The 5th International Conference on Control and Optimization with
Industrial Applications, 27-29 August, 2015, Baku, Azerbaijan
ON WEAK SUBGRADIENTS IN NONCONVEX OPTIMIZATION AND OPTIMALITY
CONDITIONS
Refail Kasimbeyli1
1
Department of Industrial Engineering, Anadolu University, Eskisehir, Turkey
e-mail: rkasimbeyli@anadolu.edu.tr
Consider the problem of minimizing function 𝑓: 𝑆 → 𝑅 over the set 𝑆 ⊆ 𝑅 𝑛 . The following is a
well-known optimality condition in nonsmooth convex analysis which states that [1, Proposition
1.8.1, page 168] if 𝑓: 𝑅 𝑛 → 𝑅 is a convex function then vector 𝑥 minimizes 𝑓 over a convex set
𝑆 ⊆ 𝑅 𝑛 if and only if there exists a subgradient 𝑥 ∗ ∈ 𝜕𝑓(𝑥) such that
𝑥 ∗ (𝑥 − 𝑥) ≥ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑆,
(1)
𝜕𝑓(𝑥) = {𝑥 ∗ ∈ 𝑅 𝑛 ∶ 𝑓(𝑥) − 𝑓(𝑥) ≥ 𝑥 ∗ (𝑥 − 𝑥) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑅 𝑛 }
(2)
where
is a subdifferential of 𝑓at 𝑥. Although the above condition is valid for both convex and nonconvex
functions, for a variety of reasons, if the function 𝑓 is not convex, the subdifferential 𝜕𝑓(∙) is not
a particularly helpful tool. This makes it very tempting to look for definitions of generalized
derivatives and subdifferentials for a nonconvex function. The concepts of generalized
differentiability appropriate for applications to optimization were defined in convex analysis: first
geometrically as the normal cone to a convex set that goes back to Minkowski [2], and then –
much later – analytically as the subdifferential of an extended real-valued convex function. The
latter notion, inspired by the work of Fenchel [3], was explicitly introduced by Moreau [4] and
Rockafellar [5]. It is well known that every convex function 𝑓: 𝑋 → 𝑅 on a Banach space admits
the classical directional derivative
𝑓 ′ (𝑥; ℎ) = lim (
𝑡↓0
𝑓(𝑥 + 𝑡ℎ) − 𝑓(𝑥)
)
𝑡
in any direction ℎ ∈ 𝑋 at any point of its efficient domain 𝑑𝑜𝑚(𝑓). This notion was generalized
by many researchers, such as Clarke [6], Rockafellar [7,8] and others.
By using a general notation 𝑓 𝑔 for the directional derivatives mentioned above, the corresponding
subdifferential of 𝑓 at 𝑥 is defined by
𝜕 𝑔 𝑓(𝑥) = {𝑥 ∗ ∈ 𝑅 𝑛 ∶ 𝑓 𝑔 (𝑥; ℎ) ≥ 𝑥 ∗ (ℎ) 𝑓𝑜𝑟 𝑎𝑙𝑙 ℎ ∈ 𝑅 𝑛 }
The 5th International Conference on Control and Optimization with
Industrial Applications, 27-29 August, 2015, Baku, Azerbaijan
This is a standard way to introduce subgradients via directional derivatives. For convex functions
it is equivalent to the classical subdifferential of convex analysis:
𝜕𝑓(𝑥) = {𝑥 ∗ ∈ 𝑅 𝑛 ∶ 𝑓 ′ (𝑥; ℎ) ≥ 𝑥 ∗ (ℎ) 𝑓𝑜𝑟 𝑎𝑙𝑙 ℎ ∈ 𝑅 𝑛 }.
One of the main purposes of introducing all these generalizations was to obtain optimality
conditions for nonconvex problems. Note that only the necessary part of the optimality condition
given in (1) could be obtained for nonconvex case by using different subdifferential and normal
cone generalizations. Since these generalizations do not satisfy the main property of the classical
subgradient given in (2) for nonconvex functions, one cannot expect to obtain the optimality
condition similar to that given in (1) in nonconvex case. For example, let function 𝑓: 𝑅 → 𝑅 be
defined as 𝑓(𝑥) = |𝑥|. Then, Clarke's directional derivative is 𝑓 0 (0; ℎ) = |ℎ| for all ℎ ∈ 𝑅 and
𝜕 𝑜 𝑓(0) = [−1,1]. It is clear that hyperplanes with normal vectors (slopes in this case) from this
subdifferential cannot be used to support the epigraph of the function. A similar interpretation is
also valid for Mordukhovich's subdifferential, which is defined for this function as 𝜕 𝑀 𝑓(0) =
{−1,1}.
The relation given in (2) for the classical subdifferential means the existence of the supporting
hyperplane to epigraph of 𝑓 at the point under consideration. It is obvious that to use such a
hyperplane as a supporting surface for nonconvex functions may not be possible in general. This
means that the investigation of nonconvex case can be made available by changing the supporting
philosophy and using suitable nonlinear supporting surfaces.
By using a cone of concave functions (instead of linear ones used in convex analysis), Gasimov
[9] investigated duality relations and obtained optimality conditions for some classes of nonconvex
optimization problems in both single-objective optimization and vector optimization. Azimov and
Gasimov [10,11] constructed a duality scheme by using a special class of concave functions. They
used a class of superlinear conic functions. A graph of such a function is a conical surface which
can be used as a supporting surface for a certain class of nonconvex sets. By using the mentioned
class of superlinear functions, they introduced the concept of weak subdifferential and derived a
collection of optimality conditions and duality relations for a wide class of nonconvex optimization
problems. The superlinear conic functions were applied to construct the so-called sharp augmented
Lagrangian functions for nonconvex constrained optimization problems and to derive zero duality
The 5th International Conference on Control and Optimization with
Industrial Applications, 27-29 August, 2015, Baku, Azerbaijan
gap conditions. In this paper, we study optimality conditions for nonconvex problems involving a
special class of directionally differentiable functions. By using the weak subgradient notion, we
generalize the necessary and sufficient optimality condition given in (1) to a nonconvex case. We
show that the point 𝑥 minimizes 𝑓 over a set 𝑆 ⊆ 𝑅 𝑛 if and only if there exists a weak subgradient
(𝑥 ∗ , 𝛼) ∈ 𝜕 𝑤 𝑓(𝑥) such that
𝑥 ∗ (𝑥 − 𝑥) + 𝛼‖𝑥 − 𝑥‖ ≥ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑆.
Keywords: directional derivative, weak subgradient, nonconvex optimization, duality, optimality
conditions.
AMS Subject Classification: 90C26, 90C30, 90C46.
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