Quadratic Fractional Programming

• Quadratic Fractional Programming is a framework for optimizing problems with ratio of quadratic functions, using specialized techniques to handle nonconvex structures.
• It employs methods like parametric analysis, quadratic transforms, and semidefinite relaxation to decouple complex ratio objectives into tractable subproblems.
• Its applications span engineering domains, including robust beamforming, radar waveform design, and SVM margin maximization, ensuring practical and efficient optimization.

Quadratic fractional programming (QFP) studies optimization problems whose objective or constraint functions are ratios of quadratic forms. These arise extensively in signal processing, communications, control, and combinatorial optimization, due to natural occurrence of figures of merit like signal-to-noise/interference ratios, Cramér-Rao bounds, and eigenvalue or margin-based metrics in machine learning. The nonconvex, nonhomogeneous rational structure fundamentally distinguishes QFP from classical quadratic programming, necessitating a specialized toolkit that spans parametric analysis, convexification, conic duality, minorization–maximization, semidefinite relaxations, and combinatorial enumeration.

1. Canonical Problem Formulations

The prototypical scalar QFP takes the form

$$\min_{x\in X} \frac{f(x)}{g(x)} \qquad \text{with } f(x) = x^\top A x + 2a^\top x + \alpha, \;\; g(x) = x^\top B x + 2b^\top x + \beta,$$

subject to $g(x) > 0$ for all $x \in X \subseteq \mathbb{R}^n$ and possibly additional constraints (polyhedral, quadratic, combinatorial, or constant-modulus) (Nguyen et al., 2014). Multi-objective or vector QFPs generalize this to simultaneous minimization of $m$ coordinate-wise ratios $f_i(x)/g_i(x)$ over a common feasible region (Oliveira et al., 2013).

Problem classes of central importance include:

• Scalar single-ratio minimization/maximization
• Max–min or sum-of-ratios objectives
• QFP with multi-dimensional matrix terms (e.g., $\operatorname{Tr}\left(A(x)B(x)^{-1}A(x)^H\right)$)
• QFP under quadratic, polyhedral, or discrete constraints ($x \in \{-1,1\}^n$, $|x_k|=1$)
• Fractionally constrained fractional programming (FCFP), where both objective and constraints are ratio structured (Liu et al., 13 Mar 2025)

2. Classical and Semidefinite Programming Methodologies

For convex feasible sets, the standard approach replaces the ratio objective by a parametric family: $$F(\lambda) = \inf_{x \in X} \left\{ f(x) - \lambda g(x) \right\},$$ with the optimal value $\lambda^* = \min_{x \in X} f(x)/g(x)$ characterized as the unique solution to $g(x) > 0$0 (the Dinkelbach root) (Nguyen et al., 2014, Gao et al., 2012). For single-ratio with quadratic numerator and denominator, this links QFP to the generalized trust region subproblem and invites strong duality and convex relaxation techniques.

Key results include:

• For two-sided quadratic constraints, $g(x) > 0$1, the exact solution for $g(x) > 0$2 is computable by semidefinite programming (SDP), using a suitably generalized S-lemma:

$g(x) > 0$3

with $g(x) > 0$4 or $g(x) > 0$5 (Nguyen et al., 2014).

• In the case of a single quadratic constraint $g(x) > 0$6, the standard S-lemma suffices and the SDP relaxation remains exact and polynomial-time (Nguyen et al., 2014).
• Absence of positive-definite pencils (i.e., “hard cases” where no Slater point exists for some $g(x) > 0$7) can be overcome by exploiting this extended S-lemma and associated constraint qualification (“Assumption B”) (Nguyen et al., 2014).
• Nonconvex quadratic-fractional programs (e.g., with nonconvex sets or concave denominator) can be globally addressed by canonical duality. The original problem is parametrized, and the dual reduces to a low-dimensional (often 2D) concave maximization, enjoying no duality gap under mild conditions (Gao et al., 2012).
• For discrete QFP ($g(x) > 0$8), e.g.,

$g(x) > 0$9

with fixed spectral ranks, the global optimum can be found in $x \in X \subseteq \mathbb{R}^n$0 via an accelerated Newton–Dinkelbach scheme, provided the diagonal patterns meet appropriate sign conditions (Yang et al., 2024).

3. Quadratic Transform and Fractional Programming Algorithms

The "quadratic transform" (QT) offers a general-purpose, highly modular approach for scalar and matrix QFP with convex (or nonconvex) constraint sets (Shen et al., 13 Mar 2025, Shen et al., 2023, Krishtal et al., 2023). The basic device is to decouple each ratio into a jointly biconcave surrogate: $x \in X \subseteq \mathbb{R}^n$1 Alternating optimization over auxiliary $x \in X \subseteq \mathbb{R}^n$2 (closed form) and $x \in X \subseteq \mathbb{R}^n$3 (convex subproblem when $x \in X \subseteq \mathbb{R}^n$4 is concave and $x \in X \subseteq \mathbb{R}^n$5 convex) yields monotonic ascent and eventual convergence to a stationary point (Shen et al., 13 Mar 2025, Shen et al., 2023). This framework extends to maximization and minimization of sums of (functions of) ratios, matrix ratios (by trace–Schur surrogates), and to mixed max–min formulations (Chen et al., 2023).

For multi-term and matrix-valued ratios, the quadratic transform yields subproblems amenable to standard convex or discrete programming techniques, or even matching and assignment solvers (for graph clustering and SVM margin maximization) (Shen et al., 13 Mar 2025).

