https://doi.org/10.1007/s43069-021-00085-w ORIGINAL RESEARCH
On Continuous Codifferentiability of Quasidifferentiable Functions
Igor M. Prudnikov1
Received: 14 June 2021 / Accepted: 8 July 2021
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021
Abstract
The author studied codifferentiable functions introduced by Professor V.F.
Demyanov and a method for calculating their subdifferentials and codifferen- tials. The proven theorems give the rules for calculating the subdifferentials. It was shown that the subdifferential of the first order, introduced by the author, coincides with the Demyanov’s difference applied to the subdifferential and superdifferential of a Lipschitz quasidifferentiable function. The author proved that any quasidifferentiable function is continuously codifferentiable under the condition of upper semicontinuity for the subdifferential and superdifferential mappings. The author suggested constructing a continuous extension for the subdifferential of the first order and introduced 𝛼 -optimal points.
Keywords Quasidifferentiable functions · Generalized gradients · Codifferentiable functions · Subdifferential of the first order · The Clarke subdifferential · The Michel-Penot subdifferential · Necessary and sufficient conditions of optimality · Continuous codifferentiability
Mathematics Subject Classification 2000 49J52 · 90C30 · 90C31
1 Introduction
Many authors introduced generalized gradients and matrices for Lipschitz functions in different ways [1–9]. Unlike smooth functions, the Clarke and Michel-Penot sub- differentials, consisting of generalized gradients, are not continuous in the Hausdorff metric as the set-valued mappings. Therefore, pairs of some sets, which are ana- logues of the subdifferentials, were introduced by V.F. Demyanov and A.M. Rubinov.
Dedicated to the memory of my teacher Prof. V.F. Demyanov, who invented codifferentials.
* Igor M. Prudnikov pim_10@hotmail.com
1 Scientific Center of Smolensk Federal Medical University, Smolensk 214000, Russia Published online: 27 August 2021
/
Quasidifferentiable functions, introduced in [4], are represented as the sum of maxi- mum and minimum of the scalar products, taken over the sets, called the subdifferen- tial and superdifferential. Quasidifferentiable functions are a generalization of convex and the difference of convex functions. All necessary definitions are given in Sect. 2.
The multi-step gradient methods were developed for optimization of quasidif- ferentiable functions [12]. The list of numerous publications of quasidifferentiable functions can be found in [13–15].
Later, V.F. Demyanov introduced codifferentiable functions [5]. Two sets, the hypodifferential and hyperdifferential, are used for description of such functions.
The author answers the question, formulated in [5], how to construct the quasidif- ferential and codifferential.
The subdifferential of the first order, introduced by the author, was used for con- struction. It was shown that the subdifferential of the first order coincides with the Demyanov’s difference applied to the subdifferential and superdifferential of a Lip- schitz quasidifferentiable function. It is important to know the subdifferentials and codifferentials for practical optimization. According to the authors of [5], the ques- tion of how constructively to find the codifferentials for a convex function is signifi- cant for practical use.
As an application, the author suggested constructing a continuous extension for the subdifferential of the first order. As a consequence, we can introduce 𝛼 -optimal points similar to 𝜀 - optimal points for convex functions and use the developed opti- mization method for looking for 𝛼 -optimal points.
The author proved that the hypodifferential and hyperdifferential are continuous set-valued mappings in the Hausdorff metric for a wide set of functions. Finally, the author introduced 𝛼 - optimal points for Lipschitz functions like 𝜀-optimal points for convex functions. The continuous 𝛼-subdifferential for Lipschitz functions is defined.
2 Notations and Examples
Firstly, quasidifferentiable functions were introduced in [4]. A function f ∶ℝn→ℝ is called quasidifferentiable at x, if the following decomposition is correct
where ox(𝛼𝛼Δ) →0 as 𝛼⇁+0 , (v,Δ) and (w,Δ) are the scalar products of the corre- sponding vectors. The sets 𝜕f(x) and 𝜕f(x) are called the subdifferential and super- differential of the function f at x correspondingly. The pair [𝜕f(x),𝜕f(x)] is called the quasidifferential of f at x. The subdifferential and superdifferential of f(⋅) are not continuous in the Hausdorff metric for nonsmooth case. It happened historically that if the pair [𝜕f(x),𝜕f(x)] consists only of one set, then we do not use the overline.
