Projection Methods for Monotone Variational Inequalities: Fixed-Point Iteration - Prof. An, Study notes of Engineering

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Lecture 21

Algorithms for Monotone VIs

Projection Methods

November 17, 2008

Outline

• Projection Methods
  • Basic Fixed-Point Iteration
  • Extra-gradient Method
  • Hyperplane Projection Method

Basic Fixed-Point Iteration

Throughout the rest, we assume that K ⊂ Rn^ is closed and convex, and

F : K → Rn^ is continuous mapping.

• We consider the “skewed” natural map F (^) K,Dnat(x) = x − ΠK,D[x − D−^1 F (x)],

where D is a symmetric and positive definite matrix.

• Such a matrix, induces and inner product and a norm in Rn: < x, y >D = xT^ Dy, ‖x‖ =

xT^ Dx.

• The skewed projection ΠK,D on K is just a projection with respect to

norm ‖ · ‖D,

x ˆ = ΠK,D[x] ⇐⇒ xˆ solves min y∈K ‖x − y‖^2 D.

Fact

The projection mapping x 7 → ΠK,D[x] is nonexpansive in the norm ‖ · ‖D

• Fact A vector x∗^ solves V I(KF ) if and only if x∗^ is zero of the skewed

natural map, i.e.,

x∗^ = ΠK,D[x∗^ − D−^1 F (x∗)].

• Define Φ(x) = ΠK,D[x − D−^1 F (x)] and note that Φ : K → K.
• When D = I and Φ is contraction with respect to Euclidean norm), we

have seen that fixed-point method xk+1 = Φ(xk) produces a sequence

with accumulation points being fixed points of Φ (for any initial x 0 ∈ K)

• The idea of fixed-point here is similar: if we could ensure that Φ(x) =

ΠK,D[x − D−^1 F (x)] is a contraction in norm ‖ · ‖D, then we could

use “fixed-point” method to find a fixed point of Φ(x) and hence, a

solution to V I(K, F )

Convergence for Strongly Monotone Mappings

Theorem 1 Let F : K → Rn. Suppose F is strongly monotone and

Lipschitz continuous on K,

(F (x) − F (y))T^ (x − y) ≥ μ‖x − y‖^2 , ‖F (x) − F (y)‖ ≤ L‖x − y‖^2.

Also, let

λmax < 2 μ L^2

λ^2 min,

where λmax and λmin are the largest and smallest eigenvalues of D.

Then, the mapping ΠK,D[x − D−^1 F (x)] is contraction in ‖ · ‖D with

contraction factor

η = 1 − L

2 λmax λ^2 min

( 2 μλ^2 min L^2 −^ λmax

) .

Therefore, the sequence {xk} converges to the unique solution of V I(K, F ).

Proof: For any two vectors x, y in K, we have

‖ΠK,D[x − D−^1 F (x)] − ΠK,D[y − D−^1 F (y)]‖^2 D ≤ ∥∥x − D−^1 F (x) − (y − D−^1 F (y))∥∥^2 D.

Why? By expanding the last term, we have

‖ΠK,D[x − D−^1 F (x)] − ΠK,D[y − D−^1 F (y)]‖^2 D ≤ ∥∥x − y‖^2 D − 2(F (x) − F (y))T^ (x − y) + ‖F (x) − F (y)‖^2 D− 1.

Using the strong convexity, we have

(F (x) − F (y))T^ (x − y) ≥ μ‖x − y‖^2 ≥ μ λmax

‖x − y‖^2 D. (1)

From

‖F (x) − F (y)‖^2 D− 1 ≤ 1 λmin

‖F (x) − F (y)‖^2

and Lipschitz continuity of F , we obtain

‖F (x) − F (y)‖^2 D− 1 ≤ L

2 λ^2 min

‖x − y‖^2 D (2)

Co-coercive Mapping

The projection method can be used to solve V I(K, F ) with co-coercive

mapping. The reason why this work is that the (Euclidean) projection

is co-coercive, and when F is co-coercive, then the τ -natural mapping

x 7 → ΠK [x − τ F (x)] is also co-coercive for some range of values of τ.

In particular, we have the following results

Lemma 1 Let F : K → Rn^ be co-coercive with constant c. If 0 < τ < 4 c,

then F K,τnat (x) = ΠK [x − τ F (x)] is co-coercive with constant 1 − 4 τc.

Proof: See Lemma 12.1.7 of FP-II.

Convergence for Co-coercive Mapping

Theorem 2 Assume that V I(K, F ) has a solution. Let F : K → Rn^ be

co-coercive with constant c. Consider the projection method

xk+1 = Π[xk − τ F (xk)],

where τ < 2 c. Then, the sequence {xk} converges to a solution of

V I(K, F ).

Proof We have for a fixed point x∗^ of F Knat (also a fixed point of F K,τnat for

any τ > 0 ),

‖xk+1 − x∗‖^2 ≤ ‖Π[xk − τ F (xk) − Π[x∗^ − τ F (x∗)]‖^2 ≤ ‖xk − x∗^ − τ (F (xk) − F (x∗))‖^2 = ‖xk − x∗‖^2 − 2 τ (F (xk) − F (x∗))T^ (x − x∗)

• τ 2 ‖F (xk) − F (x∗)‖^2

implying F (xk) → F (x∗).

