Comparative Analysis of Two Extragradient Algorithms for Solving Variational Inequalities

Variational inequalities represent a significant research topic in mathematical optimization, with broad applications across economics, engineering, and game theory. Among various algorithms for solving variational inequalities, the Extragradient Method has gained considerable attention due to its favorable convergence properties. This article compares the classical extragradient method with its modified version, discussing optimization approaches to help readers understand their differences, similarities, and applicable scenarios.

### 1. Classical Extragradient Method The classical extragradient method, proposed by Korpelevich, is an iterative algorithm primarily designed for solving monotone variational inequalities. Its core concept involves a two-step projection mechanism: first, a prediction step calculates the approximate gradient direction at the current point; then, a correction step adjusts the search direction based on the prediction result. This dual-projection approach ensures strong convergence performance under monotonicity assumptions. In code implementation, the algorithm typically requires two gradient evaluations per iteration. The basic pseudocode structure involves: 1. Initialize x₀ and step size α 2. For k = 0,1,2,...: - Prediction: yₖ = P_X(xₖ - αF(xₖ)) - Correction: xₖ₊₁ = P_X(xₖ - αF(yₖ)) where P_X denotes projection onto feasible set X, and F is the monotone operator. The method's strength lies in its robust theoretical convergence guarantees for Lipschitz continuous monotone problems. However, its computational complexity is relatively high since each iteration requires two projection or gradient updates, making it potentially inefficient for large-scale optimization problems.

### 2. Modified Extragradient Method The modified extragradient method optimizes the classical approach by reducing computational overhead and improving convergence speed. Common modification strategies include: Adaptive step size adjustment: Implementing dynamic step size selection to reduce unnecessary gradient computations and enhance iteration efficiency. Partial projection strategies: Employing approximate projections or partial variable updates instead of strict projection operations to lower computational costs in certain scenarios. Hybrid gradient techniques: Combining with other optimization methods (such as Nesterov's accelerated gradient) to accelerate the iteration process while maintaining convergence. In programming terms, the modified version might implement condition checks to avoid redundant computations. For example, a Python implementation could feature: - Adaptive step size: αₖ = line_search(xₖ, F, params) - Lazy projection: Only update components exceeding feasibility boundaries - Momentum integration: vₖ = βvₖ₋₁ + (1-β)F(yₖ) for acceleration The modified extragradient method significantly reduces computational burden while preserving convergence, making it particularly suitable for large-scale or sparse structure problems.

### 3. Algorithm Comparison The classical extragradient method boasts mature theoretical foundations and suits strictly monotone Lipschitz continuous variational inequalities, but carries higher computational costs. The modified version enhances computational efficiency through optimized iteration strategies while maintaining convergence, though it may require stronger assumptions or finer parameter tuning. In practical applications, algorithm selection depends on problem characteristics: classical algorithms may be preferable for theoretical analysis requiring high precision, while modified methods offer advantages for large-scale problems with limited computational resources.

In summary, both algorithms have distinct strengths and limitations. Understanding their core concepts and applicability conditions facilitates appropriate selection when solving variational inequalities.