Single-Ratio Fractional Minimization

• Single-ratio fractional minimization problems are optimization tasks that minimize the quotient A(x)/B(x) under constraints ensuring A(x) ≥ 0 and B(x) > 0.
• They leverage fractional Sobolev inequalities and variational formulations to establish existence, symmetry, and stability of minimizers in both finite- and infinite-dimensional settings.
• Robust algorithmic approaches such as Dinkelbach’s method, proximal-gradient, and quadratic transforms address the challenges of nonconvexity and nonsmoothness in these problems.

Single-ratio fractional minimization problems refer to optimization tasks where the objective is a quotient of two functions, that is, to minimize $F(x) = A(x) / B(x)$ over a feasible set, typically under the conditions $A(x) \geq 0$, $B(x) > 0$ for all feasible $x$. Such problems naturally appear across analysis, nonlocal PDEs, variational formulations, signal processing, machine learning, network optimization, portfolio selection, and wireless communication. The challenging nonconvex structure of the ratio, potential nonsmoothness, and frequent lack of closed-form solutions have led to the development of a rich arsenal of mathematical tools, optimization algorithms, and theoretical frameworks targeting both finite- and infinite-dimensional settings.

1. Mathematical Formulations and Analytical Foundations

The single-ratio fractional minimization paradigm encompasses formulations where either $A$, $B$ are functionals on infinite-dimensional spaces (as in Sobolev or fractional Sobolev settings) or real-valued (possibly matrix-valued) functions over Euclidean or product spaces (as in signal processing and optimization). Analytical approaches often exploit problem structure, such as convexity, positive homogeneity, or the presence of underlying fractional operators.

A foundational example arises in nonlocal variational calculus: minimizing an energy functional of the form

$$E(u) = \int_{\mathbb{R}^n} |(-\Delta)^{s/2} u(x)|^2\, dx - \int_{\mathbb{R}^n} F(|x|, u(x))\, dx$$

subject to a mass constraint $\|u\|_{L^2} = c$, with $(-\Delta)^s$ denoting the fractional Laplacian and $F$ polynomially growing in $u$ (Hajaiej, 2011). Here, sharp functional inequalities (fractional Gagliardo–Nirenberg, fractional Polya–Szegő, sharp Sobolev) underpin the existence, uniqueness, and symmetry of minimizers. Rearrangement and compactness arguments reduce the search to radially symmetric, nonincreasing functions; interpolation controls the nonlinearity to ensure coercivity and boundedness below.

In high-dimensional algebraic or combinatorial optimization, one often addresses the generic form

$\min_{x \in \mathcal{X}} \frac{f(x)}{g(x)}$

where $f$ and $g$ are convex, semi-algebraic, and differentiable (Lin et al., 2023), or possibly nonsmooth, with additional positive homogeneity or DC (difference of convex) structure (Qi et al., 22 Oct 2025). Applications span $q$-ratio sparsity minimization for signal recovery, $\ell_1 / S_\kappa$ sparse signal recovery, and graph clustering where Lovász extensions yield equivalent continuous formulations for ratios of set functions (Bühler et al., 2013, Zhou et al., 2020).

2. Functional Inequalities and Existence Theory in Fractional Spaces

In fractional Sobolev-type settings, the analysis and solution of single-ratio minimization rely on fundamental inequalities:

• Fractional Polya–Szegő Inequality: Guarantees that the Schwarz symmetrization does not increase the fractional seminorm. For $u \in H^s(\mathbb{R}^n)$, $u^*$ its symmetric decreasing rearrangement,

$$\int_{\mathbb{R}^n} |(-\Delta)^{s/2}u^*|^2 \leq \int_{\mathbb{R}^n} |(-\Delta)^{s/2}u|^2.$$

This enables restriction to radially symmetric minimizers (Hajaiej, 2011).

• Fractional Gagliardo–Nirenberg and Sharp Sobolev Inequalities: Provide interpolation and embedding bounds, controlling nonlinear terms through the fractional kinetic energy and $L^2$ norm. These are critical for deriving boundedness and demonstrating coercivity, especially when working with critical or supercritical growth (Hajaiej, 2011).
• Capacity Compactness: For optimization in fractional order Sobolev spaces $H^s(\Omega)$, compactness and the relaxation of dual variables via capacitary measures (vanishing on sets of zero $H^s$-capacity) allow passing to the limit in measure-valued multipliers and support the derivation of stronger optimality conditions (Lentz, 16 Dec 2024).

