Dinkelbach’s Algorithm: Fractional Programming

• Dinkelbach’s Algorithm is a method for optimizing ratio-valued objectives by converting fractional programming into a series of parametric subproblems.
• The approach achieves quadratic to cubic convergence through classical, interval, and accelerated variants that ensure computational efficiency.
• Its practical applications include communications, signal processing, scheduling, and combinatorial optimization, backed by rigorous convergence analysis.

Dinkelbach’s algorithm is a foundational procedure for solving fractional programming problems—those in which the objective function is a ratio of two (often convex) functions—by reducing them to a sequence of parametric subproblems. Since its inception in 1967, the algorithm and its variants have played a major role in optimization, with applications in communications, scheduling, signal processing, and combinatorial optimization. Ongoing developments—such as interval variants, minimal-correction accelerations, and look-ahead modifications—have addressed both the theoretical convergence rate and practical applicability of the method, notably maintaining computational efficiency by solving only one subproblem per iteration (Chen et al., 30 Oct 2025, Dadush et al., 2020).

1. Fractional Programming Formulation

Fractional programming aims to optimize a ratio-valued objective: $$\min_{x \in \mathcal F} \frac{f_1(x)}{f_2(x)}, \qquad \text{with } f_2(x) > 0.$$ This can be equivalently translated into the parametric root-finding problem for

$$g(\alpha) = \max_{x \in \mathcal F} \left\{ -f_1(x) + \alpha f_2(x) \right\}, \qquad \text{with root } \alpha^* \text{ where } g(\alpha^*) = 0.$$

Here, $g$ is strictly increasing and convex in $\alpha$. In the combinatorial context, the generic problem is

$\inf_{x \in \mathcal D} \frac{c^\top x}{d^\top x},$

leading to

$$f(\delta) = \inf_{x \in \mathcal D} \left\{ c^\top x - \delta d^\top x \right\},$$

where $f$ is concave, piecewise-linear, and nonincreasing in $\delta$ (Dadush et al., 2020). The fractional problem is reduced to finding the root (largest $\delta$ so that $f(\delta)=0$).

2. Classical Dinkelbach and Newton–Dinkelbach Methods

Classical Update

The core Dinkelbach update is equivalent to a Newton method for root-finding: $$g(\alpha) = \max_{x \in \mathcal F} \left\{ -f_1(x) + \alpha f_2(x) \right\}, \qquad \text{with root } \alpha^* \text{ where } g(\alpha^*) = 0.$$0 where $$g(\alpha) = \max_{x \in \mathcal F} \left\{ -f_1(x) + \alpha f_2(x) \right\}, \qquad \text{with root } \alpha^* \text{ where } g(\alpha^*) = 0.$$1. Given $$g(\alpha) = \max_{x \in \mathcal F} \left\{ -f_1(x) + \alpha f_2(x) \right\}, \qquad \text{with root } \alpha^* \text{ where } g(\alpha^*) = 0.$$2, the update simplifies to

$$g(\alpha) = \max_{x \in \mathcal F} \left\{ -f_1(x) + \alpha f_2(x) \right\}, \qquad \text{with root } \alpha^* \text{ where } g(\alpha^*) = 0.$$3

The sequence $$g(\alpha) = \max_{x \in \mathcal F} \left\{ -f_1(x) + \alpha f_2(x) \right\}, \qquad \text{with root } \alpha^* \text{ where } g(\alpha^*) = 0.$$4 is monotonic decreasing and converges to $$g(\alpha) = \max_{x \in \mathcal F} \left\{ -f_1(x) + \alpha f_2(x) \right\}, \qquad \text{with root } \alpha^* \text{ where } g(\alpha^*) = 0.$$5. Local convergence analysis shows quadratic rate under $$g(\alpha) = \max_{x \in \mathcal F} \left\{ -f_1(x) + \alpha f_2(x) \right\}, \qquad \text{with root } \alpha^* \text{ where } g(\alpha^*) = 0.$$6 (Chen et al., 30 Oct 2025): $$g(\alpha) = \max_{x \in \mathcal F} \left\{ -f_1(x) + \alpha f_2(x) \right\}, \qquad \text{with root } \alpha^* \text{ where } g(\alpha^*) = 0.$$7 The Newton–Dinkelbach update in discrete/combinatorial settings adapts by using negative supergradients of $$g(\alpha) = \max_{x \in \mathcal F} \left\{ -f_1(x) + \alpha f_2(x) \right\}, \qquad \text{with root } \alpha^* \text{ where } g(\alpha^*) = 0.$$8 at each iterate, yielding similar monotonicity and convergence (Dadush et al., 2020).

3. Interval and Accelerated Dinkelbach Variants

Interval Dinkelbach Method

The interval Dinkelbach method (Pardalos–Phillips ‘91) maintains a bracketing sequence $$g(\alpha) = \max_{x \in \mathcal F} \left\{ -f_1(x) + \alpha f_2(x) \right\}, \qquad \text{with root } \alpha^* \text{ where } g(\alpha^*) = 0.$$9 with updates:

• Lower-bound (secant):

$g$0

• Upper-bound (Dinkelbach):

$g$1

Under $g$2, the convergence rates are quadratic for the upper bound and superlinear for the lower (Chen et al., 30 Oct 2025).

Minimal Correction: Accelerated Interval Dinkelbach

An acceleration is achieved by replacing the upper-bound update with the minimum of two tangent-zeroes at $g$3 and $g$4: $g$5 where $g$6 and $g$7 are optimizers at the respective points. This requires no additional subproblem solves—only an $g$8 overhead. Under $g$9, the lower bound converges cubically, while the upper bound converges superquadratically (Chen et al., 30 Oct 2025).

