Vsevolod I. Ivanov

Second-order Kuhn-Tucker invex constrained problems

A new notion of a second-order KT invex problem (P) with inequality constraints is introduced in this paper. This class of problems strictly includes the KT invex ones. Some properties of the second-order KT invex problems are presented. For example, (P) is second-order KT invex if and only if each point, which satisfies the second-order Kuhn-Tucker necessary optimality conditions, is a global minimizer. A problem with quasiconvex data is (second-order) KT invex if and only if it is (second-order) KT pseudoconvex.

Higher-order Pseudoconvex Functions

In terms of n-th order Dini directional derivative with n positive integer we define n-pseudoconvex functions being a generalization of the usual pseudoconvex functions. Again with the n-th order Dini derivative we define n-stationary points, and prove that a point x 0 is a global minimizer of a n-pseudoconvex function f if and only if x 0 is a n-stationary point of f. Our main result is the following. A radially continuous function f defined on a radially open convex set in a real linear space is n-pseudoconvex if and only if f is quasiconvex function and any n-stationary point is a global minimizer. This statement generalizes the results of Crouzeix, Ferland, Math. Program. 23 (1982), 193–205, and Komlosi, Math. Program. 26 (1983), 232–237. We study also other aspects of the n-pseudoconvex functions, for instance their relations to variational inequalities.

Second-order invex functions in nonlinear programming

We introduce a notion of a second-order invex function. A Fréchet differentiable invex function without any further assumptions is second-order invex. It is shown that the inverse claim does not hold. A Fréchet differentiable function is second-order invex if and only if each second-order stationary point is a global minimizer. Two complete characterizations of these functions are derived. It is proved that a quasiconvex function is second-order invex if and only if it is second-order pseudoconvex. Further, we study the nonlinear programming problem with inequality constraints whose objective function is second-order invex. We introduce a notion of second-order type I objective and constraint functions. This class of problems strictly includes the type I invex ones. Then we extend a lot of sufficient optimality conditions with generalized convex functions to problems with second-order type I invex objective function and constraints. Additional optimality results, which concern type I and second-order type I invex data are obtained. An answer to the question when a kernel, which is not identically equal to zero, exists is given.

On the optimality of some classes of invex problems

In this paper we define two notions: Kuhn–Tucker saddle point invex problem with inequality constraints and Mond–Weir weak duality invex one. We prove that a problem is Kuhn–Tucker saddle point invex if and only if every point, which satisfies Kuhn–Tucker optimality conditions forms together with the respective Lagrange multiplier a saddle point of the Lagrange function. We prove that a problem is Mond–Weir weak duality invex if and only if weak duality holds between the problem and its Mond–Weir dual one. Additionally, we obtain necessary and sufficient conditions, which ensure that strong duality holds between the problem with inequality constraints and its Wolfe dual. Connections with previously defined invexity notions are discussed.

Optimality Conditions and Characterizations of the Solution Sets in Generalized Convex Problems and Variational Inequalities

We derive necessary and sufficient conditions for optimality of a problem with a pseudoconvex objective function, provided that a finite number of solutions are known. In particular, we see that the gradient of the objective function at every minimizer is a product of some positive function and the gradient of the objective function at another fixed minimizer. We apply this condition to provide several complete characterizations of the solution sets of set-constrained and inequality-constrained nonlinear programming problems with pseudoconvex and second-order pseudoconvex objective functions in terms of a known solution. Additionally, we characterize the solution sets of the Stampacchia and Minty variational inequalities with a pseudomonotone-star map, provided that some solution is known.

Second-order optimality conditions for inequality constrained problems with locally Lipschitz data

In this paper we obtain second-order optimality conditions of Fritz John and Karush–Kuhn–Tucker types for the problem with inequality constraints in nonsmooth settings using a new second-order directional derivative of Hadamard type. We derive necessary and sufficient conditions for a point

On a theorem due to Crouzeix and Ferland

{\bar x}

Characterizations of pseudoconvex functions and semistrictly quasiconvex ones

to be a local minimizer and an isolated local one of order two. In the primal necessary conditions we suppose that all functions are locally Lipschitz, but in all other conditions the data are locally Lipschitz, regular in the sense of Clarke, Gâteaux differentiable at