A Robust von Neumann Minimax Theorem for Zero ... - Semantic Scholar
A Robust von Neumann Minimax Theorem for Zero-Sum Games under Bounded Payoff Uncertainty∗ V. Jeyakumar†, G.Y. Li‡ and G. M. Lee§ Revised Version: January 20, 2011
Abstract The celebrated von Neumann minimax theorem is a fundamental theorem in two-person zero-sum games. In this paper, we present a generalization of the von Neumann minimax theorem, called robust von Neumann minimax theorem, in the face of data uncertainty in the payoff matrix via robust optimization approach. We establish that the robust von Neumann minimax theorem is guaranteed for various classes of bounded uncertainties, including the matrix 1-norm uncertainty, the rank-1 uncertainty and the column-wise affine parameter uncertainty. Key words. Robust von Neumann minimax theorem, minimax theorems under payoff uncertainty, robust optimization, conjugate functions.
1
Introduction
The celebrated von Neumann Minimax Theorem [21] asserts that, for an (n×m) matrix M, minn xT My, xT My = max minn max m m y∈S
x∈S y∈S
x∈S
n
where S is the n-dimensional simplex. It is a fundamental equality in two-person zero-sum games [19]. Due to its importance in mathematics, decision theory, economics and game theory, numerous generalizations have been given in the literature (see [9, 10, 11, 18] and ∗
The authors are grateful to the referee and the editors for their valuable comments and constructive suggestions which have contributed to the final preparation of the paper. The first and second authors were partially supported by a grant from the Australian Research Council. The third author was supported by the Korea Science and Engineering Foundation (KOSEF) NRL program grant funded by the Korea government (MEST)(No. ROA-2008-000-20010-0). † Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia ‡ Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia § Department of Applied Mathematics, Pukyong National University, Busan 608-737, Korea
1
other reference therein). However, these generalizations and their applications have so far been limited mainly to problems without data uncertainty, despite the reality of data uncertainty in many real-world problems due to modeling or prediction errors [2, 3, 5, 4, 6, 13, 14, 15]. For related recent work on incomplete-information games, see [1] and other references therein. The purpose of this paper is to present a new form of the von Neumann minimax theorem, called robust von Neumann minimax theorem, for two-person zero sum games under data uncertainty via robust optimization and to establish that the robust von Neumann minimax theorem always holds under various classes of uncertainties, including the matrix 1-norm uncertainty, the rank-1 uncertainty and the column-wise affine parameter uncertainty. minn xT My xT My and the maxmin value γ2 := max The minimax value γ1 := minn max m m y∈S
x∈S y∈S
can be calculated by the following two optimization problems: γ1 = max xT My ≤ t} and γ2 =
y∈S m
max
(y,t)∈S m ×R
x∈S
min
(x,t)∈S n ×R
{t :
{t : minn xT My ≥ t}. Whenever the cost function x∈S
is affected by data uncertainty, the effect of uncertain data on the cost matrix M can be captured by a new matrix M(u) where u is an uncertain parameter and it belongs to the compact uncertainty set U ⊆Rq . For instance, the effect of uncertain data a1 a2 (a1 , a2 , a3 ) on the cost matrix M = can be captured by the new matrix a2 a3 a1 (u) a2 (u) M(u) = where u ∈ U ⊆ R. So, the minimax value and the maxmin a2 (u) a3 (u) value in the face of cost matrix data uncertainty can be obtained by the following two uncertain optimization problems (UPI )
minn
xT M(u)y ≤ t} {t : max m
max
{t : minn xT M(u)y ≥ t}.
(x,t)∈S ×R
y∈S
and (UPII )
(y,t)∈S m ×R
x∈S
The robust counterpart [3, 15, 16] of the uncertain optimization problem (UPI ) is a deterministic optimization problem defined by (RPI )
min
(x,t)∈S n ×R
xT M(u)y ≤ t, for all u ∈ U}, {t : max m y∈S
(1.1)
and the optimistic counterpart [2, 15, 16] of the uncertain optimization problem (UPII ) is another deterministic optimization problem defined by (OPII )
max m
(y,t)∈S ×R
{t : minn xT M(u)y ≥ t, for some u ∈ U}. x∈S
(1.2)
The robust minimax theorem states that the optimal values of the robust counterpart problem (RPI ) (“worst possible loss of Player I”) and the optimistic counterpart (OPII ) (“best possible gain of Player II”) are equal. Equivalently, it asserts that minn xT M(u)y. min max max xT M(u)y = max max m
x∈S n y∈S m u∈U
u∈U y∈S
2
x∈S
(1.3)
Employing conjugate analysis [20] and Ky Fan’s minimax theorem [8], we derive the robust minimax equality (1.3) under a concave-like condition. We also show that the concave-like condition is also necessary for the robust minimax theorem in the sense that it holds if and only if inf max max{xT M(u)y + xT a} = max max inf {xT M(u)y + xT a}, ∀a ∈ Rn .
