Let
X be a real locally convex Hausdorff vector space, and
\(C\subseteq X\) be a convex set. Let
T be an index set and let
f,
g,
\(f_{t}\),
\(g_{t}\),
\(t\in T \) be proper convex functions such that
\(f-g\) and
\(f_{t}-g_{t}\),
\(t\in T\), are proper functions. Here and throughout the whole paper, following [
17, p.39], we adapt the convention that
\((+\infty)+(-\infty)=(+\infty)-(+\infty)=+\infty \),
\(0\cdot(+\infty)=+\infty\), and
\(0\cdot(-\infty)=0\). Then
$$ \emptyset\neq \operatorname{dom} f\subseteq\operatorname{dom} g \quad\mbox{and}\quad \emptyset\neq \operatorname{dom} f_{t}\subseteq \operatorname{dom} g_{t}. $$
(3.1)
Let
\(A\neq\emptyset\) be the solution set of the following system with the assumption that
\(A\cap \operatorname{dom} (f-g)\) is nonempty:
$$ x\in C;\quad f_{t}(x)-g_{t}(x)\le0, \quad\mbox{for each } t\in T, $$
(3.2)
and let
\(A^{\operatorname{cl}}\) be the solution set of the following system:
$$ x\in C;\quad f_{t}(x)-\operatorname{cl}g_{t}(x) \le0, \quad\mbox{for each } t\in T. $$
(3.3)
Then
\(A^{\operatorname{cl}}\subseteq A\). Following [
18], we use
\({\mathbb {R}}^{(T)}\) to denote the space of real tuples
\(\lambda=(\lambda_{t})\) with only finitely many
\(\lambda_{t}\neq0\), and let
\({\mathbb {R}}_{+}^{(T)}\) denote the nonnegative cone in
\({\mathbb {R}}^{(T)}\), that is,
$${\mathbb {R}}_{+}^{(T)}:=\bigl\{ \lambda=(\lambda_{t})\in {\mathbb {R}}^{(T)}: \lambda_{t}\ge 0, \mbox{for each } t\in T \bigr\} . $$
For simplicity, we denote
$$H^{\ast}:=\operatorname{dom}g^{\ast}\times\prod _{t\in T}\operatorname{dom} g_{t}^{\ast}$$
and
$$\partial H(x ):=\partial g(x )\times\prod_{t\in T} \partial g_{t}( x ), \quad\mbox{for each } x\in X. $$
To make the dual problem considered here well defined, we further assume that cl
g and
\(\operatorname{cl} g_{t}\),
\(t\in T\), are proper. Then
\(H^{\ast}\neq\emptyset\). For the whole paper, any elements
\(\lambda\in {\mathbb {R}}^{(T)}\) and
\(v^{\ast}\in \prod_{t\in T}\operatorname{dom} g_{t}^{\ast}\) are understood as
\(\lambda=(\lambda_{t})\in {\mathbb {R}}^{(T)}\) and
\(v^{\ast}=(v_{t}^{\ast})\in\prod_{t\in T}\operatorname{dom} g_{t}^{\ast}\), respectively. Following [
15], we define the characteristic set
K for the DC optimization problem (
1.1) by
$$\begin{aligned} K:= \bigcup_{\lambda\in {\mathbb {R}}_{+}^{(T)}} \biggl(\bigcap _{(u^{\ast},v^{\ast})\in H^{\ast}} \biggl(\operatorname{epi} \biggl(f+ \delta_{C}+\sum_{t\in T}\lambda_{t}f_{t} \biggr)^{\ast}-\bigl(u^{\ast},g^{\ast}\bigl(u^{\ast}\bigr)\bigr)-\sum_{t\in T}\lambda_{t} \bigl(v_{t}^{\ast}, g_{t}^{\ast}\bigl(v_{t}^{\ast}\bigr)\bigr) \biggr) \biggr), \end{aligned}$$
(3.4)
where we adopt the convention that
\(\bigcap_{t\in\emptyset}S_{t}=X\) (see [
17, p.2]). Below we will make use of the subdifferential
\(\partial h(x)\) for a general proper function (not necessarily convex)
\(h:X\to\overline{{\mathbb {R}}}\); see (
2.3). Clearly, the following equivalence holds:
$$ x_{0}\mbox{ is a minimizer of }h\mbox{ if and only if }0 \in\partial h(x_{0}). $$
(3.5)
For each
\(x\in X\), let
\(T(x)\) be the active index set of system (
3.2), that is,
$$T(x):=\bigl\{ t\in T: f_{t}(x)-g_{t}(x)=0\bigr\} . $$
Define
\(N^{\prime}(x)\) by
$$N^{\prime}(x):=\bigcup_{\lambda\in {\mathbb {R}}_{+}^{(T)}} \biggl(\bigcap_{(u^{\ast},v^{\ast})\in H^{\ast}} \biggl(\partial\biggl(f+ \delta_{C}+\sum_{t\in T(x)}\lambda _{t}f_{t}\biggr) (x)-u^{\ast}-\sum _{t\in T(x)}\lambda_{t}v_{t}^{\ast}\biggr) \biggr) $$
(3.6)
and define
\(N_{0}^{\prime}(x)\) by
$$ N_{0}^{\prime}(x):=\bigcup _{\lambda\in {\mathbb {R}}_{+}^{(T)}} \biggl(\bigcap_{(u^{\ast},v^{\ast})\in\partial H(x)} \biggl( \partial\biggl(f+\delta_{C}+\sum_{t\in T(x)} \lambda_{t}f_{t}\biggr) (x)-u^{\ast}-\sum _{t\in T(x)}\lambda_{t}v_{t}^{\ast}\biggr) \biggr). $$
(3.7)
Then, for each
\(x\in X\),
$$N^{\prime}(x)\subseteq N_{0}^{\prime}(x). $$