Later, codifferentiable and twice codifferentiable functions were introduced in [5]. f ∶ℝn→ℝ is called codifferentiable at x, if some convex compact sets
f(x+ Δ) =f(x) + max (1)
v∈𝜕f(x)(v,Δ) + min
w∈𝜕f(x)
(w,Δ) +ox(Δ),
df(x), df(x) from ℝn+1 , called the hypodifferential and hyperdifferential, respec- tively, exist, for which the decomposition
is valid, where ox(Δ)→0 as Δ→0 , ox(𝛼𝛼Δ) →0 as 𝛼→+0 . The pair of sets [df(x), df(x)] , according to the Demyanov’s terminology, is called the first codiffer-
entials of f at x. We suppose that the function ox(Δ) in Eq. (2) is uniformly infinitesi- mal with respect to Δ for small Δ. That means that ox(𝛼𝛼Δ) →0 as 𝛼→+0 uniformly with respect to Δ from a neighborhood of 0.
It was proved [5] that the set of quasidifferential functions coincides with the set of codifferential functions. Our goal here is to define and construct a con- tinuous first codifferential of f(⋅) at x, using the subdifferential of the first order, introduced by the author in [7].
We use the quasidifferentials and codifferentials for determining the necessary and sufficient conditions of optimality. It is necessary to give their construction for optimization of codifferentiable functions.
We will suppose that the function f(⋅) is Lipschitz. A set of curves was intro- duced in [7].
Definition 2.1 𝜂(x0) is a set of smooth curves r(x0,𝛼, g) =x0+𝛼g+or(𝛼) , where g∈Sn−11 (0) = {v∈ℝn ∶‖v‖=1} and the function or∶ [0,𝛼0]→ℝn,𝛼0>0, satis- fies the following conditions:
1. or(𝛼)∕𝛼 tends to zero as 𝛼↓0 uniformly in r ;
2. there is a continuous derivative o�r(⋅), and its norm is bounded for all r in the fol- lowing sense: there exists c <∞ for all curves such that
3. the derivative ∇f(⋅) exists almost everywhere (a.e.) along the curve r.
We introduce the sets
and
Since ‖∇f(⋅)‖≤L, where L is the Lipschitz constant of f(⋅) , the Lebesgue inte- gral in the definition of Ef(x0) exists according to its property [10, 11] and
f(x+ Δ) =f(x) + max (2)
[a,v]∈df(x)[a+ (v,Δ)] + min
[b,w]∈df(x)
[b+ (w,Δ)] +ox(Δ),
sup
𝜏∈(0,𝛼0)
∥o�r(𝜏) ∥≤c;
Ef(x0) = {v∈ℝn∶ ∃{𝛼k},𝛼k↓0,(∃g∈S1n−1(0)),
(∃r∈𝜂(x0)), v=lim
𝛼k↓0
𝛼−1
k ∫
𝛼k
0
∇f(r(x0,𝜏, g))d𝜏}
(3) Df(x0) = conv Ef(x0).
The set Df(x0) is called by the author [7] the subdifferential of the first order of f(⋅) at x0 . We give the examples of calculation of the subdifferentials.
It was proved in [7] that 𝜕f(x) =Df(x) for any finite convex function f ∶ℝn→ℝ with the subdifferential 𝜕f(x) at x.
We will compare Df(x) with the Clark 𝜕CLf(x) and Michel-Penot 𝜕MPf(x) subdiffer- entials which will be defined below.
1) Clarke’s method [1, 2];
Approximate functions are constructed in the following way:
It was proved that a convex compact set 𝜕Clf(x) exists that
The set 𝜕Clf(x) is called the Clarke subdifferential of the function f(⋅) at x. It is known that
N( f) is the set where the function f(⋅) is differentiable.
2) Michel-Penot’s method; Let us consider the functions
and
that are called the upper and lower Michel-Penot derivatives respectively. A com- pact set 𝜕MPf(x) exists such that
‖�
𝛼k
0
∇f(r(x0,𝜏, g))d𝜏‖≤L𝛼k.
FCl↑(x, g) = lim (4) 𝛼→+0
h→0
sup(f(x+h+𝛼g) −f(x+h))∕𝛼,
FCl↓(x, g) = lim 𝛼→+0
h→0
inf(f(x+h+𝛼g) −f(x+h))∕𝛼.
