Semi-Strictly Quasilinear Functions

• Semi-strictly quasilinear functions are real-valued functions on convex sets defined by being both semistrictly quasiconvex and semistrictly quasiconcave, ensuring strict interpolation.
• They satisfy a derivative-free segment interpolation identity where interior values are strict convex combinations of the endpoint values when these differ.
• Under additional differentiability or Clarke-Rockafellar conditions, semi-strict quasilinearity upgrades to pseudolinearity, aligning geometric properties with first-order criteria.

Semi-strictly quasilinear functions, in the sense developed in "Characterizations of Pseudolinear and Semi-strictly Quasilinear Functions" (Ivanov, 2 Sep 2025), are real-valued functions on convex sets that are simultaneously semistrictly quasiconvex and semistrictly quasiconcave. In the Banach-space setting of that work, the class is characterized by a derivative-free segment interpolation identity: values at interior points of any segment are strict convex combinations of endpoint values, with a reweighted coefficient depending on the segment and the interpolation parameter. The same paper places semi-strict quasilinearity strictly below pseudolinearity in general, and identifies the exact first-order and Clarke-Rockafellar conditions under which the two notions coincide.

1. Definition, ambient setting, and regularity hypotheses

The basic setting is a convex set $S$ in a Banach space $E$, or a convex subset $S$ of an open convex set $\Gamma\subset E$ (Ivanov, 2 Sep 2025). For the semi-strict notions, the function is defined on a convex set.

A function $f:S\to\mathbb{R}$ is semistrictly quasiconvex iff for all $x,y\in S$ and all $\lambda\in(0,1)$,

$$f(y)<f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)<f(x).$$

It is semistrictly quasiconcave iff $-f$ is semistrictly quasiconvex; equivalently,

$$f(y)>f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)>f(x).$$

Accordingly,

$E$0

The paper develops several theorem regimes under different regularity assumptions. For the basic geometric characterization of semistrict quasilinearity, $E$1 is assumed continuous on a convex set $E$2. For the differentiable criterion implying pseudolinearity, $E$3 is assumed Fréchet differentiable on an open set $E$4, and semistrictly quasiconvex and semistrictly quasiconcave on $E$5. For the nonsmooth criterion, $E$6 is assumed continuous on $E$7, semistrictly quasiconvex and semistrictly quasiconcave on $E$8, with nonempty Clarke-Rockafellar subdifferentials

$E$9

A corollary specializes this to locally Lipschitz functions on an open convex set $S$0, where $S$1 is automatic.

These hypotheses delineate the paper’s architecture. Continuity suffices for the central derivative-free characterization, while differentiability or nonsmooth first-order structure is required only for the transition from semistrict quasilinearity to pseudolinearity.

2. Derivative-free segment characterization

The principal theorem for semi-strictly quasilinear functions is a complete derivative-free characterization on convex subsets of Banach spaces (Ivanov, 2 Sep 2025). If $S$2 is a continuous function defined on a convex set $S$3, then the following are equivalent:

1. $S$4 is semistrictly quasiconvex and semistrictly quasiconcave;
2. for all $S$5 and all $S$6, there exists $S$7 such that

$S$8

and

$S$9

The identity states that every value of $\Gamma\subset E$0 on the segment $\Gamma\subset E$1 is a strict convex combination of $\Gamma\subset E$2 and $\Gamma\subset E$3, but with coefficient $\Gamma\subset E$4 rather than the geometric parameter $\Gamma\subset E$5. Because

$\Gamma\subset E$6

the interior value lies strictly between the endpoint values whenever $\Gamma\subset E$7.

A key intermediate simplification is the lemma that $\Gamma\subset E$8 is both semistrictly quasiconvex and semistrictly quasiconcave iff

$\Gamma\subset E$9

This is the direct geometric content of semi-strict quasilinearity: interior segment values are strictly ordered between endpoint values whenever the endpoint values differ.

Under additional regularity, equality of endpoint values propagates along the segment. The paper invokes quasiconvexity and quasiconcavity, obtained from semistrict quasiconvexity and semistrict quasiconcavity via standard results, to conclude that if $f:S\to\mathbb{R}$0, then

$f:S\to\mathbb{R}$1

This complements the strict-between-values description for unequal endpoints and yields a complete segmentwise geometric picture.

