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Arxiv / №5.1 · 2026 / Maqola
01.00.00 – FIZIKA-MATEMATIKA FANLARI

SHARTSIZ OPTIMALLASHTIRISH USULLARINI TAQQOSLASH

Annotatsiya
O'ZBEKCHA

Mazkur maqolada shartsiz optimallashtirish usullari nazariy va amaliy jihatdan taq-qoslanadi. Gradient, Nyuton, kvazi-Nyuton (BFGS, L-BFGS) hamda konjugat gradient algoritmlarining yaqinlashuv tezligi, hisoblash murakkabligi va barqarorlik xossalari tahlil qilinadi. Tadqiqot jarayonida turli tipdagi test funksiyalari asosida empirik sinovlar o‘tkazilib, algoritmlarning real samaradorligi baho-lanadi. Natijalar shuni ko‘rsatadiki, usul tanlovi funksiyaning tuzilishi va o‘lchamiga bog‘liq bo‘lib, kvazi-Nyuton algoritmlari tezlik va resurs tejamkorligi o‘rtasida optimal muvozanatni ta’minlaydi.

Kalit so'zlar: shartsiz optimallashtirish, gradient usuli, Nyuton usuli, BFGS, L-BFGS, konjugat gra-dient, yaqinlashuv tezligi, hisoblash murakkabligi.
РУССКИЙ

В статье проводится сравнительный анализ методов безусловной оптимизации с теоретической и практической точек зрения. Рассматриваются градиентный метод, метод Ньютона, квазиньютоновские алгоритмы (BFGS, L-BFGS) и метод сопряжённых градиентов. Исс-ледуется скорость сходимости, вычислительная сложность и устойчивость алгоритмов. На ос-нове тестовых функций различного типа проведены численные эксперименты, позволившие оце-нить эффективность методов в задачах различной размерности.

Ключевые слова: безусловная оптимизация, градиентный метод, метод Ньютона, BFGS, L-BFGS, вычислительная сложность.
ENGLISH

This paper presents a comparative analysis of unconstrained optimization methods from both theoretical and practical perspectives. The gradient method, Newton’s method, quasi-Newton
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algorithms (BFGS, L-BFGS), and the conjugate gradient method are examined. Their convergence rate, computational complexity, and stability properties are analyzed. Numerical experiments based on benchmark test functions of different types and dimensions are conducted to evaluate practical perfor-mance. The results indicate that the choice of an optimization algorithm depends on the structure of the objective function and the problem dimension. Quasi-Newton methods demonstrate an effective balance between convergence speed and computational efficiency.

Keywords: unconstrained optimization, gradient method, Newton’s method, BFGS, L-BFGS, con-jugate gradient, convergence rate, computational complexity.
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