x∈A y∈B u∈U
u∈U y∈B x∈A
Importantly, we establish that the robust minimax theorem always holds for various classes of bounded uncertainty sets, including the matrix 1-norm uncertainty set, the rank-1 matrix uncertainty set, the column-wise affine parameter uncertainty set and isotone matrix-data uncertainty set. Consequently, we also derive a robust theorem of the alternative for uncertain linear inequality systems from the robust minimax theorem.
2
A Robust Minimax Theorem under Uncertainty
In this Section, we present a concave-like condition ensuring (1.3). We also show that the condition is also necessary for (1.3) to hold for every linear perturbation. We begin this section by fixing notation and preliminaries of convex analysis. Throughout this paper, Rn denotes the Euclidean space with dimension n. The inner product in Rn is defined by hx, yi := xT y for all x, y ∈ Rn . The nonnegative orthant of Rn is denoted by Rn+ and is defined by Rn+ := {(x1 , . . . , xn ) ∈ Rn : xi ≥ 0}. For a set A in Rn , the convex hull of A is denoted by coA. We say A is convex whenever µa1 + (1 − µ)a2 ∈ A for all µ ∈ [0, 1], a1 , a2 ∈ A. A function f : Rn → R ∪ {+∞} is said to be convex if for all µ ∈ [0, 1] f ((1−µ)x+µy) ≤ (1−µ)f (x)+µf (y) for all x, y ∈ Rn . The function f is said to be concave whenever −f is convex. As usual, for any proper (i.e., domf 6= ∅) convex function f on Rn , its conjugate function f ∗ : Rn → R ∪ {+∞} is defined by f ∗ (x∗ ) = supx∈Rn {hx∗ , xi − f (x)} for all x∗ ∈ Rn . Clearly, f ∗ is a proper lower semicontinuous convex function and for any proper lower semicontinuous convex functions f1 , f2 (cf. [12, 17]), f1 ≤ f2 ⇔ f1∗ ≥ f2∗ ⇔ epif1∗ ⊆ epif2∗ .
(2.1)
The following special case of Ky Fan minimax theorem [8] plays a key role in deriving our robust von Neumann minimax theorem. Recall from Ky Fan [8] that the function f (., y) is said to be convex-like whenever (∀x1 , x2 ∈ C) (∀λ ∈ (0, 1)) (∃x3 ∈ C) (∀y ∈ D) f (x3 , y) ≤ λf (x1 , y) + (1 − λ)f (x2 , y). The function f (x, .) is said to be concave-like whenever (∀y1 , y2 ∈ D) (∀λ ∈ (0, 1)) (∃y3 ∈ D) (∀x ∈ C) f (x, y3 ) ≥ λf (x, y1) + (1 − λ)f (x, y2 ), where f : C × D → R and C and D are sets. Theorem 2.1. [8, 11] Let C be a compact subset of Rn and let D ⊂ Rm . Let f : C × D → R. Suppose that f (., y) is concave-like and f (x, .) is convex-like and that f (., y) is upper-semi-continuous. Then, max inf f (x, y) = inf max f (x, y). x∈C y∈D
y∈D x∈C
3
Theorem 2.1. (Robust von Neumann Minimax Theorem) Let A be a closed convex subset of Rn and let B be a convex compact subset of Rm . Let U be a convex compact subset of Rq . Assume that (∀λ ∈ [0, 1]) (∀(y1 , u1 ), (y2, u2 ) ∈ B × U) (∃(y, u) ∈ B × U) (∀x ∈ A) xT M(u)y ≧ λxT M(u1 )y1 + (1 − λ)xT M(u2 )y2 .
