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ON A GENERAL MINIMIZATION PROBLEM WITH CONSTRAINTS VY K. LE AND DUMITRU MOTREANU

The paper studies the existence of solutions and necessary conditions of optimality for a general minimization problem with constraints. We focus mainly on the case where the cost functional is locally Lipschitz. Applications to an optimal control problem and Lagrange multiplier rule are given. Copyright © 2006 V. K. Le and D. Motreanu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The paper deals with the following general minimization problem with constraints: inf Φ(v).

(P)

v ∈S

Here, Φ : X → R∪{+∞} is a function on a Banach space X and S is an arbitrary nonempty = ∅, where the notation dom(Φ) stands for subset of X. We suppose that S ∩ dom(Φ)  the effective domain of Φ, that is, 



dom(Φ) = x ∈ X : Φ(x) < +∞ .

(1.1)

First, we discuss the existence of solutions to problem (P). Precisely, we give an existence result making use of a new type of Palais-Smale condition formulated in terms of tangent cone to the set S and of contingent derivative for the function Φ. As a particular case, one recovers the global minimization result for a locally Lipschitz functional satisfying the Palais-Smale condition in the sense of Chang (cf. [7]). Then, by means of the notion of generalized gradient (see Clarke [8]), we obtain necessary conditions of optimality for problem (P) in the case where the cost functional Φ is locally Lipschitz. A specific feature of our optimality conditions consists in the fact that the set of constraints S is basically involved through its tangent cone. In addition, the costate variable provided by the given necessary conditions makes use essentially of the imposed tangency hypothesis. Finally, we present two applications of the necessary conditions of optimality that Hindawi Publishing Corporation Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 645–654

646

General minimization problem

demonstrate the generality of our results. The first application concerns the minimization of a locally Lipschitz functional subject to a boundary value problem for semilinear elliptic equations depending on a parameter that runs in a function space. If this parameter is a control variable, the result can be interpreted as a maximum principle for the stated optimal control problem. The second application of the abstract result concerning the necessary conditions of optimality shows that the Lagrange multiplier rule fits into this setting. In particular, the Lagrange multiplier rule for locally Lipschitz functionals is derived. The approach relies on various methods including Ekeland’s variational principle, Palais-Smale condition, tangency, generalized subdifferential calculus, orthogonality relations, Nemytskii operators, and semilinear elliptic equations. In this respect, it is worth to mention that a related work has been developed in [2–4] in the context of nonlinear mathematical programming problems. Here, the basic idea is represented by a kind of linearizing for the set of constraints S which allows to handle S locally by taking advantage of a continuous linear operator related to the tangent cone. This treatment has a unifying effect and can be applied to different problems in the optimization theory. The rest of the paper is organized as follows. Section 2 is devoted to the existence of solutions to problem (P). Sections 3 presents our necessary conditions of optimality. Section 4 contains an example in solving an optimal control problem subject to a semilinear elliptic equation. Section 5 deals with an application to the Lagrange multiplier rule. 2. Existence of solutions In the following, we make use of the notion of tangent vector to the set S at a given point v ∈ S. Precisely, the tangent cone Tv S to S at v ∈ S (Tv S is sometimes called the contingent cone to S at v) is defined as 



1 Tv S = w ∈ X : liminf d(v + tw,S) = 0 , t ↓0 t

(2.1)

where the notation d(·,S) stands for the distance function to the subset S in X. It is well known that Tv S is a closed cone in X. If S is a convex subset of X, then for every v ∈ S a very convenient description for Tv S holds: 

Tv S = cl

1 t>0

t



(S − v) ,

(2.2)

where cl means the strong closure of a set in X. For further information on the tangent cone, we refer to [5, Chapter 6]. Another useful tool in our approach is the contingent derivative ΦD (u;v) of a function Φ : X → R ∪ {+∞} at a point u ∈ dom(Φ) in any direction v ∈ X which is defined by ΦD (u;v) = limsup t ↓0 w →0



1 Φ u + t(v + w) − Φ(u) . t

The following example points out a significant particular case.

