Interval Dinkelbach Method

• The interval Dinkelbach method is an iterative bracketing algorithm for single-ratio fractional programming that constructs simultaneous lower and upper bounds converging to the optimal ratio.
• It employs a secant-based update for the lower bound and a classical Dinkelbach step for the upper bound, ensuring monotonic convergence and reliable error estimation.
• Accelerated variants integrate tangent corrections to achieve cubic or superquadratic convergence, delivering high-precision results with minimal additional computational cost.

The interval Dinkelbach method is an iterative bracketing algorithm designed to solve single-ratio fractional programming problems of the form $\max_{x \in X} \frac{p(x)}{q(x)}$ over nonempty, convex, and compact sets $X \subset \mathbb{R}^n$, with continuous numerator and denominator functions $p, q : X \to \mathbb{R}$ and $q(x) > 0$ for all $x \in X$. It constructs simultaneous, monotone lower and upper bound sequences that converge to the optimal ratio, with distinct local convergence orders under mild differentiability conditions on the associated parametric function. Recent advances introduce acceleration mechanisms leading to higher-order asymptotic convergence while preserving global bracketing and requiring only a single subproblem solution per bound update (Chen et al., 30 Oct 2025).

1. Fractional Programming and Problem Formulation

The interval Dinkelbach method addresses the single-ratio fractional program: $\max_{x \in X} \frac{p(x)}{q(x)},$ subject to $x \in X$,

where:

• $X \subset \mathbb{R}^n$ is nonempty, compact (or closed and bounded), and convex,
• $p$ and $q$ are continuous on $X \subset \mathbb{R}^n$0,
• $X \subset \mathbb{R}^n$1 for all $X \subset \mathbb{R}^n$2,
• For each $X \subset \mathbb{R}^n$3 of interest, the subproblem $X \subset \mathbb{R}^n$4 attains its maximum.

The optimal value $X \subset \mathbb{R}^n$5 is achieved at some $X \subset \mathbb{R}^n$6. The problem can be equivalently reformulated as root-finding for the strictly increasing, convex, and continuous parametric function

$X \subset \mathbb{R}^n$7

The unique $X \subset \mathbb{R}^n$8 of $X \subset \mathbb{R}^n$9 on $p, q : X \to \mathbb{R}$0 corresponds to $p, q : X \to \mathbb{R}$1.

2. Classical Dinkelbach Method and Its Properties

The Dinkelbach method iteratively updates an upper bound, starting from an initial point $p, q : X \to \mathbb{R}$2 for some $p, q : X \to \mathbb{R}$3:

1. Given $p, q : X \to \mathbb{R}$4, solve $p, q : X \to \mathbb{R}$5,
2. Update $p, q : X \to \mathbb{R}$6.

Under the assumption that $p, q : X \to \mathbb{R}$7 is twice continuously differentiable near $p, q : X \to \mathbb{R}$8 and $p, q : X \to \mathbb{R}$9, the method generates a strictly decreasing sequence $q(x) > 0$0 with $q(x) > 0$1, converging quadratically to $q(x) > 0$2: $q(x) > 0$3 This iteration, however, yields only a one-sided upper bound sequence (Chen et al., 30 Oct 2025).

3. Construction and Analysis of the Interval Dinkelbach Method

The interval Dinkelbach method, introduced by Pardalos and Phillips (1991), constructs bracketing sequences $q(x) > 0$4 and $q(x) > 0$5 such that $q(x) > 0$6 for all $q(x) > 0$7:

• Initialize $q(x) > 0$8 with $q(x) > 0$9,
• Lower (Secant) step:

$x \in X$0

• Upper (Dinkelbach) step:

$x \in X$1

Assuming $x \in X$2 on $x \in X$3:

• The lower sequence $x \in X$4 is monotonically increasing, the upper sequence $x \in X$5 is monotonically decreasing,
• The gap $x \in X$6 contracts,
• Convergence orders:
  • $x \in X$7 (quadratic),
  • $x \in X$8, specifically superlinear but order $x \in X$9.