Look-Ahead and Non-Monotone Accelerated Dinkelbach

Recent advances combine a two-step “accelerated secant–Newton” method (with order $\alpha$0) and a safeguard based on Dinkelbach’s tangent update:

• If an extrapolated (look-ahead) step is promising, it is accepted; if not, the classical update is used.
• When the method would drop below the solution, a tangent-min correction maintains global convergence and superquadratic acceleration (Chen et al., 30 Oct 2025, Dadush et al., 2020).

4. Convergence Analysis and Complexity

The convergence of Dinkelbach variants depends crucially on regularity conditions for the parametric function ($\alpha$1 or $\alpha$2):

• Classical Dinkelbach: Quadratic convergence under $\alpha$3 regularity near the solution.
• Interval Dinkelbach: Quadratic (upper), superlinear (lower bound).
• Accelerated Minimal Correction: Lower bound converges cubically, upper bound converges superquadratically.
• Non-Monotone Accelerated Dinkelbach: Incorporates two-point acceleration and periodic behavior determined by the sign of

$\alpha$4

The average asymptotic order per iteration exceeds 2 in all regimes, with $\alpha$5 for period 2 and $\alpha$6 for period 3 (Chen et al., 30 Oct 2025).

For the accelerated Newton–Dinkelbach scheme in discrete optimization, the Bregman divergence between iterates and the optimum halves every two steps, yielding $\alpha$7 complexity for $\alpha$8-approximation and strongly polynomial bounds in several combinatorial settings (Dadush et al., 2020).

Application-Specific Complexity

Problem Type	Iteration Bound	Per-Iteration Cost
Linear Fractional Combinatorial	$\alpha$9	1 linear optimization
2-Variable-Per-Inequality Systems	$\inf_{x \in \mathcal D} \frac{c^\top x}{d^\top x},$0	$\inf_{x \in \mathcal D} \frac{c^\top x}{d^\top x},$1 (general), $\inf_{x \in \mathcal D} \frac{c^\top x}{d^\top x},$2 for DMDPs
Parametric Submodular Minimization	$\inf_{x \in \mathcal D} \frac{c^\top x}{d^\top x},$3	1 submodular fn minimization

(Dadush et al., 2020)

5. Implementation and Practical Considerations

Each Dinkelbach-type iteration solves a single parametric subproblem of maximizing $\inf_{x \in \mathcal D} \frac{c^\top x}{d^\top x},$4 or minimizing $\inf_{x \in \mathcal D} \frac{c^\top x}{d^\top x},$5 subject to the original constraints. For accelerated variants, additional computations (extra tangent evaluations and screening conditions) require only $\inf_{x \in \mathcal D} \frac{c^\top x}{d^\top x},$6 overhead per iteration if the subproblem is computationally expensive (Chen et al., 30 Oct 2025).

The only algorithmic tuning typically required is a screening parameter $\inf_{x \in \mathcal D} \frac{c^\top x}{d^\top x},$7, with practical performance not unduly sensitive to its value. There are no requirements for line searches or trust-region mechanisms in accelerated frameworks.

Global convergence relies on $\inf_{x \in \mathcal D} \frac{c^\top x}{d^\top x},$8 being strictly increasing/convex and at least $\inf_{x \in \mathcal D} \frac{c^\top x}{d^\top x},$9; local superquadratic convergence needs $$f(\delta) = \inf_{x \in \mathcal D} \left\{ c^\top x - \delta d^\top x \right\},$$0. In the combinatorial context, the accelerated method leverages combinatorial monotonicity conditions, e.g., the Goemans–Radzik and subpath-monotonicity lemmas (Dadush et al., 2020).

6. Applications and Research Directions

Dinkelbach’s method and its accelerations are deployed in a broad spectrum of optimization problems, including:

• Communications: Energy-efficiency maximization in wireless networks, device-to-device (D2D), non-orthogonal multiple access (NOMA), simultaneous wireless information and power transfer (SWIPT).
• Signal Processing: Ratio-of-quadratics, total-least-squares.
• Scheduling: Cyclic scheduling.
• Combinatorial and Discrete Optimization: Fractional combinatorial auctions, label-correcting algorithms for systems with two variables per inequality, deterministic Markov Decision Processes (Chen et al., 30 Oct 2025, Dadush et al., 2020).
• Submodular Minimization: Parametric versions with strongly polynomial iteration bounds.

Further development focuses on pushing convergence guarantees towards lower iteration complexity, maintaining the essential property of single subproblem solves per iteration, and deriving problem-specific acceleration mechanisms.

7. Auxiliary Results and Structural Lemmas

The theoretical basis for the fast convergence and iteration complexity in discrete and continuous fractional programming with Dinkelbach-type methods relies on several auxiliary lemmas:

• Monotonicity: Decreasing sequence of iterates and increasing objective values.
• Bregman Divergence Halving: Each two accelerated steps halve the divergence to the optimum (Dadush et al., 2020).
• Goemans–Radzik Lemma: Halving sequences of $$f(\delta) = \inf_{x \in \mathcal D} \left\{ c^\top x - \delta d^\top x \right\},$$1-vectors have $$f(\delta) = \inf_{x \in \mathcal D} \left\{ c^\top x - \delta d^\top x \right\},$$2 length.
• Subpath Monotonicity: Governs iteration bounds in certain network flow and label-correcting settings.
• Growth of Ring Families: Limits the number of discrete changes in parametric submodular minimization.

These structural insights enable the design and analysis of strongly polynomial-time algorithms for large classes of fractional and parametric optimization problems.

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For in-depth algorithmic descriptions and proofs, see (Chen et al., 30 Oct 2025) for accelerated continuous Dinkelbach variants and (Dadush et al., 2020) for combinatorial and submodular applications of the Newton–Dinkelbach and its accelerations.