(2.3)

V. K. Le and D. Motreanu 647 Example 2.1. If the function Φ : X → R ∪ {+∞} is locally Lipschitz at a point u ∈ X, then one has ΦD (u;v) = Φo (u;v),

∀v ∈ X,

(2.4)

where Φo means the generalized directional derivative in the sense of Clarke (cf. [8]), that is, Φo (u;v) = limsup t ↓0 x →u

1 Φ(x + tv) − Φ(x) . t

(2.5)

This is clearly seen due to the locally Lipschitz property of Φ near u because then we may write limsup t ↓0 w →0



1 Φ u + t(v + w) − Φ(u) t

= limsup



1 1 Φ(u + tw + tv) − Φ(u + tw) + lim Φ(u + tw) − Φ(u) t ↓ 0 t t w →0

= limsup

1 Φ(x + tv) − Φ(x) . t

t ↓0 w →0

t ↓0 x →u

(2.6)

We now introduce a new type of Palais-Smale condition for nonsmooth functionals involving the tangent cone and contingent derivative. Definition 2.2. The functional Φ : X → R ∪ {+∞} is said to satisfy the Palais-Smale condition (for short, (PS)) at the level c (c ∈ R) on the subset S of X if every sequence (un ) ⊂ S such that

D







Φ un −→ c,

Φ un ;v ≥ −εn v ,

(2.7)

∀v ∈ Tun S,

(2.8)

for a sequence εn → 0+ , contains a strongly convergent subsequence in X. Note that in our existence result below (Theorem 2.4), we only need the (PS) condition at the level c = inf S Φ. The next example establishes that the (PS) condition in Definition 2.2 reduces to the usual Palais-Smale condition in the case of locally Lipschitz functionals. Example 2.3. If Φ : X → R ∪ {+∞} is locally Lipschitz and S = X, then Definition 2.2 becomes. Every sequence (un ) in X such that (2.7) holds and



λ un :=

inf

z∈∂Φ(un )

z −→ 0

as n −→ ∞

(2.9)

possesses a strongly convergent subsequence. This equivalence follows readily from Definition 2.2, Example 2.1, and the definition of generalized gradient











∂Φ un = z ∈ X ∗ : z,v ≤ Φo un ;v , ∀v ∈ X .

(2.10)

648

General minimization problem

We present our existence result in solving problem (P). Theorem 2.4. Let S be a closed subset of X and Φ : X → R ∪ {+∞} a function such that = ∅. Assume that Φ|S is lower semicontinuous and bounded from below, and S ∩ dom(Φ)  Φ satisfies the (PS) condition in Definition 2.2 on S at the level c = inf S Φ. Then problem (P) has at least a solution u ∈ S and it is a critical point of Φ on S in the following sense: ΦD (u;v) ≥ 0,

∀v ∈ Tu S.

(2.11)

Proof. Applying Ekeland’s variational principle (cf. [9]) to the function Φ|S yields a sequence (un ) ⊂ S such that (2.7) and  1 Φ(y) ≥ Φ un −  y − un , n

∀ y ∈ S.

(2.12)

hold. Fix any v ∈ Tun S. By (2.1) there exist sequences tk → 0+ in R and wk → 0 in X as k → ∞ such that un + tk (v + wk ) ∈ S for all k. Plugging in (2.12) gives 



1 1 Φ un + tk v + wk − Φ un ≥ − v + wk . tk n

(2.13)

Letting k → ∞ shows that liminf k→∞





1 1 Φ un + tk v + wk − Φ un ≥ − v , tk n

∀v ∈ Tun S.

