In this paper we present a method for determining some variations of a singular trajectory of an affine control system. These variations provide necessary optimality conditions which may distinguish between maximizing and minimizing problems. The generalized Legendre-Clebsch conditions are an example of these type of conditions.

where C is a convex closed cone in the Euclidean space IR, f : IR → IR and G : IR → Y is a mapping from IR into the space Y := S of m × m symmetric matrices. We refer to the above problem as a nonlinear semidefinite programming problem. In particular, if C = IR, the objective function is linear, i.e. f(x) := ∑n i=1 bixi, and the constraint mapping is affine, i.e. G(x) := A0 + ∑n i=1 xiAi where ...

The multiobjective bilevel program is a sequence of two optimization problems, with the upper-level problem being multiobjective and the constraint region of the upper level problem being determined implicitly by the solution set to the lower-level problem. In the case where the Karush-Kuhn-Tucker (KKT) condition is necessary and sufficient for global optimality of all lower-level problems near...

A basic relationship is derived between generalized subgradients of a given function, possibly nonsmooth and nonconvex, and those of a second function obtained from it by partial conjugation. Applications are made to the study of multiplier rules in finite-dimensional optimization and to the theory of the Euler-Lagrange condition and Hamiltonian condition in nonsmooth optimal control.

We consider linear optimization over a nonempty convex semialgebraic feasible region F . Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique “active” manifold, around which F is “partly smooth,” and the second-order sufficient conditions hold. Perturbing the objective results in smooth variation of...

We describe a generalization to Optimal Control of the theory of envelopes of the classical Calculus of Variations. This generalization extends our previous work, by allowing the use of \quasi-extremal" trajectories, i.e. trajectories that satisfy all the conditions of the Pontryagin Maximum Principle except for the fact that the sign of the constant 0 that appears in the Hamiltonian multiplyin...

We prove some fundamental properties of mu-differentiable functions. A new notion of local minimizer and maximizer is introduced and several extremum conditions are formulated using the language of nonstandard analysis.

In this paper we study optimization problems with infinite many inequality constraints on a Banach space where the objective function and the binding constraints are locally Lipschitz‎. ‎Necessary optimality conditions and regularity conditions are given‎. ‎Our approach are based on the Michel-Penot subdifferential.

in this paper, using clarke’s generalized directional derivative and di-invexity we introduce new concepts of nonsmooth k-α-di-invex and generalized type i univex functions over cones for a nonsmooth vector optimization problem with cone constraints. we obtain some sufficient optimality conditions and mond-weir type duality results under the foresaid generalized invexity and type i cone-univexi...

the objective of this article is to derive the necessary optimality conditions, known as pontryagin's minimum principle, for fuzzy optimal control problems based on the concepts of differentiability and integrability of a fuzzy mapping that may be parameterized by the left and right-hand functions of its $alpha$-level sets.