Relative Regularity Conditions

• Relative regularity conditions are hypotheses that assess global regularity via auxiliary structures such as boundary configurations, affine slices, and asymptotic cones.
• They are applied across fields like optimization, algebraic geometry, and inverse problems using techniques like one-dimensional slicing, spectral tests, and variational analysis.
• These conditions replace hard global criteria with local, testable properties, enabling sharper error bounds and robustness against perturbations.

Searching arXiv for recent and relevant papers on “relative regularity conditions” and closely related formulations. Searching for papers on relative regularity in optimization, algebraic geometry, inverse problems, and parametric matrix analysis. Relative regularity conditions are regularity hypotheses formulated not absolutely on an entire object but relative to auxiliary structure such as parameter intervals, affine planes, boundary divisors, level sets, active constraints, or geometric scales. In current arXiv literature, the term appears in interval parametric matrix analysis, algebraic geometry, inverse problems, vector optimization, nonlinear PDE, geometric set regularity, Banach function algebras, and variational analysis (Popova, 2021, Gryszka et al., 2021, Gerth et al., 2019, Liu, 7 Aug 2025). Across these settings, the recurring pattern is that global regularity is detected or characterized through lower-dimensional slices, boundary configurations, perturbation directions, or asymptotic regimes rather than by direct global inspection.

1. Main meanings of the term

The phrase “relative regularity conditions” is not used with a single universal definition. Instead, it denotes several structurally analogous ideas.

Area	Relative object	Representative condition
Interval parametric matrices	parameter box $\mathbf p$, extremal sign patterns	regularity via one-parameter slices, determinants, and real spectral radii
Algebraic geometry	affine planes $L\subset \mathbb K^n$	$f|_L$ regular for every affine plane implies $f$ regular
Inverse problems	Tikhonov functional $T_\alpha$	KL inequality equivalent to variational and rate conditions
Polynomial vector optimization	$S_\infty$ with $$(K_{\bar x})_\infty\subseteq S_\infty\subseteq K_\infty$$	relative weak, strong, and $I$-$\mathbf R^q_+$-zero-regularity
Set-valued mappings	parameter subset $\Omega$, active constraints	$L\subset \mathbb K^n$0-regularity under RCRCQ or RCPLD

In interval matrix analysis, the conditions are explicitly relative to the radii $L\subset \mathbb K^n$1 of the parameter intervals and to boundary sign configurations (Popova, 2021). In algebraic geometry, they are local-to-global conditions relative to lower-dimensional affine subspaces (Gryszka et al., 2021). In optimization at infinity, they are relative to sublevel asymptotic cones and intermediate cones $L\subset \mathbb K^n$2 rather than to the whole recession cone (Liu, 7 Aug 2025). In algebraic $L\subset \mathbb K^n$3-module theory, they are relative to compactification boundaries and divisorial valuations (Cailotto et al., 2015). This suggests a common meta-principle: regularity is often most tractable when rephrased in terms of a distinguished ambient structure that captures where irregularity can first appear.

2. Parametric uncertainty, error bounds, and perturbation domains

For interval parametric matrices, one starts with affine dependence

$L\subset \mathbb K^n$4

and asks for regularity of the family $L\subset \mathbb K^n$5, meaning $L\subset \mathbb K^n$6 for all $L\subset \mathbb K^n$7. The central result gives equivalent necessary and sufficient conditions in terms of one-parameter families, bounded parametric solution sets, univariate determinants, and real spectral radii (Popova, 2021). For each coordinate $L\subset \mathbb K^n$8 and sign vector $L\subset \mathbb K^n$9, one freezes $f|_L$0 parameters at signed extreme values and studies the one-dimensional slice $f|_L$1. The family is regular if and only if every such slice is regular, equivalently if the corresponding univariate determinant has no real roots in $f|_L$2, equivalently if a real spectral-radius inequality $f|_L$3 holds for all admissible extremal perturbations (Popova, 2021).

