Optimized first-order methods for smooth convex minimization.

Area of Science:

• Optimization Theory
• Numerical Analysis
• Computer Science

Background:

• First-order methods are crucial for solving large-scale optimization problems.
• Existing methods like Nesterov's fast gradient methods offer theoretical convergence rates.
• Previous numerical optimization methods for step coefficients are computationally expensive and memory-intensive for large N.

Purpose of the Study:

• To introduce novel, analytically derived optimized first-order methods for smooth unconstrained convex minimization.
• To improve upon the computational efficiency and convergence bounds of existing first-order algorithms.
• To provide practical alternatives for large-scale optimization challenges.

Main Methods:

• Analytical derivation of optimal first-order step coefficients.
• Development of new first-order algorithms based on these coefficients.
• Comparison of convergence bounds with existing methods, including Nesterov's fast gradient methods.

Main Results:

• Achieved a convergence bound that is analytically proven to be two times smaller than Nesterov's fast gradient methods.
• Developed optimized first-order methods with efficient forms similar to Nesterov's.
• Refined the numerical bound previously established by Drori and Teboulle.

Conclusions:

• The proposed optimized first-order methods offer significant theoretical and practical advantages for convex minimization.
• These methods provide a computationally efficient and analytically sound alternative for large-scale optimization.
• The findings advance the field of optimization by offering improved convergence rates and practical applicability.