(5) FCl↑(x, g) = max
v∈𝜕Clf(x)(v, g),
F↓Cl(x, g) = min
v∈𝜕Clf(x)(v, g).
𝜕Clf(x) = conv {v∈ℜn∶ ∃{xk}, xk∈N(f), v= lim
xk→x∇f(xk)},
(6) fMP↑ (x,g) = inf
q∈ℜn{lim
𝛼→+0
inf[f(x+𝛼(g+q)) −f(x+𝛼q)]∕𝛼}
fMP↓ (x,g) = inf
q∈ℜn{lim
𝛼→+0 inf[f(x+𝛼(g+q)) −f(x+𝛼q)]∕𝛼}
(7) fMP↑ (x, g) = max
v∈𝜕
MPf(x)(v, g),
The functions F↑Cl(x,⋅), F↓Cl(x,⋅), fMP↑ (x,⋅), fMP↓ (x,⋅) are used for approximation of the function f(⋅) in a neighborhood of the point x.
Also Df(x) coincides with the Clark subdifferential 𝜕CLf(x) [1] for any function f(⋅) locally represented as the difference of convex functions in a neighborhood of x.
The Michel-Penot subdifferential 𝜕MPf(x) [6] can also differ from Df(x).
Let us give the examples confirming the difference between these subdifferentials.
Example 2.1 Let us define now a function f ∶ℝ2 →ℝ for which 𝜕MPf(𝟎)≠Df(𝟎) , where 𝟎= (0, 0), graphically.
We will draw two curves r1(𝛼) =𝛼ey−𝛼22ex and r2(𝛼) =𝛼ey+𝛼22ex,oi(𝛼)∕𝛼→𝟎 for 𝛼→+0, i=1, 2, in the plane, ex= (1, 0), ey= (0, 1) are the unit basis vectors with the coordinates 1 and 0 or on the contrary. The axis OZ points toward us. The curves r1 and r2 bound the small region where f ≡0 . Above the plane XOY there is a part of the graph of the function f intersecting the plane XOY along the curve r2 . This part of the graph of the function becomes the plane z−x=0 when y→0 , and
∇f(x, y)→(1, 0) when (x, y)→(x, 0) for (x, y) from the region 2. There is a part of the graph of the function f(⋅) below the plane XOY that intersects the plane XOY along the curve r1 . This part of the graph of the function f(⋅) becomes the plane z−x=0 when y→0 , and ∇f(x, y)→(1, 0) when (x, y)→(x, 0) for (x, y) from the region 2.
Analytically, the function z=f(x, y) can be defined as follows.
For (x, y) from the region 1 f(x, y)≡0. For (x, y) from the region 2 and x>0, z>0
For (x, y) from the region 2 and x<0, z<0
It is easy to verify that the graph of the function f(⋅) intersects the XOY plane for y>0 along the curves r1 and r2 . For y<0 the equality z=x is valid.
It is easy to calculate that
though
Example 2.2 The graph of f ∶ℜ→ℜ consists of segments whose slopes are equal to ±1 and lying between the curves x4 and −x4 with the condensation point at 0.
The function f(⋅) is not represented as the difference of two convex functions in any neighborhood of zero, because the total variation
fMP↓ (x, g) = min
v∈𝜕MPf(x)(v, g).
z=f(x, y) =x−y2 2.
z=f(x, y) =x+y2 2.
𝜕CLf(𝟎) = conv {𝟎, ex},
𝜕MPf(𝟎) = {ex}.
where a is any positive number. It is easy to see that
though
It was proved [7] that the equality Df(x0) =𝜕Clf(x0) is correct for any function f(⋅) locally represented as the difference of convex functions in a neighborhood of x0 .
3 Construction of the Subdifferential for a Hypodifferentiable Function
Let f ∶ℝn→ℝ be a Lipschitz hypodifferentiable function at x, i.e. the representation
is true. It follows from (8)
where a=max{a∣ [a, v] ∈df(x)},𝜕f(x) = {v∣ [a, v] ∈df(x)}. Let us prove that
𝜕f(x) =Df(x).
For any Δ =𝛼g, g∈S1n−1(0), we take the curve r∈𝜂(x), along which we calculate the averaging integral of gradients
Let us take an arbitrary sequence {𝛼k} , 𝛼k→+0, for which there is a limit
By definition, v(g) ∈Df(x). The following is valid
Consequently,
From here we have Df(x)⊃ 𝜕f(x).