This framework is explicitly relative in three senses. First, the conditions depend on the interval radii $f|_L$4. Second, they localize singularity detection to boundary configurations of the parameter box. Third, the regularity radius

$f|_L$5

measures how far the uncertainty box can be scaled before singularity occurs (Popova, 2021). The paper also emphasizes sharpness: the maximal absolute value among real eigenvalues $f|_L$6 cannot be replaced by the full spectral radius $f|_L$7 in a necessary-and-sufficient characterization.

In variational analysis, relative regularity appears as $f|_L$8-regularity of set-valued mappings. For polyhedral moving sets

$f|_L$9

$f$0-regularity is the local error bound

$f$1

and it implies the Lipschitz-like property of the multifunction $f$2 (Bednarczuk et al., 2018). The paper proves that the Relaxed Constant Rank Constraint Qualification, together with $f$3, yields bounded KKT multipliers for projection problems, hence $f$4-regularity, hence Lipschitz-likeness (Bednarczuk et al., 2018). It also corrects an earlier incorrect multiplier argument of Minchenko and Stakhovsky.

For nonlinear parametric constraint systems

$f$5

$f$6-regularity is formulated relative to a subset $f$7: $f$8

for nearby $f$9 with $T_\alpha$0 (Mehlitz et al., 2020). The paper shows that the relaxed constant positive linear dependence constraint qualification is weaker than both MFCQ and RCRCQ, and under convexity/local boundedness or under inner semicontinuity it guarantees $T_\alpha$1-regularity of $T_\alpha$2 with respect to $T_\alpha$3 (Mehlitz et al., 2020). This is a relative regularity condition both because it is imposed with respect to $T_\alpha$4 and because it monitors only specific positive-linear dependence patterns among active gradients.

A further weakening is “almost” regularity for mappings in incomplete spaces and with non-closed graphs. Here the paper defines almost openness with a linear rate, almost metric regularity, almost metric subregularity, and almost semiregularity through limit inequalities involving $T_\alpha$5 (Cibulka, 2023). An approximate version of Ekeland’s variational principle for functions that are not necessarily lower semicontinuous then yields an analogue of Ioffe’s criterion and perturbation stability under additive single-valued and set-valued perturbations (Cibulka, 2023).

3. Local-to-global algebraic regularity and regular singularities

A clean local-to-global relative regularity theorem appears in algebraic geometry. If $T_\alpha$6 is an uncountable field of characteristic zero and $T_\alpha$7 is a function such that the restriction $T_\alpha$8 is regular for every affine plane $T_\alpha$9, then $S_\infty$0 is regular (Gryszka et al., 2021). Here “regular” means regular in the algebraic-geometric sense: locally a rational function $S_\infty$1 on a nonempty Zariski open neighborhood with denominator nonvanishing. The paper shows that regularity on all affine planes is sufficient, while regularity on affine lines is not: the standard example

$S_\infty$2

is regular on every affine line but not continuous, hence not regular, at the origin (Gryszka et al., 2021). The theorem is also sharp in its field assumptions: over countable fields, regularity on affine hyperplanes need not imply global regularity.

For algebraic connections and algebraic $S_\infty$3-modules, regularity is relative to a boundary divisor in a compactification. If $S_\infty$4 is an open dense immersion into a smooth proper variety $S_\infty$5 and $S_\infty$6 is a strict normal crossings divisor, then a coherent integrable connection $S_\infty$7 on $S_\infty$8 is regular if and only if the direct image $S_\infty$9 is a regular holonomic $$(K_{\bar x})_\infty\subseteq S_\infty\subseteq K_\infty$$0-module (Cailotto et al., 2015). On the connection side, regularity is defined by a lattice stable under a logarithmic derivation $$(K_{\bar x})_\infty\subseteq S_\infty\subseteq K_\infty$$1 at each divisorial valuation. On the $$(K_{\bar x})_\infty\subseteq S_\infty\subseteq K_\infty$$2-module side, regularity is Kashiwara-regularity of a holonomic module. The paper also gives equivalent formulations in terms of coherent $$(K_{\bar x})_\infty\subseteq S_\infty\subseteq K_\infty$$3-extensions, where $$(K_{\bar x})_\infty\subseteq S_\infty\subseteq K_\infty$$4 is generated by derivations tangent to the divisor $$(K_{\bar x})_\infty\subseteq S_\infty\subseteq K_\infty$$5 (Cailotto et al., 2015). In this setting, “relative regularity” is literally regularity relative to the boundary.