3. Position relative to pseudolinear functions

The same paper frames semi-strict quasilinearity as the exact geometric class behind a derivative-free interpolation formula that is only partially sufficient for pseudolinearity (Ivanov, 2 Sep 2025). In the nonsmooth framework, pseudoconvexity is defined by

$f:S\to\mathbb{R}$2

pseudoconcavity means that $f:S\to\mathbb{R}$3 is pseudoconvex, and pseudolinearity means both hold.

For pseudolinear functions, the paper first proves a necessary derivative-free condition: if $f:S\to\mathbb{R}$4 is locally Lipschitz and pseudolinear on a convex set $f:S\to\mathbb{R}$5, then for all $f:S\to\mathbb{R}$6 and $f:S\to\mathbb{R}$7, there exists $f:S\to\mathbb{R}$8 such that

$f:S\to\mathbb{R}$9

with

$x,y\in S$0

equivalently,

$x,y\in S$1

The similarity with semi-strict quasilinearity is immediate: both classes satisfy the same segment interpolation identity. The crucial difference lies in the strictness and in an infinitesimal compatibility requirement. For semistrictly quasilinear functions, continuity and the strict interior condition

$x,y\in S$2

already give a complete characterization. For pseudolinearity, the derivative-free equivalence additionally requires that $x,y\in S$3 be continuously differentiable and that the limit

$x,y\in S$4

exist and be strictly positive.

This makes the class relation precise: $x,y\in S$5 The converse fails in general. The paper identifies the missing ingredient as a first-order consistency condition: semistrict quasilinearity controls the geometry of segment values, whereas pseudolinearity additionally constrains how directional or subdifferential information aligns with value comparisons.

A recurrent misconception is therefore that semistrict quasilinearity is merely a reformulation of pseudolinearity. The paper’s results show otherwise: semistrict quasilinearity is the exact class captured by strict segment-value interpolation, but pseudolinearity requires more than that.

4. First-order and nonsmooth criteria for upgrading to pseudolinearity

The main transition result in the differentiable setting is an exact criterion (Ivanov, 2 Sep 2025). Let $x,y\in S$6 be a convex set in a Banach space $x,y\in S$7, and let $x,y\in S$8 be Fréchet differentiable on an open set $x,y\in S$9. Assume $\lambda\in(0,1)$0 is semistrictly quasiconvex and semistrictly quasiconcave on $\lambda\in(0,1)$1. Then $\lambda\in(0,1)$2 is pseudolinear on $\lambda\in(0,1)$3 if and only if

$\lambda\in(0,1)$4

The proof mechanism isolates the obstruction precisely. If $\lambda\in(0,1)$5, semistrict quasiconvexity implies

$\lambda\in(0,1)$6

hence differentiability gives $\lambda\in(0,1)$7. If equality were possible, condition (5) would force $\lambda\in(0,1)$8, contradicting $\lambda\in(0,1)$9. Therefore

$$f(y)<f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)<f(x).$$0

which is pseudoconvexity. Applying the same argument to $$f(y)<f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)<f(x).$$1 yields pseudoconcavity. Thus semistrict quasilinearity plus the zero-derivative/equal-value implication is exactly equivalent to pseudolinearity in the Fréchet differentiable regime.

The nonsmooth analogue is formulated through the Clarke generalized directional derivative

$$f(y)<f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)<f(x).$$2

and the Clarke subdifferential

$$f(y)<f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)<f(x).$$3

Suppose $$f(y)<f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)<f(x).$$4 is continuous, semistrictly quasiconvex and semistrictly quasiconcave on $$f(y)<f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)<f(x).$$5, and

$$f(y)<f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)<f(x).$$6

Then $$f(y)<f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)<f(x).$$7 is pseudolinear with respect to the Clarke-Rockafellar subdifferential if and only if both of the following hold: $$f(y)<f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)<f(x).$$8

$$f(y)<f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)<f(x).$$9

Together, these implications force equality whenever both apply.

A particularly clean corollary holds on open convex domains: if $-f$0 is open and convex, and $-f$1 is locally Lipschitz and semistrictly quasilinear on $-f$2, then $-f$3 is pseudolinear with respect to Clarke’s generalized directional derivative iff

$-f$4

These theorems show that semistrict quasilinearity becomes pseudolinearity precisely when first-order null signals are compatible only with value equality.