(2.14)

It turns out that (2.8) is verified with εn = 1/n. Therefore, the (PS) condition as formulated in Definition 2.2 provides a relabelled subsequence satisfying un → u in X. Moreover, we have that u ∈ S because S is closed. Taking into account that Φ is lower semicontinuous on S, we conclude Φ(u) = inf S Φ. In order to check (2.11), let v ∈ Tu S. By (2.1) there exist sequences tk → 0+ in R and wk → 0 in X as k → ∞ such that u + tk (v + wk ) ∈ S for all k. Since Φ(u + tk (v + wk )) ≥ Φ(u), we readily obtain (2.11) that completes the proof.  We illustrate the applicability of Theorem 2.4 by deriving the existence result of Chang [7, Theorem 3.5]. Corollary 2.5. Assume that Φ : X → R is a locally Lipschitz function on a Banach space X, Φ is bounded from below and satisfies the Palais-Smale condition in the sense of Chang in [7]. Then there exists u ∈ X such that Φ(u) = inf X Φ and u is a critical point of Φ, that is, it solves the inclusion problem 0 ∈ ∂Φ(u),

(2.15)

where ∂Φ(u) stands for the generalized gradient of Φ at u. Proof. By Example 2.3 we know that the (PS) condition in the sense of Definition 2.2 is fulfilled with S = X. Then it is straightforward to deduce the result by applying Theorem  2.4

V. K. Le and D. Motreanu 649 3. Necessary conditions of optimality From now on we assume that the function Φ : X → R entering problem (P) is locally Lipschitz on a Banach space X and S is an arbitrary nonempty subset of X. We formulate the following condition on the set S. (H) For every v ∈ S, there exists a (possibly unbounded) linear operator



Av : D Av ⊂ X −→ Yv ,

(3.1)

where Yv is a Banach space, such that the domain D(Av ) of Av is dense in X, Av is a closed operator (i.e., its graph is closed in X × Yv ), and



the range R Av is closed in Yv .

(3.2)

Moreover, the null space N(Av ) of Av satisfies



N Av ⊂ Tv S,

(3.3)

where Tv S stands for the tangent cone to S at v as introduced in (2.1). Theorem 3.1. Under hypothesis (H), if u ∈ S is a solution of problem (P) (at least locally), then the following necessary condition of optimality holds. There exists an element p ∈ D(A∗u ) such that A∗u (p) ∈ ∂Φ(u),

(3.4)

where A∗u : D(A∗u ) ⊂ Yu∗ → X ∗ is the adjoint operator of Au and ∂Φ(u) denotes the generalized gradient of Φ at u. Proof. Fix any z ∈ N(Au ). It follows from (3.3) that z ∈ Tu S.

(3.5)

Taking into account (2.1), we deduce from (3.5) that there exist sequences tn → 0+ in R and xn → 0 in X such that



u + tn z + xn ∈ S,

∀n.

(3.6)

Using the optimality of u ∈ S, we obtain



Φ u + tn z + xn



≥ Φ(u),

∀n,

(3.7)

that leads to liminf n→∞





1 Φ u + tn z + xn − Φ(u) ≥ 0. tn

(3.8)

650

General minimization problem

In particular, according to (2.5), inequality (3.8) implies that

∀z ∈ N Au .

Φo (u;z) ≥ 0,

(3.9)

On the basis of (3.9), we may apply the Hahn-Banach theorem to obtain the existence of some ξ ∈ X ∗ with the properties

∀z ∈ N Au ,

ξ,zX ∗ ,X = 0, ξ, y 

X ∗ ,X

o

≤ Φ (u; y),

∀ y ∈ X.

(3.10) (3.11)

We see from (3.11) that ξ ∈ ∂Φ(u),

(3.12)

while (3.10) ensures that

ξ ∈ N Au

⊥

.

(3.13)

Because [N(Au )]⊥ = R(A∗u ) (see, e.g., [6]), in view of (3.2), relation (3.13) reads





ξ ∈ R A∗u = R A∗u



(3.14)

(note that because R(Au ) = R(Au ), we also have R(A∗u ) = R(A∗u ), see again [6]). Combining relations (3.12), (3.13), and (3.14), we arrive at the desired conclusion.  Remark 3.2. (a) By Theorem 3.1, we have the system u ∈ S,

(3.15)

