A NOTE ON KKT-INVEXITY IN NONSMOOTH CONTINUOUS-TIME OPTIMIZATION VALERIANO

We introduce the notion of KKT-invexity for nonsmooth continuoustime nonlinear optimization problems and prove that this notion is a necessary and sufficient condition for every KKT solution to be a global optimal solution. AMS Subject Classification: 90C26, 90C30, 90C46.

Here X is a nonempty open convex subset of the Banach space L n ∞ [0, T ], φ : X → R, f(t, x(t)) = ξ(x)(t), g(t, x(t)) = γ(x)(t), ξ : X → Λ 1  1 [0, T ] and γ : X → Λ m 1 [0, T ], where L n ∞ [0, T ] denotes the space of all n-dimensional vector-valued Lebesgue measurable functions defined on the compact interval [0, T ] ⊂ R, which are essentially bounded, with norm k • k ∞ defined by kxk ∞ = max 1≤j≤n ess sup{|x j (t)|, 0 ≤ t ≤ T }, where for each t ∈ [0, T ], x j (t) is the jth component of x(t) ∈ R n and Λ m 1 [0, T ] denotes the space of all m-dimensional vector functions which are essentially bounded and Lebesgue measurable, defined on [0, T ], with the norm k • k 1 defined by The continuous problem was first investigated in 1953 by Bellman in [1].He studied a type of optimization problem, which is now known as a continuous-time linear problem.After that, various authors have studied more general continuous-time problems, regarding, for example, nonlinear problems.In [9], Zalmai obtained Karush-Kuhn-Tucker conditions of optimality.The results by Zalmai are natural generalizations of the KKT conditions in finite dimension.The nonsmooth problem was considered, for instance, in Brandão et al. [2] and Rojas-Medar et al. [7].A good list of references about continuous-time problems can be found in [9].
The notion of invexity was introduced in [4] by Hanson.This concept, which generalize convexity, is important on getting sufficient conditions of optimality.In the work [5], Martin relaxed invexity.He introduced the notion of KKT-invexity (in fact he called it KT-invexity), which is (like invexity) a sufficient condition for a KKT point to be a global minimizer.But what is interesting in the Martin's result is that KKT-invexity is also a necessary condition of optimality.Martin showed that every KKT point is a global minimizer if and only if the problem is KKT-invex.In [6], de Oliveira and Rojas-Medar obtained a similar result for the continuous-time problem, but with smooth functions.In this work we generalize the result of de Oliveira and Rojas-Medar for the nonsmooth case.

ASSUMPTIONS AND NOTATION
We assume that f and g i (the ith component of g), i ∈ I = {1, 2, . . ., m}, are real functions defined on V × [0, T ].
The functions t 7 → f (t, x(t)) and t 7 → g(t, x(t)) are assumed to be Lebesgue measurable and integrable for all x ∈ X.
We assume that, given a ∈ V , there exist an ε > 0 and a positive number k such that for all t ∈ [0, T ], and for all x, y ∈ a + εB (B denotes the unit ball of R n ) we have |f (t, x) − f (t, y)| ≤ kkx − yk.Similar hypotheses are assumed for g i , i ∈ I. Hence, f (t, •) and We denote by φ • (x; h) and g • i (t, x(t); h(t)), i ∈ I, the Clarke generalized directional derivative of φ and g i , i ∈ I, at x on the direction h, respectively.See Clarke [3] for more details.
Given x ∈ F, we denote by A i (x) the subset of [0, T ] where the ith constraint is active, i.e.,

INVEX CHARACTERIZATION OF KKT SOLUTIONS
In [5] Martin introduced the notion of KKT-invexity for mathematical programming problems and proved that every KKT point is a global minimizer if and only if the problem is KKT-invex.In this section we extend this concept for (CNP) and get a similar result.Definition 3.1.We say that (CNP) is Karush-Kuhn-Tucker invex (or KKTinvex) if there exists a function η : for all x, y ∈ F. Remark 3.2.By η(x, y) in (3.1) we mean the map from X × X into L n ∞ [0, T ] given by η(x, y)(t) = η(t, x(t), y(t)).
Remark 3.3.The definition of invexity differs from the KKT-invexity one by the requirement that g i (t, x(t)) − g i (t, y(t)) ≥ g • i (t, y(t); η(t, x(t), y(t)) a. e. in A i (y), i ∈ I, instead of (3.2).Definition 3.4.We say that y ∈ F is a Karush-Kuhn-Tucker solution (or KKT solution) of (CNP) if there exist Definition 3.5.We say that y ∈ F is a global optimal solution of (CNP) if φ(x) ≥ φ(y) for all x ∈ F.
In the next example we study a KKT-invex problem which is not an invex one, where hold the property that every KKT solution is a global optimal solution.So, this example shows that invexity, despite being sufficient, is not a necessary condition to hold such property.
It is clear that φ(x) ≥ φ(0) for all x ∈ F = {x ∈ L ∞ [0, 2] : x(t) ≥ 0 a.e. in [0, 2]}.Thus every KKT solution is a global optimal solution.This problem is not invex.Indeed, if we assume that it is invex we get a contradiction as follows.Suppose that the problem is invex.Then there exist η : Therefore this problem is KKT-invex.
Different of the finite dimensional case, here we need of a constraint qualification.Definition 3.7.We say that the constraint g satisfies (CQ) at y ∈ F if there do not exist u i ∈ L ∞ [0, T ], u i ≥ 0, i ∈ I, not all zero, such that Lemma 3.8.Let y ∈ F and assume that g satisfies (CQ) at y.If y is not a KKT solution of (CNP) then there exists h ∈ L n ∞ [0, T ] such that φ • (y; h) < 0, (3.9) Proof.
If the system in (3.9) and (3.10) does not have a solution, particularly, the system It follows from Corollary 3.1 on page 134 of [8], that there exist u 0 ∈ R and u i ∈ L ∞ [0, T ], i ∈ I, with u 0 ≥ 0 and u i (t) ≥ 0 a.e. in [0, T ], i ∈ I, not all zero, such that If u 0 = 0 we have a contradiction with the constraint qualification.Therefore u 0 > 0. Then dividing the expression in (3.11) by u 0 and defining Thus y is a KKT solution, what contradicts the hypothesis.Hence, there exists h ∈ L n ∞ [0, T ] satisfying (3.9) and (3.10). 2 Theorem 3.9.Assume that g satisfies (CQ) at each y ∈ F.Then, every KKT solution of (CNP) is a global optimal solution if and only if (CNP) is KKT-invex.
Proof.Necessity.Suppose that every KKT solution of (CNP) is a global optimal solution.Let x, y ∈ F. If φ(x) < φ(y), then y is not a global optimal solution, and so, by hypothesis, y is not a KKT solution of (CNP).It follows from Lemma 3.8 that there exists h ∈ L n ∞ [0, T ] satisfying (3.9) and (3.10).Set Because of (3.9) we know that  In the cases above we do not define η for x, y / ∈ F. But we can take η(t, x(t), y(t)) = 0 when x or y is not feasible.
Sufficiency.Suppose that (CNP) is KKT-invex.Let y ∈ F be a KKT solution of (CNP).It follows from (3.4) that λ i (t) = 0 a.e. in [0, T ] \ A i (y), i ∈ I. Then by (3.1), (3.2) and (3.5) we have An interesting open problem is to know if the relation (3.17) is still true when f (t, •) and g(t, •) are invex at y throughout [0, T ].When we have a finite sum instead of an integral, we verified that this is true.