5. Auxiliary results, class inclusions, and canonical example

Several intermediate statements organize the theory (Ivanov, 2 Sep 2025). First, if $-f$5 is lower semicontinuous pseudoconvex on a convex domain, then $-f$6 is quasiconvex. Second, the Karamardian proposition states that if $-f$7 is radially lower semicontinuous and semistrictly quasiconvex, then $-f$8 is quasiconvex. Applied also to $-f$9, this yields quasiconcavity from semistrict quasiconcavity under the analogous regularity. These facts explain why equal endpoint values force constant values along the connecting segment.

The paper also proves that if $$f(y)>f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)>f(x).$$0 is continuous and pseudolinear on $$f(y)>f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)>f(x).$$1, then $$f(y)>f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)>f(x).$$2 is semistrictly quasiconvex and semistrictly quasiconcave. Under the stated continuity assumptions, the inclusion

$$f(y)>f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)>f(x).$$3

is therefore explicit.

For illustration, the paper gives the standard linear-fractional example

$$f(y)>f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)>f(x).$$4

For this function,

$$f(y)>f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)>f(x).$$5

and

$$f(y)>f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)>f(x).$$6

so that

$$f(y)>f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)>f(x).$$7

This exemplifies the derivative-free pseudolinear condition.

By contrast, the paper does not provide a fully explicit worked-out counterexample of a semistrictly quasilinear function that is not pseudolinear. The separation is instead conceptual. Semistrict quasilinearity is characterized by the strict interpolation condition

$$f(y)>f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)>f(x).$$8

whereas pseudolinearity requires, in addition, either the differentiable condition

$$f(y)>f(x)\quad\Rightarrow\quad f\bigl(x+\lambda(y-x)\bigr)>f(x).$$9

or the nonsmooth subdifferential conditions (6) and (7). The absence of an explicit counterexample does not weaken the structural distinction; it simply means that the paper’s contribution is classificatory rather than example-driven at that point.

6. Related frameworks and terminological boundaries

The phrase semi-strictly quasilinear is used directly in (Ivanov, 2 Sep 2025), but neighboring arXiv literatures deploy related vocabulary differently. That distinction matters for precise classification.

In "Functions with uniform level sets" (Weidner, 2016), the term is not used, but the scalarization family

$E$00

is analyzed through geometric criteria for quasiconvexity, strict quasiconvexity, concavity, and strict quasiconcavity. In that setting,

$E$01

and, under additional hypotheses,

$E$02

while strict quasiconcavity is tied to strict convexity of $E$03. This suggests a geometric route to semistrict quasilinear behavior for the special family $E$04, but that synthesis is not stated under this terminology.

In "Axiomatizations of quasi-Lovász extensions of pseudo-Boolean functions" (Couceiro et al., 2010), the phrase also does not occur. The paper studies functions of the form

$E$05

where $E$06 is a Lovász extension and $E$07 is a nondecreasing unary map with $E$08. The characterization there is

$E$09

This is structurally adjacent to quasilinearized aggregation, but it is not the segmentwise semi-strict quasilinearity of (Ivanov, 2 Sep 2025).

In "X-convexity and Applications of Quasi-X-Convex Functions" (Ali et al., 2022), the paper introduces semi-strictly quasi-$E$10-convex functions and defines quasi-$E$11-concavity by sign reversal, but it does not define quasilinear explicitly. A natural analogue of semi-strict $E$12-quasilinearity would require both semi-strictly quasi-$E$13-convexity and semi-strictly quasi-$E$14-concavity, yet that combined notion is not developed there.

In discrete convex analysis, "Note on Minimization of Quasi $E$15-convex Functions" (Murota et al., 2023) treats semi-strictly quasi $E$16-convex functions, an exchange-based discrete class. The paper explicitly notes that it is not about quasilinear functions in the standard continuous sense. The terminological overlap is therefore superficial.

Taken together, these adjacent literatures show that “semi-strictly quasilinear” is not a generic label for transformed aggregation, generalized segment convexity, or discrete quasi-convex exchange. In the precise sense established in (Ivanov, 2 Sep 2025), it denotes a class of functions on convex subsets of Banach spaces whose defining feature is strict segmentwise interpolation between endpoint values, and whose upgrade to pseudolinearity is governed exactly by derivative or subdifferential orthogonality conditions.