Doug Ward

Calculus for parabolic second-order derivatives

In this paper, a calculus for two second-order directional derivatives is presented and then applied in the development of second-order necessary optimality conditions for a nonsmooth mathematical program. The formulae of this calculus, which include rules for sums, pointwise maxima, and certain compositions of functions, are valid for a large class of non-Lipschitzian functions and in fact subsume the sharpest results of the calculus of first-order upper and lower epiderivatives. Two methods are utilized in the derivation of these formulae. One centers around the concept of metric regularity, while the other relies upon the use of recession cones and ‘interior tangent sets’.

Isotone tangent cones and nonsmooth optimization

A simple but quite general method of formulating necessary optimality conditions for an abstract mathematical program is presented. This method centers around tangent cones which are isotone with respect to inclusion. The optimality conditions derived by this method generalize standard necessary conditions of differentiate, convex, and nonsmooth mathematical programming.

A constraint qualification in quasidifferentiable programming

One goal in quasid if Terentiable optimization is the development of optimality conditions whose hypotheses are independent of the particular choice of quasidifferentials. One such hypothesis, introduced by Demyanov and Rubinov, involves the concept of a pair of convex sets being “in a general position”. In this paper, a simple condition that implies the general position hypothesis is presented. This condition is also shown to be a constraint qualification for non-asymptotic Kuhn-Tucker conditions for a quasidifferentiable program.

A chain rule for parabolic second-order epiderivatives

Just as first-order directional derivatives can be associated with concepts of tangent cone, so second-order directional derivatives of parabolic type can be naturally and profitably associated with second-order tangent sets. In this paper, a chain rule is presented for second-order directional derivatives whose corresponding tangent sets satisfy a short list of properties. This chain rule subsumes and sharpens previous results from the calculus of first- and second-order directional derivatives. Corollaries include second-order necessary optimality conditions for nondifferentiable programs.

Chain rules for nonsmooth functions

Abstract A systematic method is presented for the derivation of chain rules for compositions of functions F ° f , where F is nondecreasing. This method is valid for directional derivatives and subgradients associated with any tangent cone having a short list of properties. Some major special cases are examined in detail; in particular, calculus rules are derived for Rockafellars epi-derivatives and Clarke generalized gradients.

Upper DSL approximates and nonsmooth optimization

For a tangent cone A, an extended-real-valued function f is said to admit an “A upper DSL approximate” at x if its “A directional derivative” at x is majorized by a difference of lower semicontinuous sublinear functions. By means of such approximates we establish necessary optimality conditions of Fritz John and Kuhn-Tucker type for a nonsmooth, inequality-constrained mathematical program. Optimality conditions involving the quasidifferentials of Demyanov, the upper convex approximates of Pshenichnyi, and the upper DSL approximates of A, Shapiro are among the special cases of these general optimality conditions.

Which subgradients have sum formulas

On donne une formule de somme generale valable pour un cone tangent avec une courte liste de proprietes