www-ai.cs.tu-dortmund.de/en/LEHRE/VORLESUNGEN/LOPT/WS14/notes/subgrad_optimality.pdf
and σ2 be the support functions of the nonempty closed convex sets S1 and S2. For positive scalars t1 and t2, t1σ1 + t2σ2 is the support function of cl (t1S1 + t2S2).
Proof Let S = cl (t1S1 + t2S2) which [...] = sup{〈t1s1 + t2s2, d〉 : s1 ∈ S1, s2 ∈ S2}.
In the above expression, s1 and s2 run independently in their index sets S1 and S2. Therefore, we have
σS(d) = t1 sup s∈S1
〈s, d〉+ t2 sup s∈S2
〈s, d〉.
The following [...] d
(h✏1)(d)
h(d)
f ′(x; 0) = 0 is obvious from definition. From convexity,
1
2 f ′(x; d) +
1
2 f ′(x;−d) ≥ f ′
( x; d
2 − d
2
) = 0
which gives the final claim.
3.3 An Alternative Definition of Subdifferential …