∨a0f�= ∞,
𝜕Clf(0) = [−1,+1],
𝜕MPf(0) =Df(0) = {0}.
f(x+ Δ) =f(x) + max (8)
[a,v]∈df(x)[a+ (v,Δ)] +ox(Δ)
f(x+ Δ) =f(x) + max
v∈𝜕f(x)(v,Δ) +ox(Δ),
𝛼−1
∫
𝛼
0
∇f(r(x,𝜏, g))d𝜏.
v(g) = lim
𝛼k→+0
𝛼−1
k ∫
𝛼k
0
∇f(r(x,𝜏, g))d𝜏.
(v(g), g) = 𝜕f(x)
𝜕g = max
u∈𝜕f(x)(u, g).
w∈Df(x)max (w, g)≥ max
u∈𝜕f(x)(u, g).
The strict inclusion is impossible, otherwise the vectors v∈Df(x), v∉𝜕f(x), and g(v) ∈Sn−11 (0) ∶
where v1=arg minw∈𝜕f(x)‖v−w‖ , would exist for which
The last is not true. This implies the equality Df(x) =𝜕f(x).
The existence of the vector v∈Df(x) , which was mentioned above, follows from the fact that for the subdifferentiable function f(⋅) the upper and lower convex approximations at the point x coincide with h(g) =maxw∈𝜕f(x)(w, g) . Subdifferential
𝜕f(x) can be obtained from the set Df(x) using the cutting method for all directions g, described in [16, 17].
Therefore, we have proved the following
Theorem 3.1 If f(⋅) is Lipschitz hypodifferentiable function at x, then Df(x) =𝜕f(x). Since the sets consisting from hypodifferentiable and subdifferentiable functions coincide with each other [5], we have from Theorem 3.1:
Corollary 3.1 If f(⋅) is Lipschitz subdifferentiable at x, then Df(x) =𝜕f(x).
4 Construction of the Subdifferentials and Superdifferentials for Codifferentiable Functions
Let f ∶ℝn→ℝ be a Lipschitz codifferentiable function, i.e. the equality Eq. (2) holds for it. As follows from Eq. (2),
where the subdifferential 𝜕f(x) and the superdifferential 𝜕f(x) can be found out in the following way:
Let us introduce the difference of convex compact sets A and B [5]:
where pA(⋅), pB(⋅) are the support functions to the sets A, B , respectively:
g(v) = v−v1
‖v−v1‖,
(v, g(v))>
𝜕f(x)
𝜕g(v).
f(x+ Δ) =f(x) + max (9)
v∈𝜕f(x)(v,Δ) + min
w∈𝜕f(x)
(w,Δ) +ox(Δ),
ā=max{a∣ [a, v] ∈df(x)},𝜕f(x) = {v∣ [a, v] ∈̄ df(x)},
b̄=min{b∣ [b, w] ∈df(x)},𝜕f(x) = {w∣ [b, w] ∈̄ df(x)}.
(10) A⇀B=co{∇pA(q) − ∇pB(q) ∣q∈Sn−1A ∩Sn−1B },
and Sn−1A , Sn−1B are the sets of the unit support vectors to the sets A, B , where the func- tions pA(⋅), pB(⋅) are differentiable. The difference Eq. (10) is called the Demyanov’s difference [5].
Couples of convex compact sets [A, B] and [C, D] are called equivalent, if A⇀B=C⇀D.
Let the equality Eq. (9) be true. Let us rewrite Eq. (2) in the form Eq. (9) and take Δ =𝛼g, g∈Sn−11 (0) , for which there is such a sequence {𝛼k},𝛼k→+0, that the fol- lowing equalities are hold for the vector g and a vector v∈Df(x) ∶
where
Consequently,
From here we have
Strict inclusion is not possible, since otherwise the vectors v∈Df(x) and g(v) ∈Sn−11 (0) would exist, for which
where
The latter cannot be correct.
