Regularized Gap Function in Game Theory: Proofs and Applications - Prof. Angelia Nedich, Study notes of Engineering

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Lecture 20

Algorithms for VIs

Merit Functions for VIs

November 16, 2008

Outline

• Merit functions for VIs
• D-gap merit functions for VIs
• Note on constructing algorithms

Loss of differentiability

Theorem 1 Let K be a nonempty closed subset and let Ω be a nonempty open set. Assume that f : Ω × K → R is continuous on Ω × K and that ∇xf (x, y) exists and is continuous on Ω × K. Define the function g : Ω → R ∪ {∞} by g(x) ≡ sup y∈K

f (x, y), x ∈ Ω and set M (x) ≡ {y ∈ K : g(x) = f (x, y)}. Let x ∈ Ω be a given vector. Suppose that a nbhd N ⊂ Ω of x exists such that M (x′) is nonempty for all x′^ ∈ N and the set ∪x′∈N M (x′) is bounded. The following two statements (a) and (b) are valid. (a) The function g is directionally differentiable at x and

g′(x; d) = sup y∈M (x)

∇xf (x, y)T^ d.

(b) If M (x) reduces to a singleton, say M (x) = {y(x)}, then g is differentiable at x and

∇g(x) = ∇xf (x, y(x)).

Proof of (a):

• Let d ∈ Rn^ be given and suppose that ¯t > 0 is small enough that x + td ∈ N for t ∈ (0, ¯t]. By definition, we may write for every y ∈ M (x), g(x + td) − g(x) ≥ f (x + td, y) − f (x, y)
• Dividing by t and passing to the limit, we have lim inf t↓ 0 g(x^ +^ td)^ −^ g(x) t

≥ sup y∈M (x)

∇xf (x, y)T^ d.

• Similarly for yt ∈ M (x + td) we have g(x + td) − g(x) ≤ f (x + td, yt) − f (x, yt) = t∇xf (¯xt, yt)T^ d, for some reasonable x¯t ∈ (x, x + td). This implies that g(x + td) − g(x) t ≤ ∇xf (¯xt, yt)T^ d, ∀yt ∈ M (x + td).
• Since the union in the definition is bounded, then for every sequence {tk} of positive numbers converging to zero, if {yk} is any sequence of vectors such that yk^ ∈ M (x + tkd) for every k, then {yk} is bounded and every limit point of this sequence belongs to M (x). It follows that lim sup t↓ 0

g(x + td) − g(x) t

≤ sup y∈M (x)

∇xf (¯xt, yt)T^ d, ∀yt ∈ M (x + td). This implies that (a) holds.

• Finally, by defining ψc(x, y) ≡ F (x)T^ (x−y)− 12 c(x−y)T^ G(x−y), (x, y) ∈ Ω×R, implying that θc(x) = supy∈K ψ(x, y).
• With K being nonempty and closed for each x ∈ Ω, there exists a unique yc(x) ∈ K that maximizes ψc(x, .) on K
• Therefore the supremum is achieved in K and we can replace it by the maximum
• By continuous differentiability of F on Ω, the function θc(x) is differentiable on Ω (follows from the continuity of yc(x)) - see next result

Differentiabilut of the regularized gap function

Theorem 2 (10.2.3) Let K be closed convex and F : Ω → Rn^ be continuous on the open set Ω. Let c be a positive scalar and let G be a symmetric positive definite matrix. Then the following four statements are valid:

(a) For every x ∈ Ω, yc(x) = ΠK,G(x − c−^1 G−^1 F (x)), where ΠK,A(x) is the unique solution to the strictly convex program   

min 12 (y − x)T^ A(y − x) subject to y ∈ K.

(b) θc(x) is continuous on Ω and nonnegative on K.

(c) θc(x) = 0, x ∈ K if and only if x ∈ SOL(K, F ).

(d) If F is continuously differentiable on Ω, then so is θc; moreover

∇θc(x) = F (x) + (JF (x) − cG)T^ (x − yc(x)).

On the other hand, by the variational principle of the maximization problem defining θc(x), we have (z − yc(x))T^ (F (x) + cG(yc(x) − x) ≥ 0 , ∀z ∈ K. Substituting z = x we have (x − yc(x))T^ (F (x) + cG(yc(x) − x) ≥ 0 , which together with (**) implies that 1 2 c(x^ −^ yc(x))

T (^) G(x − yc(x)) − 1 2 c(x^ −^ yc(x))

T (^) G(x − yc(x)) ≥ 0 =⇒ yc(x) = x. By (a), we have x = ΠK,F (x − c−^1 G−^1 F (x)) which shows that x ∈ SOL(K, F ).

