In this paper, we analyze different first-order methods of smooth convex optimization employing inexact first-order information. We introduce the notion of an approximate first-order oracle. The list of examples of such an oracle includes smoothing technique, Moreau-Yosida regularization,...

Consider a class of variational inequality problems of finding <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${x^*\in S}$$</EquationSource> </InlineEquation>, such that <Equation ID="Equa"> <EquationSource Format="TEX">$$f(x^*)^\top (z-x^*)\geq 0,\quad \forall z\in S,$$</EquationSource> </Equation> where the underlying mapping f is hard to evaluate (sometimes its explicit form is unknown), and S has the following structure <Equation ID="Equb"> <EquationSource Format="TEX">$$S=\{x\in R^n\; | \;...</equationsource></equation></equationsource></equation></equationsource></inlineequation>

Consider a class of variational inequality problems of finding $${x^*\in S}$$ , such that $$f(x^*)^\top (z-x^*)\geq 0,\quad \forall z\in S,$$ where the underlying mapping f is hard to evaluate (sometimes its explicit form is unknown), and S has the following structure $$S=\{x\in R^n\; | \; Ax\le...