The existence of the vector v∈Df(x) , which was mentioned above, follows from the fact that for the quasidifferentiable function f(⋅) the upper convex approximation at x coincides with
pA(q) =max
v∈A(v, q), pB(q) =max
w∈B(w, q)
v(g) = lim
𝛼k→+0
𝛼−1
k ∫
𝛼k
0
∇f(r(x,𝜏, g))d𝜏,
(v(g), g) = (v1, g) − (v2, g) = 𝜕f(x)
𝜕g ,
(v1, g) = max
w∈𝜕f(x)(w, g), (v2, g) = max
w∈−𝜕f(x)
(w, g).
w∈max𝜕f(x)(w, g) − max
w∈−𝜕f(x)
(w, g)≤ max
w∈Df(x)(w, g)
𝜕f(x)⇀(−𝜕f(x))⊂Df(x).
𝜕f(x)
𝜕g(v) = (v, g(v))>(v1, g(v)) − (v2, g(v)) = 𝜕f(x)
𝜕g(v),
(v1,g(v)) = max
w∈𝜕f(x)(w,g(v)),(v2,g(v)) = max
w∈−𝜕f(x)
(w,g(v).
h(g) = max
w∈𝜕f(x)⇀(−𝜕f(x))
(w, g).
Subdifferential 𝜕f(x)⇀(−𝜕f(x)) can be obtained from the set Df(x) using the cut- ting method for all directions g, described in [17].
Thus, we have proved the following theorem.
Theorem 4.1 If f(⋅) is Lipschitz quasidifferentiable function at x, then
The sets 𝜕f(x) and 𝜕f(x) are determined by the set Df(x) up to the equivalence.
The authors write: “Of course, the answer for the question of how constructively to find the set df(x) for a convex function is significant for practical use. This ques- tion must be solved for specific classes of convex functions” on page 189 [5]. Let us answer this question.
If f(⋅) is convex finite in ℝn , then it is hypodifferentiable and the representation
is true. As it follows from Eq. (11):
where a=max{a∣ [a, v] ∈df(x)},𝜕f(x) = {v∣ [a, v] ∈df(x)}.
For the more general case when the function f(⋅) is subdifferentiable and can be represented as
it was shown (Theorem 3.1) that Df(x) =𝜕f(x) . We can take as a segment of values of the parameter a any segment [−a0, 0] , a0>0, for which [0,v] ∈̄ df(x) only for v̄ belonging to the boundary of the set 𝜕f(x).
5 Construction of the Continuous First Codifferential
Although we have shown how to construct the first and second codifferentials for Lipschitz functions, the constructed codifferentials are not necessarily continuous set-valued mappings (SVM) depending on x. The main difference between codif- ferentiable and quasidifferentiable functions is that the codifferentials are continuous SVMs under rather general conditions in contrast to their predecessors, namely, the quasidifferentials. The next goal is the construction the first and second continuous codifferentials for Lipschitz functions. From the beginning we will limit ourselves to constructing the continuous first codifferentials. Further we will consider how to construct the continuous second codifferentials.
From the beginning we consider the case when the function f(⋅) is hypo-differentiable for which the decomposition
Df(x) =𝜕f(x)⇀(−𝜕f(x)).
f(x+ Δ) =f(x) + max (11)
[a,v]∈df(x)[a+ (v,Δ)]
f(x+ Δ) =f(x) + max
v∈𝜕f(x)(v,Δ) +o(Δ),
f(x+ Δ) =f(x) + max
v∈𝜕f(x)(v,Δ) +ox(Δ)
is correct. Construction of the first continuous codifferential for the Lipschitz func- tion f(⋅) will be based on the following theorem.
We will give some definitions used in the formulations of Theorems.
Definition 5.1 [4, 5] A set-valued mapping G∶ℝn→2ℝn is called upper-semicontinuous at x if from vk∈G(xk) , xk→kx , vk→kv it follows that v∈G(x).
Definition 5.2 [5] The function f(⋅) is called continuous codifferentiable at x if the SVM [df(⋅), df(⋅)] is continuous at x in the Hausdorff metric.
Theorem 5.1 Any subdifferentiable Lipschitz function f ∶ℝn→ℝ , whose subdiffer- ential 𝜕f(⋅) is upper semi-continuous as a SVM at x, is continuously codifferentiable at x.
Proof For any g∈Sn−11 (0) = {w∣∥w∥=1} the decomposition takes place:
where, by definition,
Note, since the directional derivative of the function f(⋅) is bounded in a neigh- borhood of an arbitrary point and 𝜕f(⋅) is upper semicontinuous (UP.SC), the Lip- schitz property of the function f(⋅) follows.
