In this paper, we obtain estimates for the contingent and adjacent derivatives of the epigraph of the marginal multifunction in parametric set-valued optimization. These estimates generalize some sensitivity results from scalar-valued optimization and provide new information in the setting of multiobjective nonlinear programming.

We study second-order contingent derivatives of perturbation mappings associated with families of set-valued optimization problems. After presenting some general inclusions, we focus on finite-dimensional problems whose data satisfy certain convexity assumptions. Our second-order results recover a number of known results from first order sensitivity analysis as special cases.

A chain rule is established for contingent and adjacent epiderivatives and hypoderivatives of compositions g○h, where h is assumed to be Hadamard directionally differentiable. Corollaries include a formula for the contingent and adjacent cones of an equality constraint set defined by a Hadamard directionally differentiable function. An analogous chain rule for second-order contingent and adjacent epiderivatives and hypoderivatives is also developed.

In this paper, characterizations of the existence of the directional derivative and second-order parabolic directional derivative of a locally Lipschitzian function are established. These characterizations involve the adjacent cone and second-order adjacent set of the graph of the function.

We obtain equivalences between weak Pareto solutions of vector optimization problems and solutions of vector variational inequalities involving generalized directional derivatives.

. Generalized properly efficient solutions of a vector optimization problem (VP) are defined in terms of various tangent cones and a generalized directional derivative. We study their basic properties and relationships and show that under certain conditions, a generalized properly efficient solution of (VP), defined by the adjacent cone, is a generalized Kuhn-Tucker properly efficient solution of (VP). Furthermore, using subgradients defined by closed convex tangent cones, we give a necessary optimality condition for a generalized properly efficient solution of (VP) defined by the adjacent cone.

We obtain an upper bound for the upper subderivative of the marginal function of an abstract parametric optimization problem when the objective function is lower semicontinuous. Moreover, we apply the result to a nonlinear program with right-hand side perturbations. As a result, we obtain an upper bound for the upper subderivative of the marginal function of a nonlinear program with right-hand side perturbations, which is expressed in “dual form” in terms of appropriate Lagrange multipliers. Finally, we present conditions which imply that the marginal function is locally Lipschitzian.

Upper and lower bounds are established for the Dini directional derivatives of the marginal function of a parametric mathematical program. In this program, the equality constraint functions are assumed to be strictly differentiable, but the objective and inequality constraint functions can belong to a large class of non-Lipschitzian functions. A nonsmooth version of the Mangasarian–Fromovitz constraint qualification is also assumed. The main tool in the proofs of these bounds is the calculus of tangent cones.

The first-order theory of generalized derivatives is now well developed. Significant progress toward a second-order theory has also been made. Two types of second-order directional derivatives appear to be particularly promising for applications to optimization: parabolic epiderivatives, which are well suited for necessary optimality conditions and satisfy standard calculus rules for a large class of functions; and the second-order epiderivatives of Rockafellar, which lead to general sufficient optimality conditions that are close to being necessary. In this paper, the properties of these two types of epiderivatives are surveyed and contrasted. Special results for C1,1 functions are examined in detail.

In nonlinear programming, sufficient conditions of orderm usually identify a special type of local minimizer, here termed a strict local minimizer of orderm. In this paper, it is demonstrated that, if a constraint qualification is satisfied, standard sufficient conditions often characterize this special sort of minimizer. The first- and second-order cases are treated in detail. Necessary conditions for weak sharp local minima of orderm, a larger class of local minima, are also presented.

In this paper, upper and lower bounds are established for the Dini directional derivatives of the marginal function of an inequality-constrained mathematical program with right-hand-side perturbations. A nonsmooth analogue of the Cottle constraint qualification is assumed, but the objective and constraint functions are not assumed to be differentiable, convex, or locally Lipschitzian. Our upper bound sharpens previous results from the locally Lipschitzian case by means of a subgradient smaller than the Clarke generalized gradient. Examples demonstrate, however, that a corresponding strengthening of the lower bound is not possible. Corollaries of this work include general criteria for exactness of penalty functions as well as information on the relationship between calmness and other constraint qualifications in nonsmooth optimization.

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 Rockafellar's epi-derivatives and Clarke generalized gradients.

Nonsmooth analysis has provided important new mathematical tools for the study of problems in optimization and other areas of analysis [1, 2, 6-12, 28]. The basic building blocks of this subject are local approximations to sets called tangent cones. Definition 1.1. Let E be a real, locally convex, Hausdorff topological vector space (abbreviated l.c.s.). A tangent cone (on E) is a mapping A:2 E × E → 2 E such that A(C, x) is a (possibly empty) cone for all nonempty C in 2 E and x in E. In the sequel, we will say that a tangent cone has a certain property (e.g. “A is closed” or “A is convex“) if A(C, x) has that property for all non-empty sets C and all x in C. (If A(C, x) is empty, it will be counted as having the property trivially.)

Under certain sufficient conditions for strict local optimality in a mathematical program, it is well known that a number of non-differentiable penalty functions are locally exact. With sufficient conditions involving the contingent derivative, it is shown that this local exactness is valid for programs whose objective and constraint functions need not be differentiable or even continuous.

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

The notion of subgradient, originally defined for convex functions, has in recent years been extended, via the "upper subderivative," to cover functions that are not necessarily convex or even continuous. A number of calculus rules have been proven for these generalized subgradients. This paper develops the finite-dimensional generalized subdifferential calculus for (strictly) lower semicontinuous functions under considerably weaker hypotheses than those previously used. The most general finite-dimensional convex subdifferential calculus results are recovered as corollaries. Other corollaries given include new necessary conditions for optimality in a nonsmooth mathematical program. Various chain rule formulations are considered. Equality in the subdifferential calculus formulae is proven under hypotheses weaker than the usual "subdifferential regularity" assumptions.

The tangential approximants most useful in nonsmooth analysis and optimization are those which lie "between" the Clarke tangent cone and the Bouligand contigent cone. A study of this class of tangent cones is undertaken here. It is shown that although no convex subcone of the contingent cone has the isotonicity property of the contingent cone, there are such convex subcones which are more "accurate" approximants than the Clarke tangent cone and possess an associated subdifferential calculus that is equally strong. In addition, a large class of convex subcones of the contingent cone can replace the Clarke tangent cone in necessary optimality conditions for a nonsmooth mathematical program. However, the Clarke tangent cone plays an essential role in the hypotheses under which these calculus rules and optimality conditions are proven. Overall, the results obtained here suggest that the most complete theory of nonsmooth analysis combines a number of different tangent cones.