulates pseudolinearity through the Clarke generalized directional derivative

$X$3

and the Clarke-Rockafellar subdifferential

$X$4

Within this framework, complete first-order and derivative-free characterizations are obtained for locally Lipschitz functions on convex sets in Banach spaces (Ivanov, 2 Sep 2025).

One first-order characterization states that $X$5 is pseudolinear on $X$6 if and only if there exists a function $X$7 such that

$X$8

with $X$9. A symmetric two-point version is

$E$0

for all $E$1, $E$2, $E$3 (Ivanov, 2 Sep 2025). These formulas make explicit that, for pseudolinear functions, function differences and generalized directional information are aligned up to a strictly positive scalar factor.

The same paper also gives a derivative-free characterization in the $E$4 case: $E$5 is pseudolinear if and only if there is $E$6 such that for all $E$7 and $E$8,

$E$9

with

$f$0

The related notion of semistrictly quasilinear functions uses the same interpolation formula with the stricter condition $f$1, so the value at an interior convex combination lies strictly between $f$2 and $f$3 whenever $f$4 (Ivanov, 2 Sep 2025).

The same source further states that a semistrictly quasilinear function becomes pseudolinear, in the Clarke-Rockafellar sense, precisely under a vanishing-direction condition: $f$5 In the Fréchet differentiable case this simplifies to

$f$6

This identifies pseudolinearity as semistrict quasilinearity plus a “flat direction implies constancy” property (Ivanov, 2 Sep 2025).

3. Optimization consequences and exact global optimality

In nonlinear programming, the principal operational consequence of pseudolinearity is that KKT conditions become sufficient for global optimality. The OFDMA secure cooperative communication paper states that the relevant objective function is pseudolinear on the feasible region defined by linear constraints, because the partial derivatives do not vanish on that feasible region, and that the solution obtained from the KKT conditions is the global optimal (Saini et al., 2019).

That paper considers two optimization problems: sum rate maximization subject to individual power constraints on source and relay, and sum power minimization subject to a fairness constraint in terms of per-user minimum support secure rate requirement. After reformulation, the secure-rate objective is pseudolinear in the variables $f$7 and $f$8, with monotonicity in each variable on the feasible set: increasing in $f$9 and decreasing in $$f(y) < f(x) \implies \langle x^*, y-x \rangle < 0,\ \forall x^*\in \partial f(x),$$0. The KKT analysis then yields the matched-link condition

$$f(y) < f(x) \implies \langle x^*, y-x \rangle < 0,\ \forall x^*\in \partial f(x),$$1

which the paper identifies as the optimal allocation structure (Saini et al., 2019). The practical significance is explicit in the source: pseudolinearity allows global solution of resource allocation problems that are not convex in appearance.

A different optimization setting appears in tropical algebra. There, the tropical pseudolinear objective is

$$f(y) < f(x) \implies \langle x^*, y-x \rangle < 0,\ \forall x^*\in \partial f(x),$$2

and the main constrained problem is

$$f(y) < f(x) \implies \langle x^*, y-x \rangle < 0,\ \forall x^*\in \partial f(x),$$3

The cited work translates feasibility and optimality for such problems into parametric mean-payoff games, via a value function $$f(y) < f(x) \implies \langle x^*, y-x \rangle < 0,\ \forall x^*\in \partial f(x),$$4, and develops both bisection and Newton schemes (Parsons et al., 2020). In the integer-data case, the optimal value of the tropical pseudolinear problem is stated to be a half-integer, and the Newton method terminates in a finite number of steps because the number of possible strategies is finite (Parsons et al., 2020).

Taken together, these results show that pseudolinearity supports exact optimality certification in two rather different regimes: classical KKT analysis under linear constraints, and tropical optimization through parametric game-theoretic reductions. This suggests that pseudolinearity functions as a tractable surrogate for full convexity when directional or order-theoretic structure is strong enough.