It was proved (Theorem 3.1) that
where the set Df(x) was defined earlier (see Eq. (3)).
Let us construct the hypodifferential of the function f(⋅) at x
Here
is the deviation of v(𝛽, g) from 𝜕f(x).
Let us check the expansion for 𝛼 >0 and g∈Sn−11 (0) f(x+ Δ) =f(x) + max
[a,v]∈df(x)[a+ (v,Δ)] +ox(Δ)
f(x+𝛼g) =f(x) +𝛼 max
v∈𝜕f(x)(v,g) +o(𝛼,g) = =f(x) +𝛼f�(x,g) +o(𝛼,g),
f�(x, g) = 𝜕f(x)
𝜕g , lim
𝛼→+0
o(𝛼, g) 𝛼
=0.
Df(x) =𝜕f(x),
df(x) =conv {[a,v] ∈ℝn+1∣ ∃𝛽 ∈ (0,𝛼0],∃g∈Sn−1 1(0),∃r∈𝜂(x),
a=a(𝛽,g) = −𝛽 𝜌(v(𝛽,g),𝜕f(x)),v(𝛽,g) =𝛽−1
∫
𝛽
0
∇f(r(x,𝜏,g))d𝜏}}.
𝜌(v(𝛽, g),𝜕f(x)) = min w∈𝜕f(x)‖v(𝛽, g) −w‖
Note that
where
Let us denote by
As soon as
the equality Eq. (13) can be rewritten as
where, by definition,
We will show that for all 𝛽∈ (0,𝛼0]
We have the equality
Since for any 𝛽 ∈ (0,𝛼0]
the inequality
f(x+𝛼g) =f(x) + max (12)
[a(v),v]∈df(x)[a+𝛼(v, g)] +o(𝛼, g).
(13) f(x+𝛼g) −f(x) =𝛼(𝛼−1
∫
𝛼
0
∇f(r(x,𝜏, g))d𝜏, g) +o(𝛼, g) =𝛼(v(𝛼, g), g) +o(𝛼, g),
v(𝛼, g) =𝛼−1
∫
𝛼
0
∇f(r(x,𝜏, g))d𝜏.
v(g) =arg max
w∈𝜕f(x)(w, g).
lim
𝛼→+0
𝜌(v(𝛼, g),𝜕f(x)) =0,
f(x+𝛼g) −f(x) =𝛼(v(g),g) +𝛼(v(𝛼,g) −v(g),g) +o(𝛼,g) =𝛼(v(g),g) +o(𝛼,g) = (14)
=𝛼(v(𝛼, g), g) −𝛼𝜌(v(𝛼, g),𝜕f(x)) +o(̂ 𝛼, g),
𝛼→+0lim o(̂ 𝛼, g)
𝛼
=0.
f(x+𝛼g)≥f(x) +a(𝛽, g) +𝛼(v(𝛽, g), g) +o(𝛼, g).
f(x) +𝛼(v(𝛽,g),g) −𝛽 𝜌(v(𝛽,g),𝜕f(x)) = =f(x) +𝛼(v(g),g)+
(15) +𝛼(v(𝛽, g) −v(g), g) −𝛽 𝜌(v(𝛽, g),𝜕f(x)).
𝜌(v(𝛽, g),𝜕f(x))≥(v(𝛽, g) −v(g), g),
f(x) +𝛼(v(𝛽, g), g) −𝛽 𝜌(v(𝛽, g),𝜕f(x))≤
follows from Eq. (15) for small 𝛼 and all 𝛽 > 𝛼. For 𝛽 < 𝛼
as soon as
where
Therefore, we rewrite Eq. (15) as
Note that the function o(̃ ⋅, g) in the decomposition Eq. (17) is uniformly infinitely small with respect to g∈S1n−1(0) , as soon as in the decomposition
the function o(⋅, g) is uniformly infinitely small with respect to g∈Sn−11 (0) , which follows from the Lipschitz quality of f and the upper semicontinuity of 𝜕f.
It follows from Eq. (17) that
The upper and lower semicontinuity and, consequently, continuity of the SVM df(⋅) at x [4] follows from the continuity of the vector-function v as a function of x for any fixed 𝛽 >0 and g∈Sn−11 (0) [8], the upper semicontinuity of the SVM 𝜕f(⋅) , and also from the fact that the maximum in Eq. (18) is attained for such a, v for which a(v) =0.