Representation-theoretic $$(K_{\bar x})_\infty\subseteq S_\infty\subseteq K_\infty$$6-modules generated by relative characters provide another boundary-relative usage. For a spherical homogeneous variety $$(K_{\bar x})_\infty\subseteq S_\infty\subseteq K_\infty$$7 and a spherical subgroup $$(K_{\bar x})_\infty\subseteq S_\infty\subseteq K_\infty$$8, a $$(K_{\bar x})_\infty\subseteq S_\infty\subseteq K_\infty$$9-admissible $I$0-module is finitely generated, locally $I$1-finite, and $I$2-monodromic for a reductive character $I$3 (Li, 2019). Under absolute sphericity, such modules are regular holonomic; this includes relative characters attached to two spherical subgroups, localization of Harish-Chandra modules, and generalized matrix coefficients when $I$4 is maximal compact (Li, 2019). The reductivity of the twisting character is essential: non-reductive characters, such as Whittaker characters, may yield holonomic but irregular $I$5-modules.

4. Kurdyka–Łojasiewicz inequalities as regularity conditions

In inverse problems, the KL inequality is used as a unifying regularity condition for Tikhonov regularization with linear operators in Banach spaces. For

$I$6

the paper proves equivalence between several regularity assumptions that are often treated separately: J-rate, T-rate, variational inequality, distance-function bound, dual T-rate, and a KL inequality for the noise-free Tikhonov functional (Gerth et al., 2019). In particular, Theorem 4.1 identifies the KL slope inequality

$I$7

with the T-rate condition

$I$8

Consequently, the KL framework is not stronger or weaker than these other conditions in the convex setting considered there; it is equivalent to them (Gerth et al., 2019).

This relative viewpoint is explicit in the paper’s interpretation. Source conditions, variational inequalities, and conditional stability estimates become different representations of the same underlying geometric property of the Tikhonov functional near its minimizer. For example, under the conditional stability estimate

$I$9

the paper derives a KL inequality with $\mathbf R^q_+$0 and the rate $\mathbf R^q_+$1 (Gerth et al., 2019). For classical Hilbert-space Tikhonov regularization under the spectral source condition

$\mathbf R^q_+$2

the same framework recovers the standard rate $\mathbf R^q_+$3 and encodes it through an appropriate desingularizing function $\mathbf R^q_+$4 (Gerth et al., 2019). A plausible implication is that “relative regularity” here means comparability of regularity assumptions through the index functions $\mathbf R^q_+$5 or $\mathbf R^q_+$6 rather than through a fixed hierarchy of separate smoothness classes.

5. Relative regularity at infinity in polynomial vector optimization

For polynomial vector optimization

$\mathbf R^q_+$7

with $\mathbf R^q_+$8 nonempty and closed and $\mathbf R^q_+$9 a vector polynomial, the paper defines relative regularity through an intermediate asymptotic cone $\Omega$0 satisfying

$\Omega$1

Three notions are introduced: relative $\Omega$2-$\Omega$3-zero-regularity, relative weak regularity, and relative strong regularity (Liu, 7 Aug 2025). Relative $\Omega$4-$\Omega$5-zero-regularity requires existence of $\Omega$6 such that the scalarization $\Omega$7 is regular on $\Omega$8, equivalently $\Omega$9 is bounded. Relative weak and strong regularity require boundedness of $L\subset \mathbb K^n$00 and $L\subset \mathbb K^n$01, respectively (Liu, 7 Aug 2025).