∗

Au (p) ∈ ∂Φ(u)

formed by two relations with two unknowns u and p which eventually permit to determine the optimal solution u. (b) We consider here the problem with the functional Φ being locally Lipschitz, which is a good and convenient model for our calculations. However, it could be possible to study such problem with other types of functionals and subdifferentials (see, e.g., [8, 10, 11, 13]). 4. An example For an example of the general result in Theorem 3.1, let us consider the problem of minimizing the functional Φ(v,w) subject to the following conditions expressed as a Dirichlet problem:

(v,w) ∈ H 2 (Ω) ∩ H01 (Ω) × L2 (Ω), −Δv = f (x,v) + w

v = 0 on ∂Ω.

in Ω,

(4.1)

V. K. Le and D. Motreanu 651 Interpreting the parameter w as a control variable, this is in fact an optimal control problem. Here, Φ is a locally Lipschitz functional defined on X = L2 (Ω) × L2 (Ω), f : Ω × R → R is a Carath´eodory function with f (·,0) ∈ L2 (Ω). Moreover, the partial derivative (∂ f /∂v)(x,v) exists for a.e. x ∈ Ω, all v ∈ R with ∂ f /∂v being a bounded Carath´eodory function, that is,   ∂ f   ≤c (x,v)  ∂v 

for a.e. x ∈ Ω, all v ∈ R,

(4.2)

for some constant c > 0. Notice that the considered problem is of the general form (P) in Section 1 with 



S = (v,w) ∈ X : −Δv = f (x,v) + w in Ω, v = 0 on ∂Ω .

(4.3)

Let us prove that under the above conditions, hypothesis (H) holds. Specifically, for any (v,w) ∈ S, let Y(v,w) = L2 (Ω) × L2 (Ω),

(4.4)

and let

A(v,w) : H 2 (Ω) ∩ H01 (Ω) × L2 (Ω) −→ L2 (Ω) × L2 (Ω)

(4.5)

be given by 

A(v,w) (z, q) = − Δz −

∂f (·,v)z, q ∂v



(4.6)

whenever (z, q) ∈ [H 2 (Ω) ∩ H01 (Ω)] × L2 (Ω). From (4.2) and the classical results of Agmon in [1] on linear elliptic operators , we see that the range of the operator −→ −Δz − z



∂f (·,v)z ∈ L2 (Ω) ∂v

(4.7)

is closed in L2 (Ω) (cf., e.g., [12, Theorem 8.41]). It follows immediately that the range R(A(v,w) ) of A(v,w) is closed in L2 (Ω) × L2 (Ω). Condition (3.2) is thus verified. In order to check (3.3), let (z,0) ∈ N(A(v,w) ). Because −Δv = f (x,v) + w −Δz −

in Ω,

∂f (·,v)z = 0 in Ω, ∂v

(4.8) (4.9)

by multiplying (4.9) by t > 0 and adding with (4.8), we obtain −Δ(v + tz) = f (x,v + tz) + w + tq(t),

(4.10)

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General minimization problem

where 



∂f 1 q(t) = − f (·,v + tz) − f (·,v) − t (·,v)z , t ∂v

∀t > 0.

(4.11)

Assumption (4.2) guarantees that q(t) −→ 0

in L2 (Ω) as t −→ 0+ .

(4.12)

Consequently, setting



p(t) = 0, q(t) ∈ L2 (Ω) × L2 (Ω),

(4.13)

we see from (4.10) that condition (3.3) is fulfilled since





(v,w) + t (z,0) + p(t) = (v,w) + t (z,0) + 0, q(t)



∈ S,

∀t > 0.

(4.14)

We have checked all assumptions of Theorem 3.1. According to that theorem, if Φ has a local minimum at (v,w), then there exists a pair (p1 , p2 ) ∈ L2 (Ω) × L2 (Ω) such that 

p1 ∈ H 2 (Ω) ∩ H01 (Ω), − Δp1 −



∂f (·,v)p1 , p2 ∈ ∂Φ(v,w) ∂v

(4.15)

(p2 is not significant for our purpose). Remark 4.1. If the constant c in (4.2) is less than the first eigenvalue λ1 of −Δ on H01 (Ω), then the range of A(v,w) is in fact the whole space L2 (Ω) × L2 (Ω). In this case, we observe that the auxiliary variable p1 can be explicitly determined. Consequently, the necessary condition of optimality can be expressed only with the local solution (v,w). 5. Application to Lagrange multiplier rule Assume now that the subset S in problem (P) is given by S=

 j ∈J

G−j 1 (0).

