One the Lagrange multipliers rule in problems with the equality type constraints, given by quasi-differentiable functions

In recent years, there has been a steadily growing interest in the study of extremal problems with parameters that do not satisfy the standard smoothness assumptions. This is due to both theoretical needs and important practical applications in economics, technology, physics, and other sciences. Rough objects naturally arise in a several areas of systems analysis, nonlinear mechanics, and control processes. In the theory of extremal problems, the main interest is the behavior of functions in the vicinity of points where a local extremum is attained. The local behavior of nonsmooth functions is described by subgradients, which are analogs of the derivative of differentiable functions. Using the concepts of subdifferential and subgradient F. Clarke proved the Lagrange multiplier rule in mathematical programming problems with constraints of the type of equalities and inequalities defined by locally Lipschitz functions. However, there are subclasses of locally Lipschitz functions, the simplest examples of which show that the necessary conditions for an extremum obtained by F. Clarke are rather crude and do not allow one to discard obviously non-optimal points. Such a subclass of nonsmooth functions is the subspace of quasi-differentiable functions. In this article, using the Eckland variational principle, we obtain the Lagrange multiplier rule in terms of quasi-differentials. It is shown by examples that this condition is stronger than the necessary condition of F. Clarke.

Characterizing the Lagrange multiplier rule in nonconvex set-valued optimization

In this article, by using the notions of contingent derivative, contingent epiderivative and generalized contingent epiderivative, we obtain some characterizations of the Lagrange multiplier rule at points which are not necessarily local minima.

Stress response in the presence of local constraints on the deformation

Chapter 6 develops the structure of the stress response in the presence of material constraints on the basis of the Lagrange-multiplier rule. Examples given include incompressibility and inextensibility. Further applications are examined in Problems. The concepts are presented in more detail than one usually finds in the text and monograph literatures.

Generalized Lagrange multiplier rule for non-convex vector optimization problems

In this paper a non-convex vector optimization problem among infinite-dimensional spaces is presented. In particular, a generalized Lagrange multiplier rule is formulated as a necessary and sufficient optimality condition for weakly minimal solutions of a constrained vector optimization problem, without requiring that the ordering cone that defines the inequality constraints has non-empty interior. This paper extends the result of Donato (J. Funct. Analysis261 (2011), 2083–2093) to the general setting of vector optimization by introducing a constraint qualification assumption that involves the Fréchet differentiability of the maps and the tangent cone to the image set. Moreover, the constraint qualification is a necessary and sufficient condition for the Lagrange multiplier rule to hold.