These conditions are weaker than classical regularity on the whole recession cone. The scalar example

$L\subset \mathbb K^n$02

is non-regular on $L\subset \mathbb K^n$03 because $L\subset \mathbb K^n$04 is unbounded, but it is relatively regular with a suitable $L\subset \mathbb K^n$05 for which $L\subset \mathbb K^n$06 (Liu, 7 Aug 2025). This is the paper’s main geometric point: only those recession directions compatible with a chosen sublevel set should control existence.

Under $L\subset \mathbb K^n$07-section-boundedness from below, the paper proves equivalence among $L\subset \mathbb K^n$08-properness, relative $L\subset \mathbb K^n$09-$L\subset \mathbb K^n$10-zero-regularity, relative strong regularity, relative weak regularity, the $L\subset \mathbb K^n$11-Palais–Smale condition, the weak $L\subset \mathbb K^n$12-Palais–Smale condition, and $L\subset \mathbb K^n$13-M-tameness (Liu, 7 Aug 2025). It then derives nonemptiness of the Pareto efficient solution set under relative regularity, and even under relative non-regularity if every nonzero recession-optimal direction yields descent in the original problem. As a by-product, it obtains Frank–Wolfe type theorems for non-convex polynomial vector optimization without convexity and compactness assumptions (Liu, 7 Aug 2025). The paper also studies local openness and genericity: relative $L\subset \mathbb K^n$14-$L\subset \mathbb K^n$15-zero-regularity and relative strong regularity define open sets in coefficient space, whereas relative weak regularity need not be open (Liu, 7 Aug 2025).

6. Intrinsic PDE conditions, boundary regularity, and geometric scale

For non-autonomous quasilinear elliptic equations

$L\subset \mathbb K^n$16

and variational integrals

$L\subset \mathbb K^n$17

the paper on non-Uhlenbeck regularity replaces classical structural assumptions tied to special growth functions by intrinsic conditions formulated directly in terms of $L\subset \mathbb K^n$18 or $L\subset \mathbb K^n$19 (Hästö et al., 2021). The core hypotheses are quasi-isotropic ellipticity and the weak vanishing $L\subset \mathbb K^n$20 condition, (wVA1), imposed on $L\subset \mathbb K^n$21 or on $L\subset \mathbb K^n$22. These conditions are relative because continuity is measured relative to the size of the operator itself, not relative to a prescribed model such as $L\subset \mathbb K^n$23, $L\subset \mathbb K^n$24, or $L\subset \mathbb K^n$25 (Hästö et al., 2021). Under these hypotheses, the paper proves local $L\subset \mathbb K^n$26-regularity of weak solutions or local minimizers for every $L\subset \mathbb K^n$27, and local $L\subset \mathbb K^n$28-regularity for some $L\subset \mathbb K^n$29 under an additional power-type modulus decay (Hästö et al., 2021).

For second-order boundary value problems on manifolds with boundary and bounded geometry, regularity is again made relative to a family of local models. The paper shows that the regularity property of a boundary value problem $L\subset \mathbb K^n$30 is equivalent to uniform regularity of the associated family $L\subset \mathbb K^n$31 in local coordinates, introduces a uniform Shapiro–Lopatinski regularity condition characterizing the regular problems, and proves that natural Robin boundary conditions satisfy this uniform condition provided the operator satisfies the strong Legendre condition (Große et al., 2017). It also introduces a uniform Agmon condition and proves that it is equivalent to coerciveness (Große et al., 2017). This is a boundary-relative regularity theory in which classical local conditions are upgraded to uniform conditions over a bounded-geometry atlas.

Geometric set regularity offers a scale-relative formulation. An open set $L\subset \mathbb K^n$32 is $L\subset \mathbb K^n$33-regular if both $L\subset \mathbb K^n$34 and $L\subset \mathbb K^n$35 are connected unions of balls of radius at least $L\subset \mathbb K^n$36 (Duarte et al., 2014). The main characterization says that $L\subset \mathbb K^n$37 is $L\subset \mathbb K^n$38-regular if and only if there exists a normal vector field $L\subset \mathbb K^n$39 with $L\subset \mathbb K^n$40 and $L\subset \mathbb K^n$41, together with the global intrinsic-distance condition

$L\subset \mathbb K^n$42

for all $L\subset \mathbb K^n$43 (Duarte et al., 2014). This is explicitly a regularity condition relative to the scale $L\subset \mathbb K^n$44: it encodes rolling-ball geometry, bounded curvature at scale $L\subset \mathbb K^n$45, and a no-neck condition excluding features thinner than $L\subset \mathbb K^n$46.

7. Function-algebraic transfer, bounded relative units, and sharpness

In vector-valued function algebras, regularity conditions are studied relatively between three associated algebras: a coefficient algebra $L\subset \mathbb K^n$47, an $L\subset \mathbb K^n$48-valued function algebra $L\subset \mathbb K^n$49, and the scalar-valued algebra $L\subset \mathbb K^n$50 (Barqi et al., 2023). The paper analyzes regularity, strong regularity, bounded relative units, Ditkin’s condition, and strong Ditkin’s condition under the product identification

$L\subset \mathbb K^n$51

The easy direction is inheritance from $L\subset \mathbb K^n$52 to $L\subset \mathbb K^n$53 and $L\subset \mathbb K^n$54; the substantial results are converse transfer statements. For example, if $L\subset \mathbb K^n$55 and $L\subset \mathbb K^n$56 both admit bounded relative units, then so does $L\subset \mathbb K^n$57; if $L\subset \mathbb K^n$58 and $L\subset \mathbb K^n$59 are strong Ditkin algebras, then $L\subset \mathbb K^n$60 is strong Ditkin under suitable structural assumptions (Barqi et al., 2023). Relative regularity here means that regularity of the larger algebra is assembled from regularity in the coefficient and scalar directions.

Uniform algebras provide a pointwise version of the same theme. If a separable uniform algebra $L\subset \mathbb K^n$61 on its character space $L\subset \mathbb K^n$62 has bounded relative units at every point of a dense subset of $L\subset \mathbb K^n$63, then the subalgebra

$L\subset \mathbb K^n$64

is dense in $L\subset \mathbb K^n$65 (Heath et al., 2014). The proof is local: bounded relative units at a point $L\subset \mathbb K^n$66 allow one to flatten a function near $L\subset \mathbb K^n$67 while preserving it outside a prescribed neighborhood. The same paper constructs a separable, essential, regular uniform algebra on its character space $L\subset \mathbb K^n$68 such that every point of $L\subset \mathbb K^n$69 is a peak point, bounded relative units hold at every point of a dense open subset of $L\subset \mathbb K^n$70, yet the algebra is not weakly amenable (Heath et al., 2014). This sharpness result is representative of the wider topic: strong relative regularity on a large set need not imply a global cohomological property.

Several other sharpness phenomena recur across the literature. In algebraic geometry, affine planes are sufficient but affine lines are not (Gryszka et al., 2021). In interval matrix theory, global sufficient spectral-radius tests may fail although the family is regular, and exactness is recovered only after one-parameter reduction (Popova, 2021). In nonlinear constraint systems, RCPLD alone does not guarantee $L\subset \mathbb K^n$71-regularity without convexity in $L\subset \mathbb K^n$72 or inner semicontinuity (Mehlitz et al., 2020). In polynomial vector optimization, relative strong regularity implies relative weak regularity, but the openness and genericity behavior of the corresponding coefficient sets differ (Liu, 7 Aug 2025). These examples indicate that relative regularity conditions are typically sharp descriptions of where global regularity can be recovered from restricted data, but they are not interchangeable across domains.

In the aggregate, the literature treats relative regularity conditions as mechanisms for replacing a hard global regularity question by a family of lower-dimensional, boundary-relative, asymptotic, or perturbation-relative tests. The concrete realizations differ, but the structural theme is stable: irregularity is controlled by identifying the relevant slice, boundary, cone, or active subsystem and formulating regularity exactly there.