(5.1)

Here, for each j ∈ J, G j : X → Y j is a C 1 mapping with Y j being a Banach space and 0 a regular value of G j , that is, the differential Gj (x) : X → Y j is surjective and N[Gj (x)] has a topological complement whenever G j (x) = 0. Moreover, assume that the sets G−j 1 (0) ( j ∈ J) are mutually disjoint, that is, G−i 1 (0) ∩ G−j 1 (0) = ∅ if i  = j.

(5.2)

V. K. Le and D. Motreanu 653 We check that hypothesis (H) is satisfied. In fact, for any v ∈ S, there is a unique j ∈ J such that v ∈ G−j 1 (0). Let Yv = Y j and Av = Gj (v). Then we have R(Av ) = Yv , so (3.2) is verified. Moreover, we know that







N Av = N Gj (v) = Tv G−j 1 (0) = Tv S,

(5.3)

because G−j 1 (0) is a C 1 -submanifold of X, which implies (3.3). Consequently, Theorem 3.1 can be applied, ensuring the existence of p ∈ Y j∗ with the property that whenever u ∈ S is a solution of (P) with u ∈ G−j 1 (0), then

∗

Gj (u) (p) ∈ ∂Φ(u).

(5.4)

In the particular case where J is a singleton, that is, S = G−1 (0),

(5.5)

and G is a mapping from X to R (also 0 is a regular value of G), then (5.4) becomes λG (u) ∈ ∂Φ(u)

(5.6)

for some λ ∈ R. This is the classical Lagrange multiplier rule for locally Lipschitz functionals. References [1] S. Agmon, Lectures on Elliptic Boundary Value Problems, D. Van Nostrand, New Jersey, 1965. [2] S. Aizicovici, D. Motreanu, and N. H. Pavel, Nonlinear programming problems associated with closed range operators, Applied Mathematics and Optimization 40 (1999), no. 2, 211–228. , Fully nonlinear programming problems with closed range operators, Differential Equa[3] tions and Control Theory (Athens, OH, 2000), Lecture Notes in Pure and Appl. Math., vol. 225, Marcel Dekker, New York, 2002, pp. 19–30. , Nonlinear mathematical programming and optimal control, Dynamics of Continuous, [4] Discrete & Impulsive Systems. Series A. Mathematical Analysis 11 (2004), no. 4, 503–524. [5] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, vol. 2, Birkh¨auser Boston, Massachusetts, 1990. [6] H. Brezis, Analyse Fonctionnelle. Th´eorie et Applications, Collection of Applied Mathematics for the Master’s Degree, Masson, Paris, 1983. [7] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, Journal of Mathematical Analysis and Applications 80 (1981), no. 1, 102– 129. [8] F. H. Clarke, Optimization and Nonsmooth Analysis, Classics in Applied Mathematics, vol. 5, SIAM, Pennsylvania, 1990. [9] I. Ekeland, Nonconvex minimization problems, American Mathematical Society. Bulletin. New Series 1 (1979), no. 3, 443–474. [10] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol. I, II, Springer, New York, 2006. [11] D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Nonconvex Optimization and Its Applications, vol. 29, Kluwer Academic, Dordrecht, 1999.

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General minimization problem

[12] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Texts in Applied Mathematics, vol. 13, Springer, New York, 1993. [13] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Fundamental Principles of Mathematical Sciences, vol. 317, Springer, Berlin, 1998. Vy K. Le: Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, MO 65409, USA E-mail address: [email protected] Dumitru Motreanu: D´epartement de Math´ematiques, Universit´e de Perpignan, 52 avenue Paul Alduy, 66860 Perpignan, France E-mail address: [email protected]