(2.2)
Then inf max max xT M(u)y = max max inf xT M(u)y.
x∈A y∈B u∈U
u∈U y∈B x∈A
Proof. Let z = (y, u) ∈ Rm × Rq and define F : Rn × Rm × Rq → R by F (x, z) = xT M(u)y. Then, we see that x 7→ F (x, z) is linear for any z ∈ Rn × Rq and z 7→ F (x, z) is concavelike. So, by Theorem 2.1 gives us that inf max F (x, z) = max inf F (x, z).
x∈A z∈B×U
z∈B×U x∈A
Thus, the conclusion follows.
Pn Corollary 2.1. Let A = {(x1 , . . . , xn ) ∈ Rn | xi ≧ 0, i = 1, . . . , n, i=1 xi = 1}; let m BSbe a convex compact subset of R and let U be a convex compact subset of Rq . If ( u∈U , y∈B {M(u)y} − Rn+ ) is a convex set then inf max max xT M(u)y = max max inf xT M(u)y.
x∈A y∈B u∈U
u∈U y∈B x∈A
Proof. The conclusion will follow from Theorem 2.1 if we show that (2.2) S holds. To see this, let λ ∈ [0, 1]) and let (y1 , u1 ), (y2, u2 ) ∈ B×U). Then, M(ui )yi ∈ ( u∈U , y∈B {M(u)y}− Rn+ ), for i = 1, 2. By the convexity hypothesis, we can find (y, u) ∈ B × U such that M(u)y − λM(u1 )y1 − (1 − λ)M(u2 )y2 ∈ Rn+ .
This together with the fact that x ∈ A gives us the required inequality (2.2). n It is worth noting Pn that whenever A is simplex, i.e. A = {(x1 , · · · , xn ) ∈SR | xi ≧ 0, i = 1, · · · , n, i=1 xi = 1}, (2.2) is equivalent to the convexity of the set ( u∈U , y∈B {M(u)y}− Rn+ ). As an illustration, we provide a simple numerical example verifying Corollary 2.1.
Example 2.1. Let A = B = {(x1 , x2 ) : x1 , x2 ≥ 0 and x1 + x2 = 1}. Let U = [0, 1] and let M(u) = M0 + uM1 where −1 0 0 5/6 . and M1 = M0 = 0 1 1 1/2 Then,
[
[
{M(u)y} =
u∈U , y∈B
{
u∈[0,1] (y1 ,y2 )∈co{(0,1),(1,0)}
=
[
u∈[0,1]
co{
4
1 2
−u 1
1 2
5 6
+u
−u , }. +u 1 5 6
y1 y2
}
1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 1 S Clearly, the set u∈[0,1], y∈B {(M0 +uM1 )y} which is shown by the shaded region of figure 1, is not convex; whereas the set [ 5 {(M0 + uM1 )y} − R2+ = {(a1 , a2 ) : a1 ≤ , a2 ≤ 1.5} 6 u∈[0,1], y∈B
is convex. To verify the robust minimax equality (1.3), let x2 = 1 − x1 and y2 = 1 − y1 . Then, 4 1 1 1 xT (M0 + uM1 )y = ( − u − y1 )x1 + + u + y1 − uy1. 3 3 2 2 Calculating extreme values with respect to each variable gives us 5 max max min xT (M0 + uM1 )y = max max min fu (x1 , y1 ) = . u∈U y∈B x∈A u∈[0,1] y1 ∈[0,1] x1 ∈[0,1] 6 where fu (x1 , y1) = ( 31 −u− 43 y1 )x1 + 12 +u+ 12 y1 −uy1 . Also, minx∈A maxu∈U maxy∈B xT (M0 + uM1 )y = 65 . We now show that our jointly concavelike condition of Theorem 2.1 is indeed a characterization for the robust von Neumann minimax theorem in the sense that the condition holds if and only if the robust von Neumann minimax theorem is valid for every linear perturbation, i.e., inf max max{xT M(u)y + xT a} = max max inf {xT M(u)y + xT a}, ∀a ∈ Rn .
x∈A y∈B u∈U
u∈U y∈B x∈A
Theorem 2.2. (Characterization) Let A be a closed convex subset of Rn and let B be a convex compact subset of Rm . Let U be a convex compact subset of Rq . Then, the following statements are equivalent: 5
(1) (∀λ ∈ [0, 1]) (∀(y1 , u1 ), (y2, u2 ) ∈ B × U) (∃(y, u) ∈ B × U) (∀x ∈ A) xT M(u)y ≧ λxT M(u1 )y1 + (1 − λ)xT M(u2 )y2 .
