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Necessary Optimality Conditions for Continuous-Time Optimization Problems with Equality and Inequality Constraints

The paper is devoted to obtain first and second order necessary optimality conditions 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 establish such necessary conditions.


Introduction
We are concerned with the general nonlinear continuous-time optimization problem with equality and inequality constraints in the form maximize P (z) = where φ : R n × [0, T ] → R, h : R n × [0, T ] → R p and g : R n × [0, T ] → R m . All vectors are column vectors, unless transposed when they will be denoted by a prime, and all integrals are in the Lebesgue sense.
Continuous-time problems arise often in the literature and were first proposed by Bellman [1,2] in his studies of some dynamical models of production and inventory called "bottleneck processes", which gave rise to continuous-time linear programming. Such problems can be posed as maximize P (z) = T 0 a ′ z(t)dt subject to z(t) ≥ 0, 0 ≤ t ≤ T, where B and K are m × n matrices, a is a n-vector and c is a m-vector. Considering a certain dynamic generalization of an ordinary linear programming problem, he formulated a corresponding dual problem, established a weak duality theorem, and suggested some computational procedures. Subsequently, Bellman's formulation and duality theory were substantially extended to more general forms of continuous-time linear programming problems, and also to certain classes of continuous-time nonlinear programming problems. For a summary of the results pertaining to duality theory in continuous-time programming and a fairly extensive list of relevant references the reader is referred to Zalmai [3].
Optimality conditions of the Karush-Kuhn-Tucker type were first considered in continuous-time programming by Hanson and Mond [4] for the following linearly constrained nonlinear program: maximize P (z) = T 0 φ(z(t))dt subject to z(t) ≥ 0, 0 ≤ t ≤ T, where B(t) is an m × n matrix piece-wise continuous on [0, T ], c(t) ∈ R n is piece-wise continuous on [0, T ], K(s, t) is a m × n matrix piece-wise continuous on [0, T ] × [0, T ] and φ is a given concave scalar function twice continuously differentiable. For this purpose, certain positivity conditions on B(t), c(t) and K(s, t) were imposed and the objective function was linearized in order to apply a extended version of Levinson's linear duality result [5]. A duality theorem for the nonlinear problem under consideration in then established, and the Karush-Kuhn-Tucker conditions are deduced as a consequence of this nonlinear duality theorem. In this way, many other authors obtained necessary and sufficient conditions for continuous-time problems with non-linear inequality constraints, for example, Farr and Hanson [6,7], Reiland and Hanson [8].
Roughly speaking, optimality conditions for continuous-time nonlinear programming problems have been obtained by direct methods. In Abrham and Buie [9] a certain regularity assumption is used to establish the Kuhn-Tucker conditions for a class of convex programming problems. Reiland [10], employing a continuous-time version of Zangwill's constraint qualification [11] introduced in [8], and an infinite-dimensional form of Farkas' theorem [12], established optimality conditions and duality relations for differentiable continuous-time programs.
Brando, Rojas-Medar and Silva, tackled nonsmooth continuous-time optimization problems, first in [14] where sufficient conditions were obtained and then in [13] which refers to necessary conditions. In [15], de Oliveira and Rojas-Medar generalized concepts of the KKT-invexity and WD-invexity introduzed by Martin [16] for mathematical programming problems, proving that the notion of KKT-invexity is a necessary and sufficient condition for global optimality of a Karush-Kuhn-Tucker point and that the notion of WD-invexity is a necessary and sufficient condition for weak duality. The same authors established KKT-invexity for nonsmooth continuous-time programming problems [17]. The multiobjective case was considered in [18]. de Oliveira [19] also studied multiobjective continuous-time programming problems, but without imposing any differentiability assumption. Saddlepoint type optimality conditions, duality theorems as well as results on the scalarization method were presented. The concept of pre-invexity was utilized.
In the formulation given in (1), where equality and inequality constraints are present and the feasible solutions belong to L ∞ ([0, T ]; R n ), necessary optimality conditions are not found in the literature. We believe this is due to the fact that even a few years ago, a crucial tool for the treatment of equality constraints was not available: the uniform implicit function theorem. When fixing t, we can apply the classical one. But the implicit function thereby obtained may not have good properties, such as measurebility for example, with respect to t. Such result only appear in 1997 in paper by Pinho and Vinter [20]. On the other hand, in L ∞ ([0, T ]; R n ) and with inequality constraints only, there is a vast literature, as cited above. In the case of formulations in other spaces, we cite Zalmai [21], for instance, where the feasible solutions are in a Hilbert space.
In this work, by means of the uniform implicit function theorem presented by Pinho and Vinter [20] and the use of a full rank type condition, we obtain first and second order necessary optimality conditions for continuous-time programming problems with equality and inequality constraints. The paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we consider problems with equality constraints only. Finally, in Section 4, the general case is treated.
Definition 2.1. We say thatz ∈ Ω is a local optimal solution of (1) if there exists ǫ > 0 such that P (z) ≥ P (z) for all z ∈ Ω satisfying z ∈z + ǫB, whereB denotes the closed unit ball with center at the origin.
Reiland ([10], Theorem 2) provide a condition for obtaining directions that, starting at a given point z, the values of objective function are increased.
, then there exists a number σ > 0 such that Let {F a : R n → R n | a ∈ A} be a family of maps parameterized by points a in a subset A ⊂ R n . If ∇F a is nonsingular at some point x 0 for all a ∈ A, we know by the classic inverse mapping theorem that, for each a, there exists some neighborhood of x 0 on which F a is smoothly invertible. The following uniform inverse mapping theorem ( [20], Proposition 4.1) and, consequently, the uniform implicit function theorem ( [20], Corollary 4.2), that will have important roles in the proof of the results of Sections 3 and 4, give conditions under which the same neighborhood of x 0 can be chosen for all a ∈ A. Proposition 2.2 (Uniform Implicit Function Theorem, [20]). Consider a set A ⊂ R k , a number α > 0, a family of functions is nonsingular for each a ∈ A and exists c > 0 such that Then there exist δ ≥ 0 and a family of continuously differentiable functions The numbers δ and K depend only on θ(·), c and α.

