Relaxation methods with step regulation for solving constrained optimization problems

Adaptive Conditional Gradient Method

“…where f (x) is a continuously differentiable pseudo-convex function satisfying the socalled Condition A (introduced in [8]) on a convex and closed subset D of Euclidean space R n . For solving this problem, we present a new efficient algorithm, which has the estimates of the rate of its convergence and allows one to adaptively control both the parameter of an ε-normalization of a descent direction and the step length.…”

“…Condition A describes a sufficiently broad class of functions A(μ, τ (x, y)) . It was shown in [8,14,15] that the class A(μ, x − y 2 ), in particular, is wider than C 1,1 (D)-the well-known class of functions whose gradients satisfy the Lipschitz condition on the convex set D ⊆ R n . By the way, we note that Lipschitzian properties of gradients for this class of functions have been sought as the favorable assumptions in the justification of the theoretical estimates of the convergence rate for the various modern differentiable optimization algorithms.…”

“…. The properties of elements of the sequence {θ k } were investigated, for instance, in [8,15]. Before considering the theorem on the convergence of the sequence {x k } generated by the algorithm to a solution of problem (1), we remind the following familiar fact related to convergence of some numeric sequence.…”

“…It should be pointed out that the sublinear convergence rate takes place under the following assumptions regarding the problem: 1) a feasible region is any convex and closed set; 2) an objective function is required to be satisfied with the so-called Condition A introduced in [8]. Let us note that this condition will be defined explicitly in Sect.…”

A Fully Adaptive Steepest Descent Method

“…The sublinear convergence rate takes place under the following conditions relating the original problem: 1) an objective function f (x) is pseudo-convex on some convex set D ⊆ R n (the set D may coincide, for instance, with the Lebesgue set (corresponding to a starting point of the iterates sequence) of the objective function or with the whole Euclidean space and etc), 2) the function f (x) is required to be satisfied to so-called Condition A introduced in [13]. We note that this condition will be defined explicitly below in Section 2 (see Definition 2.2).…”