(This also implies that {F (x∗) | x∗^ ∈ SOL(K, F )} is a singleton - why

would this be expected). In view of the preceding two relations, it follows

that

dist(xk, SOL(K, F )) → 0.

Note, also that (3) implies that the scalar sequence {‖xk − x∗‖} is nonin-

creasing for any fixed point x∗. Therefore, the scalar sequence {‖xk − x∗‖}

is convergent for any fixed point x∗. This, and dist(xk, SOL(K, F )) → 0

imply that {xk} is convergent and its limit point is in SOL(K, F ).

The proof in the FP-II is given in Lemma 12.1.15 for the case when

• τ is varying and τkF (xk) is replaced with F k(xk), where all V I(K, F k)

have the same solution set.

• K = Rn
• Assuming that each mapping F k^ : Rn^ → Rn^ is co-coercive with ck, and

the following condition is satisfied: infk ck > 1 / 2.

Extra-Gradient Method

The name of method comes from its equivalent version in optimization

(F = ∇f (x)).

It has two projection steps

zk+1 = ΠK [x − τ F (xk)], xk+1 = ΠK [xk − τ F (zk+1)]

The main iterate is xk+1. The extra-iterate zk+1 is used to construct the

direction for moving away from xk.

• The advantage of taking an extra step is that the algorithm performs

better than the projection method

• The convergence analysis still requires F to be Lipschitz
• It can be used to solve V I(K, F ) with a pseudo-monotone map F
• We will study it as applied to a monotone V I(K, F )

By the monotonicity of F and x∗^ ∈ SOL(K, F ), it follows

(F (zk+1)−F (x∗))T^ (zk+1)−x∗) ≥ 0 =⇒ F (zk+1)T^ (zk+1−x∗) ≥ 0.

Hence, F (zk+1)T^ (zk+1 − xk+1) + F (zk+1)T^ (xk+1 − x∗) ≥ 0 implying that

F (zk+1)T^ (zk+1 − xk+1) ≥ F (zk+1)T^ (x∗^ − xk+1)

Using this relation in (4), we see

‖xk+1 − x∗‖^2 ≤ ‖xk − x∗‖^2 − ‖xk+1 − xk‖^2 + 2τ F (zk+1)T^ (zk+1 − xk+1).

By writing xk+1 − xk = (xk+1 − zk+1) + (zk+1 − xk) and expanding the

squared-norm of this term, and then combining the terms that are in the

inner product with zk+1 − xk+1, we obtain

‖xk+1 − x∗‖^2 ≤ ‖xk − x∗‖^2 − ‖xk+1 − zk+1‖^2 − ‖zk+1 − xk‖^2 +2(xk+1 − zk+1)T^ (xk − τ F (zk+1) − zk+1).

We can further write [by adding and subtracting τ F (xk)]

(xk+1 − zk+1)T^ (xk − τ F (zk+1) − zk+1) = (xk+1 − zk+1)T^ (xk − τ F (zk) − zk+1) +τ (xk+1 − zk+1)T^ (F (xk) − F (zk+1))

Since xk+1 ∈ K and zk+1 = ΠK [xk − τ F (xk)], the first term on the right

hand side is nonnegative (by projection property). Thus, by using this and

Lipschitz continuity of F , we have

(xk+1 − zk+1)T^ (xk − τ F (zk+1) − zk+1) ≤ τ (xk+1 − zk+1)T^ (F (xk) − F (zk+1)) ≤ τ L‖xk+1 − zk+1‖ · ‖xk − zk+1‖ ≤ 12 (‖xk+1 − zk+1‖^2 + τ 2 L^2 ‖xk − zk+1‖^2 )

Convergence of Extra-Gradient Method

Theorem 4 Let F be monotone and Lipschitz continuous over K with

constant L. Let SOL(K, F ) be nonempty. Then, with τ < L^1 , the se-

quence {xk} generated by the extra-gradient method converges to a solu-

tion of V I(K, F ).

Proof The line of proof relies on the basic iterate relation, and follows a line

of analysis similar to that of Theorem for co-coercive map and projection

method. See FP-II Theorem 12.1.11.

For the estimate in basic relation to result in convergence, we need τ < 1 L.

• In practice, often L is not available
• We can use diminishing step τk at iteration k, with ∑ k τk = +∞
• But the convergence will slow down
• We next consider a modification of the method not relying on the

Lipschitz continuity

Hyperplane Projection Method

Like the extra-gradient method, this method generates an extra-iterate

yk = ΠK [xk − τ F (xk)]. However, the use of this point in constructing the

new iterate is different.

In particular, Armijo search rule is used to determine a point zk defining a

hyperplane

Hk = {x ∈ Rn^ | F (zk)T^ (xk − zk) = 0},

which separates xk^ strongly from the solution set SOL(K, F ) [for contin-

uous monotone map, this set is closed and convex - possibly empty].

In particular, zk is such that, for some positive scalars t, σ, τ ,

F (zk)T^ (xk − zk) ≥ tkσ τ

‖yk − xk‖^2 ,

which strongly separates xk from SOL(K, F ) whenever yk 6 = xk in view of

0 ≥ F (x∗)T^ (x∗^ − zk) ≥ F (zk)T^ (x∗^ − zk) for all x∗^ ∈ SOL(K, F ).