Abstract
The paper concerns with novel first-order methods for monotone variational inequalities. They use a very simple linesearch procedure that takes into account a local information of the operator. Also, the methods do not require Lipschitz continuity of the operator and the linesearch procedure uses only values of the operator. Moreover, when the operator is affine our linesearch becomes very simple, namely, it needs only simple vector–vector operations. For all our methods, we establish the ergodic convergence rate. In addition, we modify one of the proposed methods for the case of a composite minimization. Preliminary results from numerical experiments are quite promising.
Highlights
This paper considers a problem of the variational inequality in a general form find x∗ ∈ E : F(x∗), x − x∗ + g(x) − g(x∗) ≥ 0 ∀x ∈ E, (1)
We consider quite a general problem, our discussion presented below consists of two separate parts devoted to the variational inequality problems and optimization problems
B, c are generated uniformly randomly from (0, 1)d, (−1, 1)m, and (−1, 1)d respectively. For this problem we study the performance of Alg.2, Alg.3, FB method with the linesearch from [16], FBF and FISTA [4]
Summary
Where E is a finite-dimensional vector space, F : E → E is a monotone operator and g : E → This is an important problem that has a variety of theoretical and practical applications [21,22,28]. The main iteration step of the proposed methods is given as follows: yn = xn + τn(xn − xn−1), xn+1 = proxλng(xn − λnF(yn)), where we define τn, λn and yn from local properties of F(yn). For this in each iteration we run some simple linesearch procedure.
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