4. Pseudo-linear performance measures in machine learning

In learning theory, pseudo-linear functions arise as non-linear performance measures expressed in terms of error profiles. For a classifier $$f(y) < f(x) \implies \langle x^*, y-x \rangle < 0,\ \forall x^*\in \partial f(x),$$5, the error profile is

$$f(y) < f(x) \implies \langle x^*, y-x \rangle < 0,\ \forall x^*\in \partial f(x),$$6

where $$f(y) < f(x) \implies \langle x^*, y-x \rangle < 0,\ \forall x^*\in \partial f(x),$$7 and $$f(y) < f(x) \implies \langle x^*, y-x \rangle < 0,\ \forall x^*\in \partial f(x),$$8 are the per-class false negative and false positive rates (Parambath et al., 2015).

The cited paper states that many notions of $$f(y) < f(x) \implies \langle x^*, y-x \rangle < 0,\ \forall x^*\in \partial f(x),$$9-measures and Jaccard Index are pseudo-linear functions of the per-class false negatives and false positives for binary, multiclass and multilabel classification. In binary classification, with $-f$0, the $-f$1-measure is

$-f$2

and the Jaccard index is

$-f$3

Both are linear-fractional, hence pseudo-linear, in the relevant error components (Parambath et al., 2015).

The same source gives an exact reduction from optimization of any pseudolinear performance measure to cost-sensitive classification with unknown costs. If $-f$4 is the normal vector of the hyperplane defining the level set $-f$5, then

$-f$6

where $-f$7 (Parambath et al., 2015). Algorithmically, the procedure is an outer search over discretized cost vectors, with an inner cost-sensitive learner, followed by empirical selection using the target measure; the source also states that thresholding classifier scores on held-out data improves empirical $-f$8-measures in practice (Parambath et al., 2015).

The paper further interprets this reduction through the weighted-sum method of multi-objective optimization. Error profiles define multiple objectives, and the relevant cost-sensitive classifier minimizes a weighted sum of those error components. The analysis is described as valid on any dataset and any class of classifiers, and explicitly non-asymptotic (Parambath et al., 2015). Within machine learning, pseudolinearity therefore serves as the structural property that turns non-decomposable ratio metrics into exact linear optimization over error-profile space.

5. Domain-specific terminological extensions in discrete geometry

In discrete and topological graph theory, “pseudolinear” usually does not refer to scalar objective functions. Instead, it describes drawings whose edges extend to an arrangement of pseudolines. A drawing of $-f$9 is pseudolinear if there is an arrangement $F: U \subseteq \mathbb{R}^d \to \mathbb{R}$0 of pseudolines such that each edge of $F: U \subseteq \mathbb{R}^d \to \mathbb{R}$1 is contained in one of the pseudolines, and each pseudoline contains a unique edge (Arroyo et al., 2015). A related formulation says that a drawing of a graph in the plane is pseudolinear if the edges can be extended to doubly-infinite curves forming an arrangement of pseudolines, with every pair of curves crossing precisely once (Aichholzer et al., 2019).

A major structural theorem in this area states: $F: U \subseteq \mathbb{R}^d \to \mathbb{R}$2 The same paper gives an elementary, algorithmic, and self-contained proof of Levi’s Enlargement Lemma for pseudoline arrangements and proves that any face-convex drawing, hence any pseudolinear drawing, of $F: U \subseteq \mathbb{R}^d \to \mathbb{R}$3 has at least $F: U \subseteq \mathbb{R}^d \to \mathbb{R}$4 empty triangles (Arroyo et al., 2015). For general convex drawings the lower bound becomes

$F: U \subseteq \mathbb{R}^d \to \mathbb{R}$5

The crossing-number literature uses the same adjective in a closely related topological sense. The pseudolinear crossing number $F: U \subseteq \mathbb{R}^d \to \mathbb{R}$6 is the minimum number of crossings over all pseudolinear drawings, and it satisfies

$F: U \subseteq \mathbb{R}^d \to \mathbb{R}$7

For complete graphs, the asymptotic pseudolinear crossing constant is

$F: U \subseteq \mathbb{R}^d \to \mathbb{R}$8

and the cited 2019 work reports the upper bound

$F: U \subseteq \mathbb{R}^d \to \mathbb{R}$9

(Aichholzer et al., 2019). A later exact result shows that the 3-symmetric rectilinear and 3-symmetric pseudolinear crossing numbers of $U$0 coincide: $U$1 This is established via allowable sequences and a detailed analysis of 3-symmetric combinatorial configurations (Martínez et al., 14 Jan 2026).

