LECTURE 9
LECTURE OUTLINE
• Min-Max Problems • Saddle Points • Min Common/Max Crossing for Min-Max −−−−−−−−−−−−−−−−−−−−−−−−−−−−
Given φ : X × Z →, where X ⊂n, Z ⊂m consider minimize sup φ(x, z) z∈Z subject to x ∈ X and maximize inf φ(x, z) x∈X subject to z ∈ Z.
• Minimax inequality (holds always)
sup inf φ(x, z) ≤ inf sup φ(x, z) ∈ ∈ z∈Z x X x X z∈Z SADDLE POINTS
Definition: (x∗,z∗) is called a saddle point of φ if φ(x∗,z) ≤ φ(x∗,z∗) ≤ φ(x, z∗), ∀ x ∈ X, ∀ z ∈ Z
Proposition: (x∗,z∗) is a saddle point if and only if the minimax equality holds and
∗ ∗ x ∈ arg min sup φ(x, z),z∈ arg max inf φ(x, z) (*) x∈X z∈Z z∈Z x∈X
Proof: If (x∗,z∗) is a saddle point, then
∗ ∗ ∗ inf sup φ(x, z) ≤ sup φ(x ,z)=φ(x ,z ) x∈X z∈Z z∈Z ∗ = inf φ(x, z ) ≤ sup inf φ(x, z) x∈X z∈Z x∈X By the minimax inequality, the above holds as an equality holds throughout, so the minimax equality and Eq. (*) hold. Conversely, if Eq. (*) holds, then
∗ ∗ ∗ sup inf φ(x, z) = inf φ(x, z ) ≤ φ(x ,z ) z∈Z x∈X x∈X ∗ ≤ sup φ(x ,z) = inf sup φ(x, z) z∈Z x∈X z∈Z Using the minimax equ., (x∗,z∗) is a saddle point. VISUALIZATION
φ(x,z) Curve of maxima Saddle point φ(x,z(x)^ ) (x*,z*)
Curve of minima x φ(x(z),z^ )
z
The curve of maxima φ(x, zˆ(x)) lies above the curve of minima φ(ˆx(z),z), where
zˆ(x) = arg max φ(x, z), xˆ(z) = arg min φ(x, z). z x
Saddle points correspond to points where these two curves meet. MIN COMMON/MAX CROSSING FRAMEWORK
• Introduce perturbation function p : m → [−∞, ∞] p(u) = inf sup φ(x, z) − uz ,u∈m ∈ x X z∈Z
• Apply the min common/max crossing framework with the set M equal to the epigraph of p. • Application of a more general idea: To evalu- ate a quantity of interest w∗, introduce a suitable perturbation u and function p, with p(0) = w∗. • Note that w∗ = inf sup φ. We will show that: − Convexity in x implies that M is a convex set. − Concavity in z implies that q∗ = sup inf φ.
w w
φ infx supz (x,z) * = min common value w M = epi(p) M = epi(p) (µ,1) (µ,1)
φ infx supz (x,z) = min common value w* φ supzinfx (x,z) 0 u = max crossing value q* 0 u q(µ) µ φ q( ) supzinfx (x,z) = max crossing value q* (a) (b) IMPLICATIONS OF CONVEXITY IN X
Lemma 1: Assume that X is convex and that for each z ∈ Z, the function φ(·,z):X →is convex. Then p is a convex function. Proof: Let − ∈ F (x, u)= supz∈Z φ(x, z) u z if x X, ∞ if x/∈ X.
Since φ(·,z) is convex, and taking pointwise supre- mum preserves convexity, F is convex. Since
p(u) = inf F (x, u), x∈n and partial minimization preserves convexity, the convexity of p follows from the convexity of F . Q.E.D. THE MAX CROSSING PROBLEM
• The max crossing problem is to maximize q(µ) over µ ∈n, where