These principles, together with direct methods in the calculus of variations, have established existence, uniqueness (modulo symmetry or translation), and (orbital) stability for broad classes of fractional minimization problems, with extensions to variable order derivatives and isoperimetric constraints (Bourdin et al., 2012, Tavares et al., 2016).

3. Algorithmic Approaches: Classic, Proximal, and Transform-Based Methods

Single-ratio fractional minimization problems are often nonconvex, and specialized optimization algorithms are required for tractable and robust solution. Key methodologies include:

A. Dinkelbach-type and Parametric Reformulations

• Dinkelbach's algorithm iteratively solves parametrized subproblems of the form $f(x) - \lambda g(x)$, updating $\lambda$ by the current ratio $f(x)/g(x)$. This approach is optimal under quasi-convexity but may require an inner loop and can be computationally intensive for large-scale or complex objectives (Nguyen et al., 2014).
• For quadratic fractional programs with two-sided constraints, SDP reformulations and extended S-lemma with equality eliminate the need for inner loops, directly computing the optimal Lagrange multiplier (Nguyen et al., 2014).

B. Proximal-Gradient and Subgradient Methods

• The proximal-gradient approach decouples the smooth (possibly nonconvex) and nonsmooth (possibly convex) components, updating $$x^{k+1} = \operatorname{prox}_{\iota_{\Omega}}(x^k - \alpha(\nabla f(x^k) - c_k \nabla g(x^k)))$$ with $c_k = f(x^k)/g(x^k)$, yielding rapid convergence to critical points under the KL property (Lin et al., 2023, Han et al., 15 Mar 2025).
• Proximal-subgradient/difference-of-convex (DC) algorithms (PS-DCA) incorporate a DCA step after the subgradient update to help escape low-quality local minima, especially effective when both numerator and denominator are convex and positively homogeneous (Qi et al., 22 Oct 2025).

C. Quadratic Transform and MM-based Techniques

• The quadratic transform is a functional reformulation that introduces an auxiliary variable $y$ to decouple the ratio:

$$\frac{A(x)}{B(x)} = \max_{y \in \mathbb{R}}(2y\sqrt{A(x)} - y^2 B(x)),$$

with $y^* = \sqrt{A(x)}/B(x)$. Alternating maximization over $y$ and $x$ sidesteps direct ratio nonconvexity, generalizes efficiently to sums and matrix ratios, and preserves closed-form updates (Shen et al., 13 Mar 2025, Chen et al., 2023). An inverse quadratic transform is used for minimization (Chen et al., 2023).

• Majorization-minimization (MM) interpretations ensure monotonic improvement of surrogate objectives and allow generalization to complex or matrix-valued ratios (Chen et al., 2023, Shen et al., 13 Mar 2025).

D. Coordinate Descent and Splitting Strategies

• Coordinate descent frameworks solve one-dimensional surrogate problems for each coordinate, permitting global solution along that axis and yielding convergence to coordinate-wise stationary points superior to simple critical points, particularly useful for nonseparable or tightly structured objectives (Yuan, 2022).
• Splitting schemes, including ADMM variants, decouple linear and nonsmooth components, accommodating highly structured objectives (e.g., penalized worst-case robust Sharpe ratio, sparse discriminant analysis) (Yuan, 12 Nov 2024).

4. Applications Across Mathematical and Engineering Domains

Single-ratio fractional minimization has widespread use, including but not limited to:

• Nonlocal and Fractional PDEs: Existence and regularity of ground states for nonlocal Schrödinger equations, minimization of fractional perimeter-Dirichlet functionals with free boundary regularity via blow-up and monotonicity methods (Caffarelli et al., 2013, Hajaiej, 2011).
• Signal Processing and Machine Learning: $q$-ratio and $\ell_1/\ell_q$ sparsity minimization for compressed sensing, the design of robust or sparsity-promoting estimators under various matrix and norm constraints (Zhou et al., 2020, Zhang et al., 2020, Qi et al., 22 Oct 2025).
• Graph and Network Optimization: Exact continuous relaxations of normalized cut and density-based clustering via Lovász extensions, guarantee discrete feasibility via thresholding (Bühler et al., 2013).
• Wireless Communications and Sensing: SINR and CRB minimization, energy efficiency optimization, age-of-information minimization in multiratio and matrix settings via quadratic/inverse quadratic transform frameworks (Soleymani et al., 3 Feb 2025, Shen et al., 13 Mar 2025, Chen et al., 2023).
• Portfolio Optimization: Single-period Sharpe ratio maximization and robust risk-adjusted return models realized via fractional objectives on simplex-constrained weights (Lin et al., 2023, Han et al., 15 Mar 2025, Yuan, 12 Nov 2024).

