Simulaciones No Inteligentes de la Cadena de Suministro.

The objective function (2.10) in Problem (2.9) is nonsmooth and nonconvex. Most of algorithms for solving Problem (2.9) are based on the Clarke subdifferential. Let ϕ : Rn_{→ R}

be a locally Lipschitz function. The generalized directional derivative ϕ0(x, u) of ϕ at x in the direction d is defined as [47]:

ϕ0(x, u) := lim sup

y→x,α↓0

α−1[ϕ(y + αu) − ϕ(y)].

For locally Lipschitz functions the generalized directional derivative exists. The subdifferen- tial ∂ϕ(x) of the function ϕ at x is defined as follows [47]:

According to Rademacher’s theorem the locally Lipschitz function ϕ is differentiable almost everywhere and its subdifferential ∂ϕ(x) at a point x ∈ Rn can also be defined as:

∂ϕ(x) := conv  lim i→∞∇ϕ(x i_{) : x}i _{→ x and ∇ϕ(x}i_{) exists}  , (4.1)

where “conv” denotes the convex hull of a set. Each vector v ∈ ∂ϕ(x) is called a subgradient. The subdifferential ∂f (x) is a compact and convex set at any x ∈ Rn.

A function ϕ is called regular at x ∈ Rn, if it is differentiable with respect to any direction u ∈ Rnat x and ϕ0(x, u) = ϕ0(x, u) for all u ∈ Rnwhere ϕ0(x, u) is a derivative of the function ϕ at the point x in the direction u:

ϕ0(x, u) = lim

α↓0α

−1_{[ϕ(x + αu) − ϕ(x)].}

Next we will describe some properties of the objective function (2.10). For each a ∈ A consider the following function:

ψa(x) = min j=1,...,kd(x

j_{, a)}

and let

Ra(x) =j ∈ {1, . . . , k} : d(xj, a) = ψa(x) .

The function ψa is directionally differentiable and

ψ_a0(x, u) = min

j∈Ra(x)

d0[(xj, a), uj] u = (u1, . . . , uk) ∈ Rn×k. (4.2)

Here d0[(xj, a), uj] is the directional derivative of the function d at the point xj ∈ Rn _{in the}

direction uj, j ∈ Ra(x).

Proposition 1. The function fk given by (2.10) is directionally differentiable at any x =

(x1, . . . , xk) ∈ Rn×k and f_k0(x, u) = 1 m X a∈A min j∈Ra(x) d0[(xj, a), uj], u = (u1, . . . , uk) ∈ Rn×k. (4.3)

Proof: Since both distance functions d1 and d∞ are convex they are directionally differen-

tiable at any x with respect to any direction u ∈ Rn. Then the directional differentiability of the function fk follows from its representation as a sum of minima functions. The expression

(4.3) follows from the expression (4.2) for the directional derivative of the function ψa.

Corollary 4.3.1. Assume that at a point x = (x1_{, . . . , x}j_{) ∈ R}n×k _{there exist a ∈ A such}

that |Ra(x)| ≥ 2 and d0[(xj1, a), u] 6= d0[(xj2, a), u] for some j1, j2∈ Ra(x) and u ∈ Rn. Then

the function fk is not regular at x.

Proof: The generalized directional derivative of the function fk at the point x ∈ Rn×k in

the direction u = (u1_{, . . . , u}k_{) ∈ R}n×k _{is given by}

f_k0(x, u) = 1 m X a∈A max j∈Ra(x) d0[(xj, a), uj].

This obviously implies that if the conditions of the corollary are satisfied then f_k0(x, u) < f_k0(x, u) for some u ∈ Rn×k. This completes the proof.

It is well-known that the Clarke subdifferential calculus with most widely used operations (summation, maximum) exists in the form of equalities only for regular functions. Fur- thermore, for functions defined using complex compositions more restrictive conditions on component functions are required to have equalities. Calculus exists in the form of inclusions if these conditions are not satisfied which makes such calculus not applicable in numerical algorithms since it cannot be applied to calculate the subgradients. It follows from Corollary

4.3.1 that the Clarke subdifferential calculus for the function fk in some points x ∈ Rn×k

can be expressed only as inclusions and therefore, this calculus cannot be applied to com- pute its subgradients. In this situation derivative free algorithms or algorithms based on the approximation of subgradients using values of a function are only choice.

The Discrete Gradient Method introduced in [16] is one such method. It uses discrete gradients to approximate subgradients and discrete gradients are computed using only values of a function; n + 1 function evaluations are required to compute one discrete gradient where n is the number of variables. In [20] the version of the discrete gradient method is introduced where the number of function evaluations can be reduced significantly exploiting a special structure of the objective function such as piecewise partial separability. Next we recall

definitions of partial and piecewise partial separabilities (see [20], for details) and show that the function fk is piecewise separable for distance functions d1, d22 and d∞.

Definition 1. The function ϕ : Rn→ R is called partially separable iff there exists a family of n × n diagonal matrices Ui, i = 1, . . . , M, M ≥ 1 such that the function ϕ can be represented

as follows: ϕ(x) = M X i=1 ϕi(Uix).

In other terms, the function ϕ is called partially separable if it can be represented as the sum of functions of a much smaller number of variables. If M = n and diag(Ui) = ei where

ei is the i-th unit vector, i = 1, . . . , n then the function ϕ is separable.

Definition 2. The function ϕ : Rn→ R is said to be piecewise partially separable iff there exists a finite family of closed sets D1, . . . , Dm such thatSmi=1Di= Rn and the function ϕ is

partially separable on each set Di, i = 1, . . . , m.

Next we will show that the function fk is piecewise partially separable.

Proposition 2. The function fkdefined by (2.10) is piecewise separable for distance functions

d1, d22 and d∞.

Proof: It is clear that distance functions d1, d22 and d∞ are separable. Since the function

ψa(x) = minj=1,...,kd(xj, a), a ∈ A is represented as a minimum of functions each depending

on a subset of variables, it is piecewise separable. Finally the function fkas a sum of piecewise

separable functions ψa, a ∈ A is piecewise separable itself by Proposition 7 in [20].

It is shown in [20] that by exploiting piecewise partial separability it is possible to reduce the number of function evaluations by the discrete gradient method. The number of variables in Problem (2.9) is nk. This means that to compute one discrete gradient of the function fk one needs nk + 1 evaluations of this function. However, since the function fk is piecewise

separable applying the scheme from [20] we need only 2 evaluations of this function to compute one discrete gradient that is using it we can reduce the number of the objective function evaluations (nk + 1)/2 times.