Logarithmically Homogeneous Self-Concordant Barrier

• Logarithmically homogeneous self-concordant barriers are key functions in convex optimization that encode cone structures via barrier growth, logarithmic homogeneity, and self-concordance.
• They leverage affine invariance, Dikin ellipsoid-based curvature control, and scaling identities to facilitate robust step-size selection and iteration complexity bounds in algorithms.
• Their framework underpins efficient interior-point and first-order methods across applications including spectral sparsification, quantum information optimization, and advanced conic programming.

A logarithmically homogeneous self-concordant barrier (LHSCB) is a central object in the theory of convex optimization over conic domains, unifying analytic, geometric, and algorithmic properties that enable efficient interior-point and first-order methods. LHSCBs provide a canonical way to encode the structure of convex cones, and their critical parameter determines both the complexity of optimization algorithms and key geometric invariants.

1. Definition and Main Properties

Let $K \subset \mathbb{R}^n$ be a regular closed convex cone with nonempty interior. A function $f : \mathrm{int}\, K \to \mathbb{R}$ is called a $\nu$-logarithmically homogeneous self-concordant barrier (LHSCB) if:

• Barrier growth: $f(x_k) \to +\infty$ whenever $x_k \to x \in \partial K$,
• Logarithmic homogeneity: $f(t x) = f(x) - \nu \log t$ for all $x \in \mathrm{int}\, K$, $t > 0$,
• Self-concordance: $$|D^3 f(x)[h, h, h]| \le 2 \left( D^2 f(x)[h, h] \right)^{3/2}$$ for all $x \in \mathrm{int}\, K$, $h \in \mathbb{R}^n$ (Zhao, 2023, Saunderson, 26 Dec 2025, Chewi, 2021).

The parameter $\nu$ is known as the barrier parameter or complexity parameter. It tightly controls the effective "dimension" of the geometry, arises in complexity estimates, and usually cannot be made smaller than the Euclidean dimension for classical cones (e.g., $\nu = n$ for $\mathbb{R}_+^n$, $\nu = d$ for $\mathbb{S}_+^d$) (Chewi, 2021, Bubeck et al., 2014).

The LHSCB structure yields:

• Affine invariance: The definition and fundamental inequalities are invariant under linear transformations.
• Local norm: At $x \in \mathrm{int}\, K$, the "Dikin" norm is $\|h\|_x := \sqrt{ h^T \nabla^2 f(x) h }$.
• Scaling identities: $\nabla f(t x) = t^{-1} \nabla f(x)$, $\nabla^2 f(t x) = t^{-2} \nabla^2 f(x)$.

2. Fundamental Inequalities and Geometry

The self-concordant structure enforces uniform control over third derivatives in terms of the local Hessian:

$|D^3 f(x)[h,h,h]| \le 2 [D^2 f(x)[h,h]]^{3/2}$

This gives rise to the key univariate control functions, for $s \in (-1,1)$:

$$\begin{align*} \omega(s) &= s - \ln(1+s), \ \omega^*(s) &= -s - \ln(1-s), \end{align*}$$

and fundamental sandwich bounds for $\|h\|_x < 1$:

$$\begin{align*} f(x+h) &\ge f(x) + \langle \nabla f(x), h \rangle + \omega(\|h\|_x), \ f(x+h) &\le f(x) + \langle \nabla f(x), h \rangle + \omega^*(\|h\|_x). \end{align*}$$

The Dikin ellipsoid $\mathcal{E}_x(r) = \{ x + h \mid \|h\|_x < r \}$ quantifies the region of uniform curvature control (Zhao, 2023, Zhao et al., 2020).

3. Barrier Parameter and Examples

The barrier parameter $\nu$ quantifies how rapidly the barrier function blows up at the boundary and determines the computational complexity for associated algorithms. Table 1 summarizes canonical examples:

Domain	Barrier Function	$\nu$
$\mathbb{R}_+^m$	$-\sum_{i=1}^m \ln x_i$	$m$
$\mathbb{S}_+^d$	$-\ln \det X$	$d$
Entropic barrier	Fenchel dual of log-Laplace	$n$
Hyperbolicity cones	$-\ln p(x)$, deg-$d$ poly.	$d$
Root-det cone, rank $d$	custom spectral barriers	$d+1$
Sandwiched Rényi, $n$	$-\log(t-\Psi_\alpha(X,Y)) - \ln\det X - \ln\det Y$	$1+2n$

For the entropic barrier, $$f^*(x) = \sup_\theta \langle\theta, x\rangle - \ln \int_K e^{\langle\theta, y\rangle} dy$$, one has an explicit universal LHSCB with parameter $\nu = n$, which is information-theoretically optimal for arbitrary cones (Bubeck et al., 2014, Chewi, 2021).

