A Full Rank Condition for Continuous-Time Optimization Problems with Equality and Inequality Constraints

Abstract

First and second order necessary optimality conditions of Karush-Kuhn-Tucker type are established for continuous-time optimization problems with equality and inequality constraints. A full rank type regularity condition along with an uniform implicit function theorem are used in order to achieve such necessary conditions.

Duality Theory in Continuous-Time Linear Optimization

“…Let us note that the existence of a constant K A > 0 such that det(A(t) A(t)) ≥ K A a.e. in [0, T ] implies that the full rank assumption in Monte and de Oliveira [4] is satisfied. It follows from Theorem 4.1 in [4], that there exist u ∈…”

“…Optimality conditions for (CLP) can be found, for example, in de Oliveira [3] and in Monte and de Oliveira [4]. Particularly, in [4] we can find optimality conditions for a more general case: problems with nonlinear equality and inequality constraints. However, the constraint qualifications used in [3] and [4] are different.…”

“…Particularly, in [4] we can find optimality conditions for a more general case: problems with nonlinear equality and inequality constraints. However, the constraint qualifications used in [3] and [4] are different. In what follows, we reproduce the optimality conditions for (CLP) developed in [3].…”

“…Proof. If (CDP) has an optimal solution, sayw, it follows from Theorem 4.1 in Monte and de Oliveira [4], that there exist u ∈ L ∞ ([0, T ]; R n ) and v ∈ L ∞ ([0, T ]; R m ) such that b(t)+A(t)u(t)−v(t) = 0, v(t) ≥ 0 and v(t) w(t) = 0 a.e. in [0, T ] (the existence of K A > 0 such that det(A(t) A(t)) ≥ K A a.e.…”

“…in [0, T ] (the existence of K A > 0 such that det(A(t) A(t)) ≥ K A a.e. in [0, T ] implies that the full rank assumption in [4] is satisfied). The pair (z,ȳ) = (−u, v) satisfies (4).…”