(2.3)
(2) inf max max{xT M(u)y + xT a} = max max inf {xT M(u)y + xT a} ∀a ∈ Rn . x∈A y∈B u∈U
u∈U y∈B x∈A
Proof. [(1) ⇒ (2)] Let z = (y, u) ∈ Rm × Rq and define F˜ : Rn × Rm × Rq → R by F˜ (x, z) = xT M(u)y + aT x. Then, we see that x 7→ F˜ (x, z) is linear for any z ∈ Rm × Rq and z 7→ F˜ (x, z) is concavelike. So, the Ky Fan’s minimax theorem (Theorem 2.1) gives us the statement (2). [(2) ⇒ (1)] We will establish this implication by the method of contradiction and suppose that (1) fails. Then, there exist λ ∈ [0, 1], y1 , y2 ∈ B and u1 , u2 ∈ U such that for all (y, u) ∈ B × U, there exists x ∈ A such that xT M(u)y < λxT M(u1 )y1 + (1 − λ)xT M(u2 )y2 .
(2.4)
Let a0 = λM(u1 )y1 + (1 − λ)M(u2 )y2 and let a = −a0 . Then, by (2.4) and statement (2), inf max max xT M(u)y − a0 = max max inf xT M(u)y − a0 < 0. x∈A y∈B u∈U
u∈U y∈B x∈A
Let h(x) := max max xT M(u)y + δA (x). Then, h is convex and y∈B u∈U
−h∗ (a0 ) = inf max max xT M(u)y − a0 < 0. x∈A y∈B u∈U
Thus, (a0 , 0) ∈ / epih∗ . Let hy,u (x) = xT M(u)y, for (y, u) ∈ B × U. As h ≥ hy,u , for each (y, u) ∈ B × U, we see from (2.1) that epih∗ ⊇ epih∗y,u for each (y, u) ∈ B × U. This together with the convexity of epih∗ gives us that [ [ epih∗ ⊇ co epih∗y,u = co {M(u)y} × [0, +∞). y∈B,u∈U
y∈B,u∈U
Since (a0 , 0) ∈ / epih∗ , it follows that λM(u1 )y1 + (1 − λ)M(u2 )y2 = a0 ∈ / co
[
M(u)y,
y∈B,u∈U
which is impossible. It is easy to see that, the jointly concavelike condition (2.3) holds if the classical condition “(u, y) 7→ xT M(u)y is concave” is satisfied, and so, robust von Neumann minimax theorem holds under the classical condition. However, we shall see, in the following simple example, that this classical condition is hard to satisfy even in the case of a linear perturbation: 6
m1 m2 Example 2.2. Let M0 = , and consider M(∆) = M0 + ∆ where ∆ is a m2 m3 (2 × 2) symmetric matrix (which can be equivalently regarded as a vector in Rq with q = 3). Let n = m = 2. Now, we show that (∆, y) 7→ xT (M0 + ∆)y is not concave for any fixed x ∈ R2+ \{0}. To see this, fix x = (x1 , x2 )T ∈ R2+ \{0} and let a1 a2 ∆= a2 a3 Then, for each fixed x = (x1 , x2 ), the mapping (∆, y) 7→ xT (M0 + ∆)y can be equivalently rewritten (up to an invertible linear transformation) as f (a1 , a2 , a3 , y1, y2 ) = (m1 + a1 )x1 y1 + (m2 + a2 )x1 y2 + (m2 + a2 )x2 y1 + (m3 + a3 )x2 y2 . Note that an invertible linear transformation preserves concavity, we only need to show f is not concave. To see this, note that, for each (a1 , a2 , a3 , y1 , y2 ) ∈ R5 , ∇2 f (a1 , a2 , a3 , y1 , y2) is a constant (5 × 5) matrix 0 0 0 x1 0 0 0 0 x2 x1 0 0 0 0 x C= 2 x1 x2 0 0 0 0 x1 x2 0 0 As x = (x1 , x2 )T ∈ R2+ \{0}, eT5 Ce5 = 4x1 + 4x2 > 0 where e5 = (1, 1, 1, 1, 1)T . So, f is not concave.
