15,064 research outputs found
The Forward-Backward-Forward Method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces
Tseng's forward-backward-forward algorithm is a valuable alternative for
Korpelevich's extragradient method when solving variational inequalities over a
convex and closed set governed by monotone and Lipschitz continuous operators,
as it requires in every step only one projection operation. However, it is
well-known that Korpelevich's method converges and can therefore be used also
for solving variational inequalities governed by pseudo-monotone and Lipschitz
continuous operators. In this paper, we first associate to a pseudo-monotone
variational inequality a forward-backward-forward dynamical system and carry
out an asymptotic analysis for the generated trajectories. The explicit time
discretization of this system results into Tseng's forward-backward-forward
algorithm with relaxation parameters, which we prove to converge also when it
is applied to pseudo-monotone variational inequalities. In addition, we show
that linear convergence is guaranteed under strong pseudo-monotonicity.
Numerical experiments are carried out for pseudo-monotone variational
inequalities over polyhedral sets and fractional programming problems
On the Convergence of Classical Variational Inequality Algorithms
In this paper, we establish global convergence results for projection and relaxation algorithms for solving variational inequality problems, and for the Frank-Wolfe algorithm for solving convex optimization problems defined over general convex sets. The analysis rests upon the condition of f-monotonicity,which we introduced in a previous paper, and which is weaker than the traditional strong monotonicity condition. As part of our development, we provide a new interpretation of a norm condition typically used for establishing convergence of linearization schemes. Applications of our results arize in uncongested as well as congested transportation networks
Beyond Monotone Variational Inequalities: Solution Methods and Iteration Complexities
In this paper, we discuss variational inequality (VI) problems without
monotonicity from the perspective of convergence of projection-type algorithms.
In particular, we identify existing conditions as well as present new
conditions that are sufficient to guarantee convergence. The first half of the
paper focuses on the case where a Minty solution exists (also known as Minty
condition), which is a common assumption in the recent developments for
non-monotone VI. The second half explores alternative sufficient conditions
that are different from the existing ones such as monotonicity or Minty
condition, using an algorithm-based approach. Through examples and convergence
analysis, we show that these conditions are capable of characterizing different
classes of VI problems where the algorithms are guaranteed to converge.Comment: 29 page
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