• Conversely if x ∈ SOL(K, F ) we have θc(x) ≤ 0 implying that θc(x) = 0. This establishes (c).
• Finally, the gradient formula follows from theorem 10.2.1. By part (b),

∇θc(x) = ∇xψx(x, yc(x)).

Note that the nonnegativity of the regularized gap function is only on K; for a zero of θc to be a solution of VI(K,F), the zero needs to belong to K. Essentially θc is a valid merit function only on K.

• Since K is closed and convex, Tc(x; K) is a closed convex cone containing the vector yc(x) − x; Tc(x; K, F ) is also a closed convex cone.
• When x ∈ SOL(K, F ), the space given by Tc(x; K, F ) reduces to a familiar space. When x ∈ SOL(K, F ), we have θc(x) = 0 and thus x = yc(x) since yc(x) is the unique vector in K satisfying θc(x) = F (x)T^ (x − yc(x)) + 12 c(yc(x) − x)T^ G(yc(x) − x).
• Therefore Tc(x; K) = T (x; K) ∩ (−T (x; K)) which is equal to the lineality space of the tangent cone of K at x.∗
• Moreover, T (x; K) ∩ (−F (x))∗^ = T (x; K) ∩ (F (x))⊥.
• The critical cone C(x; K, F ) of the pair (K,F) is defined as

C(x; K, F ) ≡ T (x; K) ∩ F (x)⊥.

. The elements of the critical cone are called the critical vectors of the pair (K,F) at x. In specifying the local uniquness requirements of a solution to a VI, the critical cone is an essential object. ∗Recall that the intersection C ∩ (−C) is called the lineality subspace of C.

• With a little more effort, we may show that Tc(x; K, F ) = C(x; K), F ∩ (−C(x; K, F )).
• Essentially, if x ∈ SOL(K, F ) we have that the Tc(x; K, F ) is the lineality space of the critical cone C(x; K, F ) of VI(K,F).
• In the next result, which we state without proof = the cone Tc(x; K, F ) needs to be contained in F (x)⊥^ for the a vector x ∈ K to be a solution to V I(K, F ).

Theorem 3 Let K be closed convex and F is continuously differentiable on the open set Ω. Let c be a positive scalar and G be symmetric positive definite. Suppose that x is a stationary point of min θc(x) subject to x ∈ K. Then the following are equivalent:

(a) x solves VI(K,F)

The D-Gap Merit Function

• We conssider a closed convex K without assuming that it is finitely representable
• The domain of definition of θc coincides with that of F - however, the regularized gap program is a constrained minimization problem
• Is there an equivalent unconstrained minimization problem - yes!

Definition 2 Let F : Rn^ → Rn^ be a given mapping and K a closed convex set. Let a, b be positive scalars such that b > a > 0. Then the D-gap function is defined as θab ≡ θa(x) − θb(x), ∀x ∈ Rn, whhere D stands for difference.

• Lemma 1 For every x ∈ Rn^ it holds that b − a 2 ‖x − yb‖^2 G ≤ θab(x).

Proof: By the definition of the D-gap function, theorem 10.2.3 and simple majoriza- tions we have θab(x) = sup x∈K

{F (x)T^ (x − y) − 12 a(x − y)T^ G(x − y)}

− sup x∈K

{F (x)T^ (x − y) − 12 b(x − y)T^ G(x − y)}

≥ {F (x)T^ (x − yb(x)) − 12 a(x − yb(x))T^ G(x − yb(x))} − {F (x)T^ (x − yb(x)) − 12 b(x − yb(x))T^ G(x − yb(x))} = 12 (b − a)‖x − yb‖^2 G.

• The next result shows that the D-gap function is truly an unconstrained merit function of the VI(K,F).

Theorem 5 (10.3.4) Let F be continuous and K closed convex. Let a and b be scalars with b > a > 0 and G be a symmetric positive definite matrix. Suppose that x is a stationary point of θab(x). Then the following are equivalent:

• x ∈ SOL(K, F )
• Tab(x; K, F ) is contained in F (x)⊥
• The following implication holds: d ∈ Tab(x; K, F ), JF (x)T^ d ∈ −Tab(x; K, F )∗^ =⇒ dT^ F (x) = 0

Merit function-based Algorithms

• A natural question is whether we can use the ideas discussed earlier to construct iterative descent type methods.
• However, we face some problems in that regard:
  • The evaluation of θa(x) and its gradient requires the computation of a Euclidean projection on the set K
  • The evaluation of θab(x) and its gradient requires the computation of two Euclidean projections on the set K
  • Neither is naturally associated with a set of nonsmooth equations (specifically neither of these is associated with a systems of nonsmooth equations as the two-norm squared).
• If K is polyhedral, some of the computational questions are easier to deal with (projection is a strictly convex QP)
• Third point has implications in terms of constructing locally fast methods - thats the effort - find a family of Newton approximation that allow for obtaining directions