So, we have considered all possible cases and have proved the equality Eq. (12).
Thus, we have proved the theorem.
We will formulate another theorem. Its proof follows from Theorem 5.1.
Theorem 5.2 Any quasidifferentiable Lipschitz function f ∶ℝn→ℝ , the SVMs
𝜕f(⋅) , 𝜕f(⋅) of which are upper semi-continuous at x and, consequently, the SVM Df() is upper semicontinuous at x, is continuously codifferentiable at x.
≤f(x) +𝛼(v(g), g) +o(𝛼, g) =f(x+𝛼g) (16)
𝛼(v(𝛽,g) −v(g),g) −𝛽 𝜌(v(𝛽,g),𝜕f(x)) =o(̃ 𝛼,g),
lim
𝛽→0(v(𝛽,g) −v(g),g) =0,
lim
𝛽→0
𝜌(v(𝛽,g),𝜕f(x)) =0,
lim
𝛼→+0
o(̃ 𝛼,g) 𝛼
=0.
f(x) +𝛼(v(𝛽, g), g) −𝛽 𝜌(v(𝛽, g),𝜕f(x))≤f(x) +𝛼(v(g), g) +o(̃ 𝛼, g) =f(x+𝛼g).(17)
f(x+𝛼g) =f(x) +𝛼f�(x, g) +o(𝛼, g)
f(x) + max
𝛽∈[0,𝛼0][𝛼(v(𝛽, g), g) −𝛽 𝜌(v(𝛽, g),𝜕f(x))] +o(𝛼, g) =
=f(x) + max (18)
[a,v]∈df(x)[a+𝛼(v, g)] +o(𝛼, g).
Let us give an example of a Lipschitz quasidifferentiable function f(⋅) for which Df(⋅) is not upper semicontinuous and the function f(⋅) is not continuously codifferentiable.
Example 5.1 The graph of the function f ∶ℝ→ℝ consists of the segments, located between the curves −x2,+x2 , with the slopes ±1 . The function f(⋅) is not representable as the difference of two convex functions in a neighborhood of the point zero, since ∨a0f�= ∞ for an arbitrary a>0. The following is valid
𝜕Clf(0) = [−1,+1], Df(0) = {0}. The SVM Df(⋅) is not upper semi-continuous at zero. The function f(⋅) is quasidifferentiable, but not continuously codifferentiable at zero.
6 Applications
Let us consider the problem of finding a minimum of a Lipschitz quasidifferenti- able function f(⋅) . Quasidifferentiable functions are used in optimization widely.
The optimization methods of such functions are described in the list of numerous publications, which can be found in [4, 5, 13, 14]. These methods are based on cal- culations of the subdifferentials and superdifferentials. The necessary minimum con- dition at x∗ in ℝn [4, 5] is
The above-mentioned condition is written in the following form
for a subdifferentiable function f(⋅) . We can conclude from Theorem 4.1 Theorem 6.1 The inclusion
is the necessary condition of optimality.
If the inclusion
is correct, then the point x∗ is a minimum of f. Therefore, Eq. (19) is the sufficient condition for the minimum of f(⋅) at x∗ [4]
Basing on the result of Theorem 4.1 and described above, we have Corollary 6.1 If Eq. (19) is correct, then
−𝜕f(x∗)⊂ 𝜕f(x∗).
0∈𝜕f(x∗)
0∈Df(x∗)
(19)
−𝜕f(x∗)⊂ int 𝜕f(x∗)
0∈ int Df(x∗).
We need to know the subdifferentials and superdifferentials for optimization of quasidifferentiable functions. Therefore, the methods of finding for the subdifferen- tials and superdifferentials are as important for practical optimization of quasidif- ferentiable functions as the rules for calculations of the derivatives for differentiable functions.
Let us define for some 𝛼0>0 the set
We will call V(x) the 𝛼0-subdifferential of f(⋅) at x. The set V(x) is bounded, which follows from the Lipschitz property of the function f(⋅).
It follows from the equality
that the SVM V ∶ℝn→2ℝn is a continuous extension of the mapping Df(⋅) if f(⋅) is a locally DC (difference of convex) function in a neighborhood of x [7, 8]. The SVM V(⋅) can be used in optimization similar to using the 𝜀 - subdifferential mapping for
convex functions [4].