From the preceding example, we see that the classical sufficient condition “(∆, y) 7→ xT (M + ∆)y is concave“ is somewhat limited from the application viewpoint. However, we shall see in the next section that our condition (2.2) can be satisfied under various types of simple and commonly used data uncertainty sets, and hence produces various classes of the robust von Neumann minimax theorems in the face of payoff matrix data uncertainty.
3
Classes of Robust Minimax Theorems
In this Section, we establish that robust von Neumann’s minimax theorem always holds under various classes of uncertainty sets by verifying the joint concavelike condition in Theorem 2.1.
3.1
Matrix 1-Norm Uncertainty
In the first case, we assume that the matrix data in the bilinear function of the von Neumann’s minimax theorem is uncertain and the uncertain data matrix belongs to the matrix 1-norm uncertainty set U 1 = {M0 + ∆ ∈ Rn×m | k∆k1 ≤ ρ} where M0 ∈ Rn×m , k∆k1 is the matrix 1-norm defined by k∆k1 = sup k∆xk1 and k∆xk1 is the x∈Rm ,kxk1 =1
l1 -norm of the vector ∆x ∈ Rn .
7
Theorem 3.1. (Robust Minimax Theorem I) Let M0 ∈ Rn×n and let U 1 = {M0 + ∆ ∈ Rn×mP| k∆k1 ≤ ρ}. ρ > 0. Let S n = {(x1 , . . . , xn ) ∈ Rn | xi ≧ n m 0, i = 1,P . . . , n, = {(x1 , . . . , xm ) ∈ Rm | xi ≧ 0, i = i=1 xi = 1} and let S m 1, . . . , m, i=1 xi = 1}. Then, we have minn xT My. min max max xT My = max max m
x∈S n y∈S m M ∈U1
M ∈U1 y∈S
x∈S
(3.5)
Proof. Let A = S n , B = S m . Consider U1 = {∆ : k∆k1 ≤ ρ} ⊆ Rn×m as a subset of Rq with q = mn and let M(∆) = M0 + ∆, ∆ ∈ U1 . Note that (3.5) is equivalent to minn xT M(∆)y max xT M(∆)y = max max minn max m m
x∈S y∈S
∆∈U1
∆∈U1 y∈S
x∈S
Thus, to see the conclusion from Theorem 2.1, it suffices to show that: for any λ ∈ [0, 1], y1 , y2 ∈ B and ∆1 , ∆2 ∈ U1 , there exists (y, ∆) ∈ B × U1 such that xT M(∆)y ≧ λxT M(∆1 )y1 + (1 − λ)xT M(∆2 )y2
∀ x ∈ A.
(3.6)
To see this, fix λ ∈ [0, 1], y1 , y2 ∈ B, ∆1 ∈ U1 and ∆2 ∈ U1 . Let y = λy1 +(1−λ)y2 ∈ S m and a = λ∆1 y1 + (1 − λ)∆2 y2 . Now, consider a matrix ∆ defined by ∆ = aeT , where e ∈ Rm with each coordinate is equal to 1. As y ∈ S m , we have ∆y = a and kyk1 = 1. Moreover, as kyk1 = 1, we have k∆k1 = sup k∆xk1 = kak1 sup |eT x| ≤ kak1 sup kxk1 = kak1 . kxk1 =1
kxk1 =1
kxk1 =1
Note that kak1 = kλ∆1 y1 + (1 − λ)∆2 y2 k1 ≤ λk∆1 k1 ky1 k1 + (1 − λ)k∆2 k1 ky2 k1 ≤ ρ. So, ∆ satisfying ∆y = a = λ∆1 y1 + (1 − λ)∆2 y2 and k∆k1 ≤ ρ. Now, for any x ∈ A, from ∆y = a, we have λxT M(∆1 )y1 + (1 − λ)xT M(∆2 )y2 = = = =
λxT (M0 + ∆1 )y1 + (1 − λ)xT (M0 + ∆2 )y2 xT M0 y + λxT ∆1 y1 + (1 − λ)xT ∆2 y2 xT M0 y + xT a xT (M0 + ∆)y = xT M(∆)y.
So, the conclusion follows from Theorem 2.1.