These geometric usages are terminologically linked to pseudoline arrangements rather than to generalized convexity. A plausible implication is that the shared adjective emphasizes linear-like combinatorial behavior without implying a common analytic definition.

6. Coding-theoretic and field-theoretic usages

The term also appears in coding theory. The adversarial wiretap-channel paper studies pseudolinear codes, a family of non-linear codes with efficient encoders and succinct representations. For blocklength $U$2, rate $U$3, and parameter $U$4, an $U$5-pseudolinear code is built by first mapping a message $U$6 to a syndrome-like vector $U$7, taken as the $U$8-th column of a parity-check matrix $U$9, and then linearly mapping that vector to a codeword through

$$\forall t \in \mathrm{im}(F),~\exists ~a(t) \in \mathbb{R}^d,~b(t) \in \mathbb{R},$$0

The overall encoder is thus

$$\forall t \in \mathrm{im}(F),~\exists ~a(t) \in \mathbb{R}^d,~b(t) \in \mathbb{R},$$1

(Ruzomberka et al., 2023).

The cited paper states that, when $$\forall t \in \mathrm{im}(F),~\exists ~a(t) \in \mathbb{R}^d,~b(t) \in \mathbb{R},$$2 is random and $$\forall t \in \mathrm{im}(F),~\exists ~a(t) \in \mathbb{R}^d,~b(t) \in \mathbb{R},$$3 is fixed, the codewords of a pseudolinear code are uniformly distributed and $$\forall t \in \mathrm{im}(F),~\exists ~a(t) \in \mathbb{R}^d,~b(t) \in \mathbb{R},$$4-wise independent. It further proves that random pseudolinear codes can achieve rates up to the binary symmetric channel capacity

$$\forall t \in \mathrm{im}(F),~\exists ~a(t) \in \mathbb{R}^d,~b(t) \in \mathbb{R},$$5

for any $$\forall t \in \mathrm{im}(F),~\exists ~a(t) \in \mathbb{R}^d,~b(t) \in \mathbb{R},$$6 in the less noisy region

$$\forall t \in \mathrm{im}(F),~\exists ~a(t) \in \mathbb{R}^d,~b(t) \in \mathbb{R},$$7

and calls them the first known optimal-rate binary code family for the less noisy AWTC that admit efficient encoders (Ruzomberka et al., 2023). Here, “pseudolinear” denotes a specific nonlinear code construction rather than a generalized-convexity property of a scalar function.

In spin-2 field theory, the 2014 paper uses “pseudolinear functions” to denote nonlinear functionals of the metric perturbation $$\forall t \in \mathrm{im}(F),~\exists ~a(t) \in \mathbb{R}^d,~b(t) \in \mathbb{R},$$8 that remain invariant under linearized gauge symmetries. The central example is the unique cubic pseudolinear derivative term in $$\forall t \in \mathrm{im}(F),~\exists ~a(t) \in \mathbb{R}^d,~b(t) \in \mathbb{R},$$9, written as

$F(e) \geq t \iff a(t)^\top e + b(t) \leq 0,$0

Up to cubic order, the most general Lorentz-invariant two-derivative interaction with at most five propagating degrees of freedom is a linear combination of terms from the Einstein-Hilbert expansion and this cubic pseudolinear derivative term, together with compatible dRGT mass-term expansions (Gao, 2014). The same source also states that no viable ghost-free nonlinear completion is known for the pseudolinear derivative interaction.

Across these latter domains, the label “pseudolinear” is therefore strongly context-dependent. In optimization and learning it denotes a precise generalized-convexity class; in tropical algebra it names a max-plus objective form; in discrete geometry it refers to realizability by pseudoline arrangements; in coding it denotes a structured nonlinear encoder; and in field theory it labels nonlinear interaction terms that preserve linearized gauge invariance (Parsons et al., 2020, Arroyo et al., 2015, Ruzomberka et al., 2023, Gao, 2014).