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Abderrahim, Guemmaz. (1405). A Multi-Parameter Kernel Function for Primal–Dual Interior-Point Methods in Linear Optimization: Complexity Analysis and Numerical Performance. فصلنامه علمی زیست فناوری گیاهان زراعی, (), -. doi: 10.30473/coam.2026.75869.1339
Guemmaz Abderrahim. "A Multi-Parameter Kernel Function for Primal–Dual Interior-Point Methods in Linear Optimization: Complexity Analysis and Numerical Performance". فصلنامه علمی زیست فناوری گیاهان زراعی, , , 1405, -. doi: 10.30473/coam.2026.75869.1339
Abderrahim, Guemmaz. (1405). 'A Multi-Parameter Kernel Function for Primal–Dual Interior-Point Methods in Linear Optimization: Complexity Analysis and Numerical Performance', فصلنامه علمی زیست فناوری گیاهان زراعی, (), pp. -. doi: 10.30473/coam.2026.75869.1339
Abderrahim, Guemmaz. A Multi-Parameter Kernel Function for Primal–Dual Interior-Point Methods in Linear Optimization: Complexity Analysis and Numerical Performance. فصلنامه علمی زیست فناوری گیاهان زراعی, 1405; (): -. doi: 10.30473/coam.2026.75869.1339

A Multi-Parameter Kernel Function for Primal–Dual Interior-Point Methods in Linear Optimization: Complexity Analysis and Numerical Performance

Control and Optimization in Applied Mathematics
مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 29 خرداد 1405    اصل مقاله (435.83 K)
نوع مقاله: Research Article
شناسه دیجیتال (DOI): 10.30473/coam.2026.75869.1339
نویسنده
Guemmaz Abderrahim*
Laboratory of Science for Mathematics, Computer Science and Engineering Applications, Department of Mathematics, University Center of Barika, Amdoukal Road, Barika, 05001, Algeria
چکیده
This paper introduces a novel parameterized kernel function for primal–dual interior-point methods (IPMs) applied to the linear optimization (LO) problem. The proposed kernel, governed by two parameters — a base u > 1 and a term-count ℓ ∈ N \ {0} — employs a multi-exponent power-sum structure (i.e., a sum of power terms with geometrically spaced exponents (u, u2, ..., uℓ) that decouples local curvature near the central path from asymptotic growth far from it. This decoupling is impossible with single-parameter kernels such as self-regular or trigonometric variants. Under mild, verifiable conditions, the resulting IPM attains a worst-case iteration complexity of O(uℓ n((uℓ +1)/2uℓ ) log(n/ϵ )) for large-update strategies, and O(u2ℓ √n log(n/ϵ )) for small-update strategies. Choosing u = (log n/2 )1/ℓ recovers the best-known large-update bound O(√n log(n) log(n/ϵ )). Numerical experiments on benchmark LP instances with dimensions up to n = 1000 confirm that the ℓ = 1, u = 3.5 variant consistently outperforms the classical self-regular kernel of Peng et al. in both iteration count and CPU time. The study demonstrates that self-regularity is sufficient but not necessary for optimal complexity, broadening the theoretical foundations of kernel-based interior-point frameworks.

تازه های تحقیق
  • A novel multi-parameter kernel function φ_ℓ(x) is introduced for primal–dual interior-point methods (IPMs) applied to linear optimization (LO), governed by two parameters: a base u > 1 and a term-count ℓ ∈ ℕ\{0}.
  • The proposed kernel employs a multi-exponent power-sum structure that decouples local curvature near the central path from asymptotic growth far from it — a property unachievable by any single-parameter kernel such as self-regular or trigonometric variants.
  • Worst-case iteration complexity bounds of O(u^ℓ · n^((u^ℓ+1)/(2u^ℓ)) · log(n/ε)) and O(u^(2ℓ) · √n · log(n/ε)) are rigorously established for large-update and small-update strategies, respectively.
  • By selecting u = (log n / 2)^(1/ℓ), the method recovers the best-known O(√n log n · log(n/ε)) large-update complexity bound, demonstrating that self-regularity is sufficient but not necessary for optimal complexity.
  • Numerical experiments on benchmark LP instances with dimensions up to n = 1000 confirm that the ℓ = 1, u = 3.5 variant consistently outperforms the classical self-regular kernel of Peng et al. in both iteration count and CPU time.
کلیدواژه‌ها
Primal-dual interior-point algorithms؛ Large and small-update methods؛ Parameterized kernel function؛ Central path؛ Complexity bounds؛ Linear optimization
مراجع

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