Kurdyka–Łojasiewicz Properties

• Kurdyka–Łojasiewicz properties are a framework defining explicit desingularizing inequalities that link subgradient behavior with function value gaps near critical points.
• They offer precise convergence and error bounds in optimization algorithms by quantifying the geometric and variational structure of real-valued functions.
• Extensions to semialgebraic, Nash, and non-smooth settings highlight their application in regularization, singularity theory, and determining algorithmic rates.

The Kurdyka-Łojasiewicz (KL) properties form a unifying framework to quantify the geometric and variational structure of real-valued functions, connecting subgradient behavior with value gaps near critical points via explicit desingularizing inequalities. The classical inequalities, originating from the study of real-analytic functions by Łojasiewicz, have been extended, refined, and generalized in the non-smooth, semi-algebraic, and variational analytic settings, and are now ubiquitous in the analysis of convergence rates for optimization algorithms, error bounds, and stability theorems across modern mathematical optimization and singularity theory.

1. Foundational Definitions and Classical Inequalities

Let $f:\mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$ be a proper, lower semicontinuous function, and let $\bar{x}\in\operatorname{dom} f$ be a critical point ($0\in\partial f(\bar{x})$), where $\partial f$ denotes the limiting (Mordukhovich) subdifferential. The function $f$ is said to satisfy the Kurdyka-Łojasiewicz (KL) property at $\bar{x}$ if there exist a neighborhood $U\ni\bar{x}$, a number $\eta>0$, and a desingularizing function $\varphi : [0,\eta) \to [0,\infty)$, continuous, concave, and $C^1$ on $(0,\eta)$, and strictly increasing with $\varphi(0)=0$, such that for all $x\in U$ with $0<f(x)-f(\bar{x})<\eta$,

$$\varphi'\bigl(f(x)-f(\bar{x})\bigr)\cdot\operatorname{dist}(0,\partial f(x)) \geq 1.$$

Frequently, the desingularizing function is chosen as $\varphi(s) = c s^{1-\theta}$ for some $c>0, \theta\in[0,1)$, giving the equivalent Hölder-type bound: $$\operatorname{dist}(0, \partial f(x)) \geq c (f(x)-f(\bar{x}))^{\theta}.$$ The smallest such $\theta$ is called the KL exponent, or Łojasiewicz exponent, at $\bar{x}$ (Li et al., 2023).

For functions smooth near $\bar{x}$, the inequality becomes the classical Łojasiewicz gradient inequality for real analytic functions: $$\|\nabla f(x)\| \geq C|f(x)|^{\theta}, \quad \text{for some } 0<\theta<1,$$ in a neighborhood of $\bar{x}$ (Osińska-Ulrych et al., 2018, Cibotaru et al., 2021).

2. Explicit Exponent Bounds: Degrees and Algebraic Complexity

Nash Functions and Semialgebraic Case

For Nash functions (real analytic with semialgebraic graphs) on a compact semialgebraic set $X \subset \mathbb{R}^n$, if $f: X \to \mathbb{R}$ is a non-constant Nash function and there exists a polynomial $P(x, f(x)) \equiv 0$ on $X$ of degree $d$, effective explicit bounds on the KL exponent can be given (Osińska-Ulrych et al., 2018). Define: $$S(n,d) = 2(2d-1)^{3n+1}, \qquad R(n,d) = \max\{2d(2d-1), d(3d-2)^n\} + 1.$$

• In the general Nash case (without additional assumptions), the bound is

$|\nabla f(x)| \geq C |f(x)|^{1-1/S(n,d)}.$

• Under the transversality assumption ($\partial P/\partial y \neq 0$), the exponent improves to

$|\nabla f(x)| \geq C |f(x)|^{1-1/R(n,d)}.$

These bounds quantify the vanishing rate of the gradient near zeros of $f$ and govern the convergence rate for gradient descent (Osińska-Ulrych et al., 2018). For polynomial $f$ with isolated zeros, one recovers sharper previous bounds such as $\rho \leq 1 - 1/(d-1)^n$ or $\rho \leq 1 - 1/[d(3d-3)^{n-1}]$.

Polynomial and Semialgebraic Mappings

For polynomial maps $F: X\to\mathbb{R}^m$ on closed semialgebraic sets $X \subset \mathbb{R}^N$, Grzelakowski (Grzelakowski, 2021) refines the earlier bounds of Kurdyka-Spodzieja-Szlachcińska. Let $d:=\max\{\deg F, K(X)\}$, $r=r(X)$ the minimal maximal number of defining inequalities in a decomposition of $X$, and $m$ the codomain dimension. Then at $0 \in X$ and $F(0)=0$,

$\Loj_{0}(F|X) \leq d(6d-3)^{N+r+m-1}.$

Global in $X$, one obtains similar power-law lower bounds on $\|F(x)\|$ in terms of distance to the fiber and size at infinity. These exponents control uniform error bounds and the sharpness of gradient flows on semialgebraic sets (Grzelakowski, 2021).

3. Variational, Generalized, and Nonsmooth KL Properties

Generalized Concave KL and Exact Modulus

The classical smooth-concave desingularizing function may not be optimal, especially in non-smooth or composite settings. The generalized concave KL property relaxes to strictly increasing, right-continuous, concave functions (possibly non-$C^1$), yielding: $$\varphi_-'\big(f(x)-f(\bar{x})\big) \cdot \inf\{ \|u\| : u\in\partial f(x) \} \geq 1.$$ For any function with the generalized KL property, the exact modulus is constructed: $$\tilde{\varphi}(t) = \int_0^t h(s) ds, \qquad h(s) = \sup\left\{ \min \left\{ \|u\| : u \in \partial f(x) \right\}^{-1} : f(x) - f(\bar{x}) \geq s \right\}.$$ $\tilde{\varphi}$ is minimal among all concave desingularizers. This provides the sharpest quantitative control on the length of iterates of first-order methods (e.g., PALM) and enables calculus rules such as for sums, minima, and compositions, beyond functions with power-type moduli (Wang et al., 2020, Wang et al., 2021).