Accelerated variants leveraging Nesterov-type extrapolation or Steffensen's fixed-point schemes can achieve $x \in X \subseteq \mathbb{R}^n$6 local convergence rates (Shen et al., 2023, Park et al., 2 Jan 2026). Theoretical guarantees and practical performance are established for diverse applications (massive MIMO, radar, ISAC, AoI minimization, SVMs, graph cuts).

4. Convexification, Copositive and Combinatorial Approaches

For QFP over polytopes or discrete sets, convexification via homogenization and copositive programming forms a central paradigm (He et al., 2023). The convex hull of the feasible set can be characterized in lifted (moment or completely positive) spaces: $x \in X \subseteq \mathbb{R}^n$7 where $x \in X \subseteq \mathbb{R}^n$8 is the cone of completely positive $x \in X \subseteq \mathbb{R}^n$9 matrices. This reformulation is exact under mild conditions and admits strong relaxations via the RLT or moment hierarchies.

For 0-1 QFP over cardinality-constrained polytopes, hierarchy-based relaxations close optimality gaps several times faster than McCormick or CEF relaxations and are computationally efficient up to $m$0 (He et al., 2023). In low-rank convex–convex cases, global optimality can be ensured by explicit region checking on sign patterns of rank vectors, yielding $m$1 worst-case complexity, but far better practical scaling (Krishtal et al., 2023).

5. Pareto Optimality, Vector QFP, and Multi-objective Extensions

For vector quadratic fractional programming (VQFP), necessary and sufficient Pareto optimality conditions generalize KKT by exploiting the structure of the ratio gradients. For $m$2, the stationarity reads

$m$3

with appropriate feasibility and complementarity (Oliveira et al., 2013, Oliveira et al., 2013).

Strong duality holds under Guignard–generalized constraint qualification (GGCQ). Second-order sufficient optimality reduces to positive definiteness of a weighted sum of Hessians on the critical cone. Additional geometric criteria (“radius of efficiency”) certify global vs. local Pareto solutions: efficient computational tests are available for algorithmic termination and local–global optimality distinction (Oliveira et al., 2013).

6. Specialized Domains: Fractionally Constrained Problems and Applications

In advanced scenarios such as FCFP—fractionally constrained fractional programming—when both the objectives and constraints are ratios (e.g., RIS-aided ISAC), the quadratic transform can be recursively applied to both constraint and objective ratios (Liu et al., 13 Mar 2025). For constant-modulus or combinatorial constraints, penalty-based or dual variable-based linearization delivers tractable convex or linear subproblems, preserving polynomial solvability in problem dimension. This framework is essential for modern joint sensing–communications optimization, robust MIMO, and statistical waveform design.

The quadratic-fractional paradigm underlies a wide spectrum of engineering and data science problems:

• Power/rate allocation in wireless networks (SINR-based QFP) (Shen et al., 13 Mar 2025)
• Robust beamforming under imperfect CSI (Iimori et al., 2020)
• Radar waveform optimization (CRLB/KLD minimization) (Shen et al., 13 Mar 2025, Park et al., 2 Jan 2026)
• Graph partitioning/object clustering (discrete matrix QFP) (Shen et al., 13 Mar 2025)
• SVM margin maximization and multi-class classifier design (Shen et al., 13 Mar 2025)
• Joint sensing-communication (RIS-based, ISAC) (Liu et al., 13 Mar 2025)
• Age-of-information minimization and queueing systems (Chen et al., 2023)

7. Algorithmic and Theoretical Summary

Quadratic fractional programming unifies a range of nonconvex, nonhomogeneous optimization tasks for which global stationarity can be certified via majorization-minimization, parametric analysis, or semidefinite duality. The critical techniques include:

• Parametric Dinkelbach-type root-finding for single-ratio cases
• Semidefinite relaxation and S-lemma-based SDP for quadratically constrained QFP, even in “hard cases” without Slater points (Nguyen et al., 2014)
• Canonical duality transforming nonconvex QFP into low-dimensional concave dual maximization (Gao et al., 2012)
• Quadratic transform for sum-of-ratios or matrix-ratio FP, yielding efficient AO/MM style alternations with convergence guarantees (Shen et al., 13 Mar 2025, Shen et al., 2023, Chen et al., 2023)
• Hierarchical convexification and combinatorial enumeration in discrete/low-rank settings (He et al., 2023)
• Pareto and multi-objective optimality criteria enabling certified local and global efficiency (Oliveira et al., 2013, Oliveira et al., 2013)

A directed selection of approaches is provided in the following table:

Formulation Type	Representative Method	Key Reference
Single-ratio, quadratic	Parametric, S-lemma + SDP	(Nguyen et al., 2014)
Multi-term (sum-of-ratios)	Quadratic Transform	(Shen et al., 13 Mar 2025, Shen et al., 2023)
Matrix-ratio QFP	Matrix Quadratic Transform	(Shen et al., 13 Mar 2025, Park et al., 2 Jan 2026)
Mixed max/min (multi-task)	Unified QT + MM	(Chen et al., 2023)
Discrete/low-rank/fractional	Copositive/convexification	(He et al., 2023, Krishtal et al., 2023)
Pareto multi-objective	KKT/critical cone analysis	(Oliveira et al., 2013, Oliveira et al., 2013)
FCFP (objective + constr. FP)	Recursive QT + penalty/LP	(Liu et al., 13 Mar 2025)

Despite the diversity of formulations, a recurring backbone is the reduction of ratio structures to biconcave or convexified surrogates amenable to efficient numerical solution, and the availability of theoretically sound optimality and duality guarantees for verifying stationarity or globality.