3.2
Rank-1 Matrix Uncertainty
Secondly, we derive the robust minimax theorem in terms of rank-1 uncertainty sets U 2 = {M0 + ρuv T | u ∈ Rn , v ∈ Rm , kuk∞ ≤ 1 and kvk∞ ≤ 1} where kuk∞ (resp. kvk∞ ) is the l∞ -norm of u = (u1 , . . . , un ) ∈ Rn (resp. v = (v1 , . . . , vm ) ∈ Rm ) defined by kuk∞ = max1≤i≤n |ui | (resp. kvk∞ = max1≤i≤m |vi |).
8
Theorem 3.2. (Robust Minimax Theorem II) Let M0 ∈ Rn×n . Let S n = Pn m n = {(x1 , . . . , xm ) ∈ {(x1 , · · · , xn ) ∈ R | xi ≧ 0, i P = 1, · · · , n, i=1 xi = 1} and let S m m T R | xi ≧ 0, i = 1, . . . , m, | u ∈ Rn , v ∈ i=1 xi = 1}. Let U 2 = {M0 + ρuv m R , kuk∞ ≤ 1 and kvk∞ ≤ 1} where ρ > 0. Then, minn xT My. min max max xT My = max max m
x∈S n y∈S m M ∈U 2
M ∈U 2 y∈S
x∈S
(3.7)
Proof. Let A = S n , B = S m . Consider U2 = {ρuv T : u ∈ Rn , v ∈ Rm , kuk∞ ≤ 1 and kvk∞ ≤ 1} ⊆ Rn×m as a subset of Rq with q = mn and let M(∆) = M0 + ∆, ∆ ∈ U2 . Note that (3.7) is equivalent to minn xT M(∆)y. min max max xT M(∆)y = max max m
x∈S n y∈S m ∆∈U2
∆∈U2 y∈S
x∈S
The conclusion will follow from Theorem 2.1, if we show that for any λ ∈ [0, 1], y1 , y2 ∈ B and ∆1 , ∆2 ∈ U2 , there exists (y, ∆) ∈ B × U2 such that xT M(∆)y ≧ λxT M(∆1 )y1 + (1 − λ)xT M(∆2 )y2
∀ x ∈ A.
(3.8)
To see this, fix λ ∈ [0, 1], y1 , y2 ∈ B and ∆1 , ∆2 ∈ U2 . Then, we can find u1 , u2 ∈ Rn and v1 , v2 ∈ Rm such that ku1 k∞ ≤ 1, ku2 k∞ ≤ 1, kv1 k∞ ≤ 1, kv2 k∞ ≤ 1, ∆1 = ρu1 v1T and ∆2 = ρu2 v2T . Now, consider a matrix ∆ defined by ∆ = ρaeT , where e ∈ Rm with each coordinate is equal to 1 and a = λu1 v1T y1 + (1 − λ)u2 v2T y2 . Letting y = λy1 + (1 − λ)y2 , we see that y ∈ S m and ∆y = ρa. Moreover, as ky1 k1 = 1 and ky2k1 = 1, it follows that kak∞ = ≤ ≤ ≤
kλu1 v1T y1 + (1 − λ)u2 v2T y2 k∞ λku1 k∞ |v1T y1 | + (1 − λ)ku2 k∞ |v2T y2 | λ|v1T y1 | + (1 − λ)|v2T y2 | λkv1 k∞ ky1 k1 + (1 − λ)kv2 k∞ ky2k1 ≤ 1.
So, ∆ = ρaeT ∈ {ρuv T : u, v ∈ Rn , kuk∞ ≤ 1 and kvk∞ ≤ 1}. Then, we see that ∆ ∈ U2 and it satisfies ∆y = ρa = ρ(λu1 v1T y1 + (1 − λ)u2v2T y2 ). Now, for each x ∈ A, we have xT M(∆1 )y1 + (1 − λ)xT M(∆2 )y2 = = = =
λxT (M0 + ρu1 v1T )y1 + (1 − λ)xT (M0 + ρu2 v2T )y2 xT M0 y + ρxT (λu1 v1T y1 + (1 − λ)u2 v2T y2 ) xT M0 y + ρxT a xT (M0 + ∆)y = xT M(∆)y.
So, the conclusion follows from Theorem 2.1.
