A NOTE ON THE GRADIENT OF THE INTEGRAL ENTROPY FUNCTION

In this paper, we consider a parametric minimization problem and use the integral entropy function to approximate its value function. We show that the repeated limits of the gradient of the integral entropy function may be strictly contained in the Clarke generalized gradient of the value function in some cases. We also apply the result to bilevel programs. AMS Subject Classification: 65K10, 90C26


Introduction
The parametric minimization is a problem in m variables that depends on n parameters which can be specified by min y∈Y f (x, y) for each vector x ∈ X, where X ⊆ R n , Y ⊆ R m are closed convex sets and f : R n × R m → R is continuously differentiable.
We denote the value function of the parametric minimization problem by V (x) := inf y∈Y f (x, y).
While the major difficulty encountered is the nondifferentiability of the value function.[5] proposed to use the integral entropy function to approximate the value function and also showed the uniformly convergence as ρ → ∞. [6, Theorem 5.1 and 5.5] showed that {γ ρ (x) : ρ > 0} is a smoothing function of V (x), i.e., for any x ∈ X, and satisfies the gradient consistency property, i.e., for any x ∈ X, In this paper, we discuss the gradient of the integral entropy function.

Preliminaries
In this section, we present some background materials which will be used later on.
We first adopt the following standard notation in this paper.Given a function G : R n → R m , we denote its Jacobian by ∇G(z) ∈ R m×n and, if m = 1, the gradient ∇G(z) ∈ R n is considered as a column vector.For a set Ω ⊆ R n , we denote by int Ω, ri Ω, co Ω, and dist(x, Ω) the interior, relative interior, the convex hull, and the distance from x to Ω respectively.In addition, we let N be the set of nonnegative integers and exp[z] be the exponential function.
Let ϕ : R n → R be Lipschitz continuous near x.The Clarke generalized directional derivative of ϕ at x in direction d is defined by The Clarke generalized gradient of ϕ at x is a convex and compact subset of R n defined by Detailed discussions of the Clarke generalized gradient and its properties can be found in [2,3].For a nonempty closed set Ω ⊆ R n and a point x ∈ Ω, the Clarke tangent cone [2,3] of Ω at x is given by and the Clarke normal cone [2,3] of Ω at x is given by Proposition 2.1 (Danskin's Theorem).([3, Page 99] or [4]) Let Y ⊆ R m be a compact set and f (x, y) be a function defined on R n × R m that is continuously differentiable at x. Then the value function is Lipschitz continuous near x and its Clarke generalized gradient at x is where S(x) is the set of all minimizers of f (x * , y) over y ∈ Y .
In the end of this section, we review the definition of Lebesgue Measure [10].
where |Q| denotes the volume of a closed cube Q and the infimum is taken over all countable closed cubes For a measurable set E, m * (E) is called the Lebesgue measure of E.
Theorem 3.1.Let x * be a limiting point of {x k } and ρ k → ∞, if S(x * ) = {y * }, then we have lim Proof.From the gradient consistent property of γ ρ (•), for any vector v and subset K ⊂ N such that Since v and K are arbitrary accumulation point and subset, ∂V (x * ) is a singleton set, we have lim Theorem 3.2.For any point x * and any subset X 0 ⊆ X such that x * ∈ cl X 0 and m * (S(x)) = 0, ∀x ∈ X 0 .We have where S * 0 = lim sup x→x * ,x∈X 0
Due to the fact that S * 0 is compact, there exists 0 < δ < δ 2 such that y∈S * 0 B(y, δ 2 ) ⊇ (S * 0 + δB), where B denotes the closed unit ball.We get from the Heine-Borel covering theorem that there exist N > 0 and y * i ∈ S * 0 such that From the Lemma 3.1 and the Theorem 4.10 (b) [9], we know that for δ > 0, ∀x ∈ X 0 such that x − x * sufficiently small, we have Thus there exists an index set For each i ∈ I(x), there exists y x i ∈ B(y * i , δ 2 )∩S(x).Note that I(x) may not equal to the whole index set {1, • • • , N } since S * 0 is the outer limit.Let B(y x i , δ) = B(y x i , δ) ∩ S(x), i ∈ I(x), for each x ∈ X 0 such that x − x * sufficiently small.Without loss of generality, we assume that It is obvious that We complete the proof.

Applications to Bilevel Program
Many scientific problems such as a very important model in economics called the principal-agent problem [7] can be formulated as the following simple bilevel program: where S(x) denotes the set of solutions of the lower level program where X ⊆ R n , Y ⊆ R m are closed convex sets and F, f : R n × R m → R are twice continuously differentiable.For a numerical purpose, Outrata [8] proposed to reformulate (SBP) as the following single level optimization problem: Since the value function V (x) is generally nonsmooth even when the function f (x, y) is smooth, the problem (V P ) is a nonsmooth problem.To copy with such difficulty, [6] approximated the value function by its integral entropy function and proposed a smoothing gradient projection algorithm to solve (V P ).
From the definition of V (x), for any (x, y) ∈ X ×Y , we always have f (x, y)− V (x) ≥ 0. Hence for any feasible point (x * , y * ) of the problem (V P ), (x * , y * ) is an optimal solution of the problem min Therefore, the GMFCQ never holds for the problem (V P ) and hence the nonsmooth KKT condition may not hold at a local optimal solution.While for a sequence of iteration points {(x k , y k )} which convergent to (x * , y * ), the set lim sup k→∞ ∇γ ρ k (x k ) may strictly contain in ∂V (x).Therefore while (4.2) holds, the following inclusion may not hold: which guarantees a sequence of bounded multipliers and thus (x * , y * ) is a stationary point of the problem (V P ) [11].
In the case where S(x * ) is a singleton set, we know that the condition (4.3) always holds from Theorem 3.1.While in this case, the value function is continuously differentiable around x * and we suggest to use the first order approach which replace the lower level program by its Kurash-Kuhn-Tucker (KKT) condition to solve such problem.
For any feasible point (x * , y * ), since S * 0 is possibly a proper subset of S(x * ), which implies that U (x * ) is strictly contained in ∂V (x * ), the following inclusion may fails: