Accelerated Dinkelbach method

TL;DR

The paper addresses fractional programming by developing two accelerated Dinkelbach-type schemes. The accelerated interval Dinkelbach method introduces a small correction to the upper-bound update, achieving cubic convergence for the lower-bound sequence and superquadratic convergence for the upper-bound sequence while preserving a single subproblem per iteration. The accelerated Dinkelbach method employs a Newton-type acceleration with two prior iterates, generating a globally convergent, non-monotone sequence with an asymptotic average convergence order of at least $\sqrt{5}$ per iteration under sufficient differentiability, with potential higher orders in special cases. Together, these methods significantly improve convergence rates over classical Dinkelbach-type algorithms without increasing per-iteration computational burden, offering practical advantages for large-scale fractional programming problems where evaluating subproblems is costly.

Abstract

The classical Dinkelbach method (1967) solves fractional programming via a parametric approach, generating a decreasing upper bound sequence that converges to the optimum. Its important variant, the interval Dinkelbach method (1991), constructs convergent upper and lower bound sequences that bracket the solution and achieve quadratic and superlinear convergence, respectively, under the assumption that the parametric function is twice continuously differentiable. However, this paper demonstrates that a minimal correction, applied solely to the upper bound iterate, is sufficient to boost the convergence of the method, achieving superquadratic and cubic rates for the upper and lower bound sequences, respectively. By strategically integrating this correction, we develop a globally convergent, non-monotone, and accelerated Dinkelbach algorithm-the first of its kind, to our knowledge. Under sufficient differentiability, the new method achieves an asymptotic average convergence order of at least the square root of 5 per iteration, surpassing the quadratic order of the original algorithm. Crucially, this acceleration is achieved while maintaining the key practicality of solving only a single subproblem per iteration.

Figures (6)

• Figure 1: The iterate $\alpha_{k+1}$ is defined as the root of the tangent line to $g$ at $\alpha_k$.
• Figure 2: The interval Dinkelbach method generates monotonic and convergent upper and lower bound sequences. Specifically, the upper bound sequence $\{\alpha_k\}$ is derived from the original Dinkelbach method \ref{['eq:upper bound sequence']}, while the lower bound sequence $\{\gamma_k\}$ is computed using the Secant method \ref{['eq:lower bound sequence']}.
• Figure 3: The accelerated interval Dinkelbach method only modifies the upper bound iterations, as shown in \ref{['eq:upper bound sequence next']}. In the illustrated specific case, the zero-crossing of the tangent line to $g$ at $\gamma_{k+1}$ may provide a better approximation to $\alpha^{*}$ than that from the tangent at $\alpha_{k}$.
• Figure 4: The accelerated method provides more accurate estimate of $x _{k+1}$ than the standard Newton method near the optimal solution $x^{*}$.
• Figure 5: In Case 1.1, the proposed method demonstrates better approximation than the Dinkelbach method. In Case 1.2, the approximation error is bounded by $M|\alpha_k - \alpha^*|$. The subsequent iterate $\alpha_{k+2}$ is determined by the tangents at $\alpha_{k}$ and $\alpha_{k-1}$.

Theorems & Definitions (41)

• Lemma 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof