Error Bound and Convergence Analysis of Matrix Splitting Algorithms for the Affine Variational Inequality Problem
By Zhi-Quan Tom Luo, Paul Tseng. 1992
Consider the affine variational inequality problem. It is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point. This bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem. This latter result improves upon a recent result of Luo and Tseng that further assumes the problem to be monotone.
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Citations in the corpus
- Convergence Rates in Forward-Backward Splitting (1997, background): “Renaud in his thesis [32] got a contraction rate based on strong monotonicity constants for both F and F−1.”
- On M-stationary points for mathematical programs with equilibrium constraints (2005, unknown): “See [16, 10, 21] for proofs using this approach.”
- Error bounds and convergence analysis of feasible descent methods: a general approach (1993, methodology): “condition. It is known that this error bound holds when f is strongly convex [57] or when f is quadratic and X is polyhedral [ 37 ] or when f is a certain convex (not necessarily strictly convex) function and X is polyhedral [35, 38].”
- Randomized Methods for Linear Constraints: Convergence Rates and Conditioning (2008, unknown): “This approach generalizes broadly, for example, to gradient projection and reduced gradient schemes and matrix splitting algorithms (see Luo and Tseng [28, 29]), to nonconvex minimization (see Luo and Tseng [31]), to composite and linearly constrained convex minimization (see Luo and Tseng [27, 30])”
- Calculus of the Exponent of Kurdyka–Łojasiewicz Inequality and Its Applications to Linear Convergence of First-Order Methods (2016, methodology): “It is known that this property is satisfied for many choices of h and P , and we refer to [28, 29, 30, 40, 41, 46] and references therein for more detailed discussions.”