Monotonicity and bracketing are preserved, supporting strong a posteriori error estimation during the iterative process (Chen et al., 30 Oct 2025).

4. Accelerated Interval Dinkelbach Algorithm

An accelerated variant of the interval Dinkelbach method incorporates a minimal "correction" in the upper bound update by utilizing the tangent from the updated lower bound, computed at no extra subproblem cost: $\max_{x \in X} \frac{p(x)}{q(x)},$0 where $\max_{x \in X} \frac{p(x)}{q(x)},$1 results from the lower bound computation.

For $\max_{x \in X} \frac{p(x)}{q(x)},$2 until $\max_{x \in X} \frac{p(x)}{q(x)},$3 is sufficiently small:

• Update $\max_{x \in X} \frac{p(x)}{q(x)},$4 via the secant step,
• Compute both $\max_{x \in X} \frac{p(x)}{q(x)},$5 and $\max_{x \in X} \frac{p(x)}{q(x)},$6,
• Update $\max_{x \in X} \frac{p(x)}{q(x)},$7 via the minimum projection.

Key results under $\max_{x \in X} \frac{p(x)}{q(x)},$8:

• Cubic convergence for the lower bound: $\max_{x \in X} \frac{p(x)}{q(x)},$9,
• Superquadratic and, under strict convexity ($x \in X$0), cubic convergence for the upper bound: $x \in X$1,

The accelerated variant preserves global monotonic bracketing and requires only one subproblem solve per bound per iteration–the same as the unaccelerated version–while dramatically enhancing the rate of local convergence (Chen et al., 30 Oct 2025).

5. Comparative Properties and Theoretical Guarantees

A comparison of methods is summarized as follows:

Method	Solves per Iteration	Lower Convergence	Upper Convergence
Classical Dinkelbach	1	N/A	Quadratic
Interval Dinkelbach (original)	2	Superlinear ($x \in X$2)	Quadratic
Accelerated Interval Dinkelbach	2	Cubic	Superquadratic–Cubic

The interval Dinkelbach method—original and accelerated—maintains global bracketing: $x \in X$3, $x \in X$4, and contracts the interval $x \in X$5 at increased rates under the accelerated variant. The lower–upper gap closes cubically once the iterates are sufficiently close to the optimum, facilitating highly accurate enclosures in a minimal number of iterations. Practical implications favor the accelerated bracketed approach when extremely high precision is required from fractional programs (Chen et al., 30 Oct 2025).

6. Practical Applications and Implications

The interval Dinkelbach method is particularly effective in scenarios where global bracketing, high accuracy, and robust a posteriori error control are essential, such as in numerical optimization, control, economics, and other areas where fractional programs arise. The accelerated algorithm produces near-optimal solutions with enclosures of $x \in X$6 in very few iterations without requiring additional oracle (subproblem) calls per bound. This suggests that fractional programming practitioners seeking both reliability and efficiency in high-precision regimes can benefit from the algorithm's structure and convergence properties (Chen et al., 30 Oct 2025).

This suggests broad applicability, especially given the minimal algorithmic overhead compared to existing bracketed approaches.

7. Discussion and Future Directions

The interval Dinkelbach method, especially in its accelerated form, offers a rigorous and efficient approach to fractional programming, advancing beyond classical and secant-based bracketing strategies. Its asymptotic convergence order—at least the square root of 5 per iteration under sufficient differentiability—surpasses quadratic rates and is achieved without additional subproblem evaluations. A plausible implication is that further algorithmic refinements or generalizations may be possible for broader classes of parametric nonlinear programs. Open questions concern extensions to nonconvex or nonsmooth settings, multi-ratio objectives, and integration with advanced convex optimization or oracle frameworks (Chen et al., 30 Oct 2025).