E. R. Avakov

Directional Regularity and Metric Regularity

For general constraint systems in Banach spaces, we present the directional stability theorem based on the appropriate generalization of the directional regularity condition, suggested earlier in [A. V. Arutyunov and A. F. Izmailov, Math. Oper. Res., 31 (2006), pp. 526-543]. This theorem contains Robinsons stability theorem but does not reduce to it. Furthermore, we develop the related concept of directional metric regularity which is stable subject to small Lipschitzian perturbations of the constraint mapping, and which is equivalent to directional regularity for sufficiently smooth mappings. Finally, we discuss some applications in sensitivity theory.

Covering mappings and their applications to differential equations unsolved for the derivative

We continue to study the properties of covering mappings of metric spaces and present their applications to differential equations. To extend the applications of covering mappings, we introduce the notion of conditionally covering mapping. We prove that the solvability and the estimates for solutions of equations with conditionally covering mappings are preserved under small Lipschitz perturbations. These assertions are used in the solvability analysis of differential equations unsolved for the derivative.

Necessary Conditions for an Extremum in a Mathematical Programming Problem

For minimization problems with equality and inequality constraints, first-and second-order necessary conditions for a local extremum are presented. These conditions apply when the constraints do not satisfy the traditional regularity assumptions. The approach is based on the concept of 2-regularity; it unites and generalizes the authors’ previous studies based on this concept.

Necessary optimality conditions for constrained optimization problems under relaxed constraint qualifications

We derive first- and second-order necessary optimality conditions for set-constrained optimization problems under the constraint qualification-type conditions significantly weaker than Robinson’s constraint qualification. Our development relies on the so-called 2-regularity concept, and unifies and extends the previous studies based on this concept. Specifically, in our setting constraints are given by an inclusion, with an arbitrary closed convex set on the right-hand side. Thus, for the second-order analysis, some curvature characterizations of this set near the reference point must be taken into account.

Stability Theorems for Estimating the Distance to a Set of Coincidence Points

Coincidence points of two set-valued mappings of metric spaces are analyzed. Uniform estimates are obtained for the distance to the set of coincidence points and to the set of intersections of the graphs of two set-valued mappings. Sufficient conditions for the existence of double fixed points are derived as a consequence of the results obtained. In addition, estimates are obtained for the distance between the sets of coincidence points of two pairs of set-valued mappings.

An Implicit Function Theorem for Inclusions Defined by Close Mappings

The paper deals with the question of the solvability of inclusions F(x, σ) ∈ Q for mappings F close, in some metrics, to a given mapping ̂F.

Pontryagin maximum principle, relaxation, and controllability

The relations between the necessary minimum conditions in an optimal control problem (Pontryagin maximum principle), the minimum conditions in the corresponding relaxation (weakened) problem, and sufficient conditions for the local controllability of the controlled system specifying the constraints in the original formulation are studied. An abstract optimization problem that models the basic properties of the optimal control problem is considered.

Exact penalties for optimization problems with 2-regular equality constraints

A new first-order sufficient condition for penalty exactness that includes neither the standard constraint qualification requirement nor the second-order sufficient optimality condition is proposed for optimization problems with equality constraints.