Characterizations of Asymptotic Cone of the Solution Set of a Composite Convex Optimization Problem

We characterize the asymptotic cone of the solution set of a convex composite optimization problem. We then apply the obtained results to study the necessary and sufficient conditions for the nonemptiness and compactness of the solution set of the problem. Our results generalize and improve some known results in literature.


Introduction
In this paper, we consider the following extended-valued convex composite optimization problem: min f g x , s.t.

CCOP
where S ⊂ R n is closed and convex.The outer function f : R m → R ∪ { ∞} is a convex function; denote by dom f the effective domain of f, that is, dom The inner function g : R n → R m is a vector-valued function such that g S ⊆ dom f.It is known that convex composite optimization model provides a unifying framework for the convergence behaviour of some algorithms.Moreover, it is also a convenient tool for the study of first-and second-order optimality conditions in constrained optimization.The study

Preliminaries
To begin, we must develop our basic definitions and assumptions, to describe the class of convex composite optimization problem which this paper will consider.
Definition 2.1 see 14 .Let K be a nonempty set in R n .Then the asymptotic cone of the set K, denoted by K ∞ , is the set of all vectors d ∈ R n that are limits in the direction of the sequence {x k } ⊂ K, namely:

2.1
In the case that K is convex and closed, then, for any When φ is a proper convex and lower semicontinuous lsc in short function, we have or equivalently

2.5
For the indicator function δ K of a nonempty set K, we have that

2.6
Definition 2.6 see 17 .The function φ : R n → R ∪ { ∞} is said to be coercive if its asymptotic function φ ∞ d > 0, for all d / 0 ∈ R n , and it is said to be countercoercive if its asymptotic function φ ∞ d −∞, for some d / 0 ∈ R n .
Since the inner function is vector valued, we may define some partial order in objective and decision spaces.Let C R m ⊂ R m .We define, for any y 1 , y 2 ∈ R m , Through these partial orders, we introduce some definitions in vector optimization theory.
Definition 2.7 see 18 .Let K ⊂ R n be convex, and a map for any x, y ∈ K and λ ∈ 0, 1 .F is said to be strictly C-convex if for any x, y ∈ K with x / y and λ ∈ 0, 1 .
Definition 2.8 see 18 .A map f : K ⊂ R m → R ∪ { ∞} is said to be C-monotone if, for any x, y ∈ K and x ≤ C y.There holds Next we give an example to show the C-convex and C-monotone of a function.
Then, g is a C-convex function.Let f : R 2 → R be defined by

2.12
Then, f is C-monotone.

Main Results
Before discussing the main results, we propose the following proposition for continuity and convexity of the composite function.
Proposition 3.1.In the problem CCOP , one assumes f is proper, lsc, convex, and C-monotone, and g is continuous and C-convex.Then, the composite function f g • is proper, lsc, and convex.
Proof.Since f is proper and lsc, g is continuous, we derive f g • is proper and lsc by virtue of Proposition 1.40 in 17 .Next we will check the convexity of f g • .Let x 1 , x 2 ∈ S and λ ∈ 0, 1 .By the C-convexity of g, we have 3.1 By the C-monotonicity of f, we have from 3.5 that From the convexity of f, we know 3.3 Combining 3.2 with 3.3 , we may conclude the composite function f g • is convex.The proof is complete.
We denote

3.4
Theorem 3.2.Let the assumptions in Proposition 3.1 hold.Further one assumes the solution set X / ∅.Then, the asymptotic cone of X can be formulated as follows: From the definition of asymptotic cone, we know there exist some sequences {x k } ⊂ X and 3.6 Since f g x is convex, for any fixed λ > 0 when t k is sufficiently large, we get Combining 3.6 with 3.7 , we have x k ≤ f g y , ∀y ∈ S.

3.8
Taking limit in 3.8 as k → ∞, we obtain f g y λu ≤ f g y , ∀y ∈ S.

3.9
That is, u ∈ S 1 and ⇐ For any d ∈ S 1 ∩ S ∞ .By the assumption that X is nonempty, we have where x ∈ X is fixed and t k → ∞.For any y ∈ S, we know Proof.The necessity part follows from the statements in Theorem 3.2 and in Lemma 2.3.Now we prove the sufficiency.We may define a function ϕ : R n → R ∪ { ∞} as ϕ x f g x .Clearly, ϕ is proper, lsc, and convex.By virtue of Proposition 3.1.3of 14 , we know the coercivity of ϕ is a sufficient condition for the nonemptiness and compactness of X. From 3.4 , for all y ∈ S we have

3.17
Consequently This is ϕ ∞ u > 0, for all u / 0 and ϕ is coercive.Thus, X is nonempty and compact.The proof is complete.

Conclusion
In this paper, we characterized the asymptotic cone of the solution set of a convex composite optimization problem CCOP .We obtained the analytical expression of the asymptotic cone of the solution set.Furthermore, we studied the necessary and sufficient conditions for the nonemptiness and compactness of the solution set of the problem via the analytical expression of the asymptotic cone.Our results generalized some known results in 14 and firstly studied the compactness of the solution set of convex composite optimization problems.
2 ∈ R n × R | φ x ≤t} is the epigraph of φ.Consequently, we can give the analytic representation of the asymptotic function φ ∞ :