High-Order Approximations for

Nonregular Optimization Problems  and

Necessary Conditions for Optimality

of Abnormal Trajectories

(joint research with Urszula Ledzewicz of Southern Illinois University in Edwardsville)

If X and Y are Banach spaces and F is an operator from X into Y, then the classical Lyusternik Theorem describes the tangent space to the set Q = {x in X : F(x) = 0} at x* in Q as the set of all h in X for which F'(x*)h = 0 provided the Frechet-derivative F'(x*) is onto. This theorem provides a key tool in deriving first-order necessary conditions for optimality in extremum problems with equality constraints. However, if the Frechet-derivative F'(x*) is not onto, the corresponding conditions take a trivial or degenerate form.

Based on earlier results by A.A. Avakov for the quadratic case, in our research we developed a scheme that allows high-order polynomial approximations to the set Q in the absense of a surjectivity condition on F'(x*). This condition is replaced by surjectivity assumptions on suitably constructed operators which use higher order derivatives to make up for the lack of surjectivity of the differential (called p-regularity in our papers). The corresponding approximations can then be used to derive stronger necessary conditions for optimality in a variety of optimization and optimal control problems where the classical Lyusternik theorem only yields trivial necessary conditions. 

Selected Publications: (all papers are joint publication with Urszula Ledzewicz)

• A High-Order Generalized Local Maximum Principle, SIAM J. on Control and Optimization, 38, (3), 2000, pp. 823-854

• High-Order Approximations and Generalized Necessary Conditions for Optimality, SIAM J. on Control and Optimization, 37, (1), 1999, pp. 32-52

• A High-Order Generalization of the Lyusternik theorem,  Nonlinear Analysis, 34, 1998, pp. 793-815,

• Second Order Conditions for Extremum Problems with Nonregular Equality Constraints, J. of Optimization Theory and Applications, 86, (1), 1995,  pp. 113-144,