KKT conditions for problems with equality constraints
In this section, we will consider the continuous-time programming problem with equality constraints only. The general case is postponed to the next section. We start with the continuous-time problem without constraints, instead. The necessary optimality conditions for unrestricted problems will be used later in the proof of the main result of this section. Consider the unrestricted continuous-time problem, namely, Assume that Proposition 3.1. Ifz is a local optimal solution for (2), then Proof. Let γ ∈ L ∞ ([0, T ]; R n ). From the local optimality ofz, there existsτ > 0 such that P (z) ≥ P (z + τ γ) for all τ ∈ (0,τ ). By first order Taylor expansion in Banach spaces [22] we have that where (ε(τ )/τ ) → 0 when τ → 0. Dividing both sides by τ > 0 and taking limits as From the last equality we see that ∇φ(z(t), t) = 0 a.e. t ∈ [0, T ]. By the second order Taylor expansion, we can write where (ε(τ )/τ 2 ) → 0 when τ → 0. Dividing both sides by τ 2 > 0 and taking limits as τ → 0, we obtain The proof is complete. Now, consider the continuous-time problem with equality constraints: In this case, Let us remember that φ : Given ǫ > 0 andz ∈ Ω, we assume that, in addition to (H1), the following hypothesis are valid: (H3) There exists an increasing functionθ : (0, ∞) → (0, ∞), θ(s) ↓ 0 as s ↓ 0, such that for allz, z ∈z(t) + ǫB and a.e. t ∈ [0, T ], where ∇h(t) = ∇h(z(t), t).
Remark 3.1. Hyphotesis (H4) guarantees that the rows of ∇h(t), formed by the gradient vectors ∇h i (t), i ∈ I, are linearly independent for almost every t ∈ [0, T ]. Moreover, along with (H3), it guarantees also that the norm of [∇h(t)∇h ′ (t)] −1 is uniformly bounded, a required property in the application of the uniform implicit theorem. See proposition below.
for some K, L > 0. Then there exists C > 0 such that Proof. Consider the singular values decomposition are diagonal matrices with singular values ordered, without loss of generality, in decreasing order which, in turn, imply that On the other hand, Therefore, which concludes the proof with C = L p−1 /K.
We are now in position to state and prove the main result of the section. In the sequel, Karush-Kuhn-Tucker type necessary optimality conditions of first and second order are provided for problem (3) under the full rank condition (H4).
where N is given by Proof. Letz be a local optimal solution of (3) onz + ǫB. The proof is divided in several steps.

KKT conditions for problems with equality and inequality constraints
The general case is now tackled. Consider the problem (1) with equality and inequality constraints. Given ǫ > 0 andz ∈ Ω, we assume that, in addition to (H1), the following hypotheses are valid: (H5) h(z, ·) and g(z, ·) are measurable for each z, h(·, t) and g(·, t) are continuously differentiable onz(t) + ǫB a.e. t ∈ [0, T ].
The lemma below will be used in the proof of the main result of this section.