[Paper Review] Conditional extragradient algorithms for solving variational inequalities

Yunier Bello-Cruz, R. Díaz Millán+1 more|arXiv (Cornell University)|Nov 17, 2014

Optimization and Variational Analysis31 references4 citations

TL;DR

This paper introduces conditional extragradient algorithms for solving constrained variational inequality problems by incorporating nonzero normal vectors of the feasible set to enhance convergence. By using two Armijo-type linesearch strategies—one based on boundary projections and another on feasible directions—the method achieves convergence under mild assumptions, including continuity of the operator and a weaker condition than pseudomonotonicity, thus broadening the class of solvable problems beyond classical extragradient methods.

ABSTRACT

In this paper, we generalize the classical extragradient algorithm for solving variational inequality problems by utilizing nonzero normal vectors of the feasible set. In particular, conceptual algorithms are proposed with two different linesearchs. We then establish convergence results for these algorithms under mild assumptions. Our study suggests that nonzero normal vectors may significantly improve convergence if chosen appropriately.

Motivation & Objective

• To develop more efficient variants of the extragradient algorithm for solving constrained variational inequality problems (VIPs).
• To leverage nonzero normal vectors of the feasible set to improve convergence behavior in iterative methods.
• To establish convergence under weaker assumptions than pseudomonotonicity, such as continuity and solution existence.
• To generalize classical extragradient schemes by incorporating conditional linesearch strategies that use normal vector information.

Proposed method

• Introduces two conceptual conditional extragradient algorithms that use nonzero normal vectors to define halfspaces containing the solution set.
• Employs an Armijo-type linesearch on the boundary of the feasible set to determine stepsize via projection-based linesearch in an inner loop.
• Uses an alternative Armijo-type linesearch along feasible directions, requiring only one projection per outer iteration.
• Applies a conditional linesearch strategy that dynamically selects stepsize based on descent-like conditions involving the normal vector and operator evaluation.
• Formulates the extragradient scheme with three steps: prediction (z^k), convex combination (y^k), and final projection (x^{k+1}) onto the feasible set.
• Utilizes two distinct linesearch mechanisms: one based on the norm of the difference in operator values and another based on inner product descent conditions.

Experimental results

Research questions

• RQ1 Can nonzero normal vectors of the feasible set be effectively used to improve convergence in extragradient-type algorithms for variational inequalities?
• RQ2 How can linesearch strategies be adapted to incorporate normal vector information while maintaining convergence under weak assumptions?
• RQ3 What is the impact of using conditional linesearches based on normal vectors compared to standard extragradient methods?
• RQ4 Can convergence be established without requiring Lipschitz continuity or strong monotonicity of the operator?
• RQ5 To what extent does the proposed method generalize existing extragradient algorithms in terms of applicability to non-monotone or weakly pseudomonotone operators?

Key findings

• The proposed algorithms achieve convergence for variational inequality problems under the assumptions of operator continuity and existence of at least one solution, without requiring pseudomonotonicity.
• The use of nonzero normal vectors enables the construction of halfspaces that contain the solution set, facilitating long steplengths and improved convergence behavior.
• The Armijo-type linesearch based on boundary projections ensures convergence under monotonicity and continuity, even without Lipschitz continuity.
• The alternative linesearch strategy based on feasible directions requires only one projection per outer iteration, improving computational efficiency.
• Numerical illustrations show that incorporating normal vectors accelerates convergence in early iterations, suggesting potential for hybrid schemes.
• The method broadens the class of solvable VIPs by relaxing the need for strong monotonicity or Lipschitz continuity.