5. Key Theoretical Insights and Comparative Analysis

A consolidation of optimality and convergence theory for single-ratio problems reveals several critical phenomena:

• Critical Point Characterization: In the convex positively homogeneous setting, global optimality reduces to inclusion relations $$E(x^*) \partial B(x^*) \cap \partial T(x^*) \neq \emptyset$$; in nonsmooth settings, lifted stationary points (arising from split or surrogate formulations) generalize classical Lagrange conditions (Qi et al., 22 Oct 2025, Han et al., 15 Mar 2025).
• Hierarchy of Solution Concepts: Coordinate-wise stationary points are strictly stronger than critical or directional points. For concave denominators, all critical points are global minimizers (Yuan, 2022).
• Comparative Algorithmic Features:

Method	Curvature Requirements	Inner Loops	Applicability	Special Features
Dinkelbach	Quasi-convex/pseudo-convex	Yes	Single-ratio, smooth	Superlinear convergence
Proximal/PGSA	KL property, Lipschitz	No	Nonconvex, composite	Global convergence
Quadratic Tx	None (via MM)	No	Sums, matrices, mixed ratios	Closed-form updates
PS-DCA	Positive homogeneity, convex	No	DC-structured, nonsmooth	Escape poor local minima
ADMM (FADMM)	Weak convexity of denominator	No	Structured, nonsmooth	Splitting+Lyapunov proof

• Limitations and Open Problems: Critical and supercritical fractional growth often pose compactness challenges, and problems with highly nonconvex, nonseparable, or discrete structure may require further algorithmic innovation (Hajaiej, 2011, Lasserre et al., 2020).

6. Future Directions and Emerging Paradigms

Ongoing research in single-ratio fractional minimization is focused on several directions:

• Beyond Standard Smoothness and Convexity: Extending algorithms and theory to handle general non-differentiable or non-polyhedral structures, as well as variable order and time-dependent fractional operators (Tavares et al., 2016).
• Hybrid and Accelerated Algorithms: Combining quadratic transform, MM acceleration, block-coordinate, or Nesterov-type extrapolations for faster convergence, especially in matrix or large-scale settings (Shen et al., 13 Mar 2025).
• Integration with Discrete and Combinatorial Optimization: Developing frameworks that efficiently combine ratio minimization with discrete constraints or machine learning models (e.g., clustering, community detection).
• Applications to Emerging Areas: Fractional minimization formulations increasingly appear in federated learning, integrated sensing and communications, and dynamic networked systems, motivating customized surrogates and scalable solvers (Chen et al., 2023, Soleymani et al., 3 Feb 2025).

7. Summary Table of Representative Problem Structures and Algorithmic Techniques

Setting / Problem	Objective Structure	Dominant Algorithm(s)	Reference
Fractional Sobolev Variational Problems	$\int|\nabla^s u|^2$ / $L^2$-norm(+nonlin)	Rearrangement, interpolation	(Hajaiej, 2011)
Quadratic Ratio Minimization	$x^{\top}A_1x + 2b_1^{\top}x + c_1$ / ...	SDP (S-lemma), Dinkelbach	(Nguyen et al., 2014)
$q$-ratio Sparsity Signal Recovery	$\|\cdot\|_1 / \|\cdot\|_q$	PM, CCP	(Zhou et al., 2020)
L₁/Sκ Sparse Recovery, CT	$(\lambda\|\cdot\|_1 + \text{LS}) / S_{\kappa}$	Single-loop proximal subgradient	(Han et al., 15 Mar 2025)
Matrix Fractional Metric Minimization	$\operatorname{Tr}(AX)/\operatorname{Tr}(BX)$ etc.	Quadratic/inverse quadratic transform, MM	(Shen et al., 13 Mar 2025, Chen et al., 2023)

This framework, under ongoing theoretical and computational expansion, continues to provide critical tools and structural insights for both foundational analysis in fractional spaces and applied optimization across data science and engineering.