Variational Characterizations: Outer Limiting Subdifferential, Modulus, and Quadratic Growth

For a fixed exponent $\theta$, the KL property at $\bar{x}$ can be equivalently characterized using the outer limiting subdifferential of the auxiliary function $g(x) = (\max\{f(x) - f(\bar{x}), 0\})^{1-\theta}$: $$\partial^>g(\bar{x}) = \limsup_{x\to\bar{x}, f(x)>f(\bar{x})}(1-\theta)\partial f(x).$$ The function $f$ has KL property of exponent $\theta$ at $\bar{x}$ if and only if $0 \notin \partial^>g(\bar{x})$; the best possible modulus is given by the distance from zero to this set (Li et al., 2023). In prox-regular, twice epi-differentiable settings, quadratic growth of $f$ is equivalent to KL exponent $\frac{1}{2}$, with the modulus determined via the second subderivative.

4. Structural and Topological Implications

Classes of KL Functions

KL functions encompass (Cibotaru et al., 2021, Wu et al., 24 Nov 2025, Ahookhosh et al., 13 Nov 2025):

• Real-analytic and subanalytic functions (all such functions are KL).
• Semialgebraic and globally subanalytic (o-minimal) functions.
• Morse and Morse-Bott functions (via Morse lemma).
• Composite mappings with analyticity or definability in a tame structure.

Topological Consequences

The zero locus of a KL function has geometric and topological restrictions—admitting mapping cylinder neighborhoods and excluding pathological sets such as the Alexander horned sphere. The KL property guarantees that the complement of the zero locus harbors a tubular-like neighborhood, imposing constraints on singularity structure and local geometry (Cibotaru et al., 2021).

5. Algorithmic Rate Implications and Transfer Rules

KL exponents (or the generalized modulus) govern the convergence of a wide array of optimization algorithms, via the following rate relationships (Ahookhosh et al., 13 Nov 2025, Qian et al., 15 Apr 2025, Wu et al., 24 Nov 2025, Bento et al., 30 Jun 2024, Fatkhullin et al., 2022):

• Finite-step convergence: $\theta=0$.
• Linear convergence: $$0<\theta\leq\frac{1}{2} \implies \|x^k-\bar{x}\| = O(\rho^k)$$.
• Sublinear convergence: $$\frac{1}{2}<\theta<1 \implies \|x^k-\bar{x}\| = O(k^{-(1-\theta)/(2\theta-1)})$$.

Extensions to inf-projection and canonical transformations:

• Inf-projection: The KL exponent is preserved when marginalizing over auxiliary variables (e.g., in semidefinite-programming representable, Bregman envelope, or rank-constrained models) (Yu et al., 2019).
• Reparametrizations (e.g., square or Hadamard parametrization): The KL exponent of the transformed model can be expressed in terms of the exponent of the original problem, often as $\max\{\alpha,\frac{1}{2}\}$ or related weighted averages, depending on strict complementarity and error-bound properties (Ouyang et al., 1 Feb 2024, Ouyang, 11 Jun 2025).

6. Applications and Impact in Optimization, Regularization, and Singularity Theory

KL properties and exponents are central in (Gerth et al., 2019, Jiang et al., 2019, Dinh et al., 2015, Aktas et al., 30 Apr 2025):

• Optimization complexity: Explicit iteration bounds for stochastic and deterministic algorithms through the KL exponent or modulus, including generalized descent methods, GLL-type schemes, and decentralized nonconvex algorithms.
• Error bounds and regularization: KL inequalities are equivalent to and unify standard regularity conditions (variational inequalities, distance functions, source conditions) in inverse problems and Tikhonov regularization, determining optimal convergence rates (Gerth et al., 2019).
• Finite determinacy and singularity theory: The finiteness of the KL exponent gives explicit degrees of determinacy for Nash and analytic singularities, quantifying when jets or local perturbations are topologically indistinguishable (Osińska-Ulrych et al., 2018).
• Non-smooth spectral problems: KL inequalities, with explicit exponents from the D’Acunto-Kurdyka scheme, hold for the largest eigenvalue of real symmetric polynomial matrices, enabling global error guarantees and stability for semidefinite constraints (Dinh et al., 2015).

7. Open Directions, Extensions, and Limitations

Significant directions include sharpening exponent estimates to match the true Łojasiewicz exponent in concrete cases; extending the effective theory to other analytic categories (e.g., complex Nash functions); and developing calculus and transfer rules for the exact modulus in generalized concave settings (Osińska-Ulrych et al., 2018, Wang et al., 2021, Wang et al., 2020). Fundamental limitations are present in the estimation of exponents in non-isolated or non-analytic cases and in the universality of power-type moduli, which may fail for non-smooth composite constructions.

In summary, the Kurdyka-Łojasiewicz properties and exponents provide a quantitative geometric link between variational structure and algorithmic rate in nonconvex, nonsmooth, and algebraic settings—connecting algebraic-measure-theoretic complexity, algorithmic convergence, and topological constraints in a uniform analytic framework (Osińska-Ulrych et al., 2018, Cibotaru et al., 2021, Li et al., 2023, Grzelakowski, 2021, Ouyang, 11 Jun 2025, Ouyang et al., 1 Feb 2024, Wu et al., 24 Nov 2025, Wang et al., 2020, Wang et al., 2021, Dinh et al., 2015, Yu et al., 2019).