The concept of a min-max function was introduced in [75] in order to eciently characterize and classify min-max optimization problems and corresponding iterations.
Denition 25 ([75]) We say that a function (x,y) 7→ F(x,y), (x,y) ∈ X × Y, is a min-max function of x,y, ifF is twice Frechet-dierentiable and
∇2xF(x,y)− ∇02x,yF(x,y)(∇2yF(x,y))−1∇2x,yF(x,y)º0, ∇2yF(x,y)≺0, (x,y)∈ X × Y. A max-min function is dened analogously. The denition of a strict min-max function is a straightforward extension of Denition 25.
A min-max function has the following key property.
Proposition 45 ([75]) If (x,y)7→F(x,y), (x,y)∈ X × Y, is a min-max function of x, y, then it has either no stationary points or only min-max points (¯y,x)¯ ∈ X × Y such that [75]
(¯x,y) = arg min¯
x∈Xmax
y∈Y F(x,y), (A.10)
and (¯x,y)¯ is unique ifF is a strictly min-max function.
The class of min-max functions generalizes/ contains the class of twice Frechet-dierentiable convex-concave functions, for which the second inequality in (A.8) is strict (note that it does not generalize the class of twice Frechet-dierentiable convex-concave functions which are strictly convex in x ∈ X [88]). Consequently, a min-max point (A.10) becomes a saddle point (A.9) if F is also strictly convex-concave.
A.5 Some notes on polymatroids
Denition 26 ([154]) A set functionA 7→f(A)∈RCard(E)+ , A⊆E, withE ={1, . . . ,Card(E)}
is referred to as a rank function if i.) f(∅) = 0 (f is normalized),
ii.) if A⊆B ⊆E, then f(A)≤f(B) (f is increasing),
iii.) if A, B⊆E, then f(A) +f(B)≥f(A∪B) +f(A∩B) (f is submodular).
The denition of a polymatroid is the following.
Denition 27 ([154]) A polymatroid is a polytope dened as P(f) ={x≥0 :X
i∈A
xi ≤f(A), A⊆E}, (A.11)
with E={1, . . . ,Card(E)} and a rank function A7→f(A)∈RCard(E)+ ,A⊆E.
For completeness it has to be noted that a polytope characterized by (A.11) but with reversed inequality is sometimes referred to as a contra-polymatroid.
By Denition 27, a polymatroid is a polytope representing an intersection of2Card(E)−1 half-spaces and the nonnegative orthant RCard(E)+ . Every polymatroid hasCard(E)! vertices which lie in the interior of the nonnegative orthant and each one of such vertices corresponds to a dierent permutation of elements in E. Precisely, given a permutationi7→πk(i),i∈E,1≤k≤Card(E)!, the components of the corresponding vertexxπk = (xπ1k, . . . , xπCard(E)k ) are
xπik =f({πk(j)}ij=1)−f({πk(j)}i−1j=1), 1≤i≤Card(E). (A.12) We have a following interesting feature of linear programs dened on a polymatroid.
Lemma 21 ([154]) Given any c∈RCard(E)+ , the solution to the optimization problem
x∈Pmax(f)c0x,
where P(f) denotes the polymatroid determined by the rank function f, is the vertex xπk of P(f), where πk orders the elements of c decreasingly, that is,
cπk(1) ≥cπk(2) ≥...≥cπk(K).
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