A point x0 is called 𝛼0-optimal if 0∈V(x0) . We can develop the algorithms for finding 𝛼0-optimal points of Lipschitz functions, as described in [4] for finding 𝜀 -optimal points of convex functions.
7 Conclusion
In the article the author is studying codifferentiable functions, introduced by V.F.
Demyanov, and some methods for calculating their codifferentials.
The author’s first subdifferential is used for calculating the first codifferen- tial for any codifferentiable function f(⋅) . The connections among the author’s first subdifferential, the Clark and Michel-Penot subdifferentials are considered.
Finally, the theorem was proven that every Lipschitz quasidifferentiable func- tion, whose subdifferential and superdifferential are upper semicontinuous as the SVMs, is continuously codifferentiable. The continuous 𝛼-subdifferential for Lipschitz functions, similar to the 𝜀-subdifferential for convex functions, is introduced.
The proven theorems, that give the rules for calculating the subdifferentials and codifferentials, are important for practical optimization. Using the continuous exten- sion of the Clark subdifferential, we are able to construct the numerical optimization methods for finding the stationary points [18].
V(𝛼,x) = conv {v∈ℝn∣ ∃g∈Sn−1 1(0),∃r∈𝜂(x),
v=𝛼−1
∫
𝛼
0
∇f(r(x,𝜏,g))d𝜏},V(x) = conv ⋃
𝛼∈(0,𝛼0]
V(𝛼,x).
V(0, x) =Df(x)
Funding No funding was received.
Declarations
Conflicts of interest On behalf of all authors, the corresponding author states that there is no conflict of interest.
Informed consent All data and material are available for public.
References
1. Clarke FH (1975) Generalized Gradients and Applications. Trans Amer Math Soc 205:247–262 2. Clarke FH (1988) Optimization and nonsmooth analysis. Nauka, Moscow
3. Michel P, Penot J-P (1984) Calcul sous-differential pour les fonctions lipschitzienness et non- lipschitziennes. C R Acad Sc Paris Ser I(298):269–272
4. Demyanov VF, Vasilyev LV (1981) Nondifferentiable optimization. Nauka, Moscow
5. Demyanov VF, Rubinov AM (1990) Foundation of nonsmooth analysis and quasidifferentiable calcu- lus. Nauka, Moscow
6. Mordukhovich BS (2006) Variational Analysis and Generalized Differentiation, I: Basic Theory, II:
Applications. Springer, Berlin
7. Prudnikov IM (2003) New constructions for local approximation of Lipschitz functions I. Nonlinear Anal 53(3):373–390
8. Prudnikov IM (2007) New constructions for local approximation of Lipschitz functions. Nonlinear Anal 66(7):1443–1453
9. Mordukhovich BS, Rockafellar RT (2012) Second-order subdifferential calculus with applications to tilt stability in optimization. SIAM J Optim 22:953–986
10. Kolmogorov AN, Fomin CV (1976) Elements of function theory and functional Analysis. Nauka, Moscow, p 542
11. Craven B.D. (1982) Lebesgue measure and integral. Pitman Book Limited. London, p 221 12. Demyanov VF, Gamidov S, Sivelina TI (1986) An algorithm for minimizing a certain class of qua-
sidifferentiable functions. Math Program Study 29:74–85
13. Demyanov VF, Dixon LCW (eds) (1986) Quasidifferential calculus. North-Holland, Amsterdam 14. Demyanov VF, Pallaschke D (eds) (1985) Nondifferentiable optimization: Motivations and applica-
tions. Lecture Notes in Economics and mathematical systems, vol 255. Berlin, Springer Verlag 15. Bagirov AM (2000) Numerical methods for minimizing quasidifferentiable functions: A survey
and comparison. In: Demyanov V, Rubinov A (eds) Quasidifferentiability and related topics.
Nonconvex optimization and its applications, vol 43. Springer, Boston
16. Prudnikov I.M. (2000) Lower convex approximations for Lipschitz functions. J Calcul Math Phys 40(3):378–386
17. Prudnikov IM (2013) Construction method of Exhauster of upper convex approximations. Journal Vestnik St. Peterburg State University Series 10 1:37–51
18. Polak E, Mayne DQ, Wardi Y (1983) On the extention of constrained optimization algorithms from differentiable to nondifferentiable Problems. SIAM J Control Optim 21:179–203
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