4. Algorithmic Role in Optimization

LHSCBs are the analytic foundation for interior-point methods (IPMs) on cones, providing:

• Path-following central paths: The central path $x_\mu$ is defined via the optimality condition $\nabla f(x_\mu) + s/\mu = 0$, ensuring proximity of iterates to the analytic center.
• Polynomial complexity: An LHSCB with parameter $\nu$ leads to an iteration bound $O( \sqrt{\nu} \log(1/\varepsilon) )$ for achieving $\varepsilon$-accuracy in conic linear programming (Chewi, 2021, Nesterov et al., 2014).
• Curvature control: The control provided by LHSCBs in the Dikin ellipsoid enables robust step-size selection in Newton or quasi-Newton IPMs, and precise prediction of convergence rates, including local superlinear convergence under mild sharpness and curvature assumptions (Nesterov et al., 2014).
• Affine-invariant first-order methods: LHSCBs enable Frank-Wolfe variants and mirror-descent schemes with step-size and progress analysis based entirely on the local barrier norm, and not on global Lipschitz constants (Zhao, 2023, Zhao et al., 2020).

In Frank-Wolfe methods with composite barriers, the LHSCB structure guarantees clean affine-invariant iteration complexity bounds, with global linear phases in the regime where the Frank-Wolfe gap exceeds $\nu$, and improved local rates upon facial identification (Zhao, 2023, Zhao et al., 2020).

5. Structural Refinements and Generalizations

Negative Curvature

For certain cones (notably hyperbolicity cones and symmetric cones), the barrier may possess negative curvature: $\nabla^3 F(x)[h] \preceq 0$ for $h \in K$. This yields monotonicity of the Hessian along feasible rays and simplifies the analysis of both primal and dual path-following dynamics (Nesterov et al., 2014).

Pairwise Self-Concordance

To achieve dimension-optimal "sparsification" bounds (generalizing spectral sparsification), a stronger two-dimensional pairwise self-concordance is imposed: for all $x \in \mathrm{int}\, K$ and $u,v \in K$

$$0 \le -D^3 F(x)[v,u,u] \le 2 D^2 F(x)[v,u] \, |u|_x,$$

leading to $O(\nu / \varepsilon^2)$ size $\varepsilon$-sparsifiers for conic sums (Saunderson, 26 Dec 2025).

Spectral and Noncommutative Barriers

LHSCBs have been systematically constructed for epigraph and perspective cones of spectral functions, particularly on Euclidean Jordan algebras, with optimal or near-optimal barrier parameters ($d+1$ or $d+2$), supporting numerically robust and efficient oracles for large-scale nonsymmetric conic programming (Coey et al., 2021). LHSCBs have also been identified for sophisticated noncommutative trace functions (e.g., sandwiched Rényi entropies), using operator convexity along lines as a verification technique, enabling the conic modeling of quantum information quantities in interior-point frameworks (He et al., 8 Feb 2025).

Projective Self-Concordance

LHSCBs naturally correspond to projectively self-concordant barriers when restricted to affine slices. This equivalence enables tighter Hessian-approximation bounds, improved step-size control, and thus more aggressive (less conservative) tuning of predictor-corrector parameters in contemporary IPM implementations (Hildebrand, 2019).

6. Applications and Impact

LHSCBs serve as the fundamental analytic infrastructure for a broad spectrum of algorithms and domains:

• Interior-point methods: Underpinning both polynomial-time path-following and modern variants, with optimality of the entropic barrier yielding the best possible iteration bounds (Bubeck et al., 2014, Chewi, 2021).
• First-order and primal-dual methods: Enabling step-size selection and convergence analysis in terms of barrier geometry rather than fixed Euclidean norms (Zhao et al., 2020, Zhao, 2023).
• Deterministic sparsification and matrix approximation: Specifying optimal size of $\varepsilon$-sparsifiers for conic sum problems, such as the generalizations of spectral sparsification to hyperbolic and other cones (Saunderson, 26 Dec 2025).
• Quantum information optimization: Facilitating the implementation of conic constraints for trace functions related to entropy, divergence, and matrix monotonicity, now integrated into solvers such as QICS (He et al., 8 Feb 2025).
• Symmetric and nonsymmetric spectral cones: Supporting natural formulations in symmetric cones and efficient computation for root-determinant and matrix-monotone-derivative cones, as implemented in Hypatia (Coey et al., 2021).

7. Optimality and Open Directions

It is established that $\nu \ge n$ is required for any LHSCB on a full-dimensional $n$-dimensional cone or convex domain, and constructions achieving this (notably the entropic barrier) are now available for all domains (Bubeck et al., 2014, Chewi, 2021). The interplay between operator convexity and self-concordance has yielded new families of LHSCBs, particularly for noncommutative and quantum domains (He et al., 8 Feb 2025). Open directions include further reductions of the barrier parameter for structured cones, extension of these techniques for alpha-parameter ranges beyond current results in the sandwiched Rényi case, and optimal lifted representations for advanced trace-function constraints (He et al., 8 Feb 2025).