9
3.3
Column-wise Affine Parameter Uncertainty
Thirdly, we obtain our robust minimax theorem in the case where the matrix data is uncertain and the uncertain data matrix is columnwise affinely parameterized, i.e., the matrix data M belongs to the uncertainty set U3 = {
a10
+
q1 X
u1i a1i
,...,
am 0
+
qm X i=1
i=1
m um : (uj1 , . . . , ujqj ) ∈ Z j , j = 1, . . . , m}, i ai
where Z j , j = 1, . . . , m is a compact convex set in Rqj , aji ∈ Rn , i = 0, 1, . . . , qj , j = 1, . . . , m. To begin with, we first derive the following proposition as a preparation. Proposition 3.1. Let A be P a closed convex set in Rn and let S m = {(x1 , . . . , xm ) ∈ m m qj n R | xj ≧ 0, j = 1, . . . , m, j=1 xi = 1}. Let aj : R → R , j = 1, · · · , m, be affine functions and let U = {(a1 (u1 ), . . . , am (um )) : uj ∈ Uj } where Uj ⊆ Rqj is a convex compact set, j = 1, . . . , m. Then, inf max max xT My = max max inf xT My. m
x∈A y∈S m M ∈U
M ∈U y∈S
x∈A
(3.9)
Q Pm q Proof. Let B = S m and consider U = m j=1 Uj as a subset of R with q = j=1 qj and define M(u) = (a1 (u1 ), . . . , am (um )) , u = (u1 , . . . , um ) ∈ U. Note that (3.9) is equivalent to inf xT M(u)y. inf max max xT M(u)y = max max m
x∈A y∈S m u∈U
u∈U y∈S
x∈A
As it was seen earlier, the conclusion will follow from Theorem 2.1 if we show that for any λ ∈ [0, 1], y 1, y 2 ∈ B and u1 , u2 ∈ U, there exists (y, u) ∈ B × U such that xT M(u)y ≧ λxT M(u1 )y 1 + (1 − λ)xT M(u2 )y 2
∀ x ∈ A.
(3.10)
To see this, fix λ ∈ [0, 1], y 1 , y 2 ∈ B, u1 = (u11 , . . . , u1m) ∈ U and u2 = (u21, . . . , u2m ) ∈ U. So, for any x ∈ A, we have λxT M(u1 )y 1 + (1 − λ)xT M(u2 )y 2 = λxT a1 (u11 ), . . . , am (u1m ) y 1 + (1 − λ)xT a1 (u21 ), . . . , am (u2m ) y 2 m m X X 1 1 T yj aj (uj ) x + (1 − λ) = λ yj2 aj (u2j )T x =
j=1 m X
j=1
λyj1aj (u1j ) + (1 − λ)yj2aj (u2j )
j=1
T
x.
Let y = λy 1 + (1 − λ)y 2. Then, y = (y1 , . . . , ym ) with yj = λyj1 + (1 − λ)yj2. Let u = (u1 , . . . , um ) where each uj , j = 1, . . . , m, is given by ( 1 1 λyj uj +(1−λ)yj2 u2j if yj 6= 0 yj uj = 1 uj else. 10
So, uj ∈ Uj and yj uj = λyj1u1j + (1 − λ)yj2u2j (this equality is straightforward by the construction of uj when yj 6= 0. On the other hand, if yj = 0, then yj1 = yj2 = 0 so the equality again follows). This implies that u ∈ U and yj a(uj ) = λyj1 aj (u1j ) + (1 − λ)yj2aj (u2j ). Thus, T
1
1
T
2
2
λx M(u )y + (1 − λ)x M(u )y =
m X
yj aj (uj )T x = xT M(u)y.
j=1
Thus, the conclusion follows. Remark 3.1. Using a similar method of proof, if we further assume that A ⊆ Rn+ , then the assumption “each aj is an affine function” can be relaxed to “each aj is a concave function”. Now, we establish the robust minimax theorem for columnwise affine parameterization case. Theorem 3.3. (Robust Minimax Theorem III) Let S n = {(x1 , · · · , xn ) ∈ Rn | xi ≧ Pn S m = {(x1 , . . . , xm ) ∈ Rm | xi ≧ 0, i = 0, i = 1, · · · , n, i=1 xi = 1} and let Pm P Pk m m 1, . . . , m, i=1 xi = 1}. Let U 3 = { a10 + ki=1 u1i a1i , . . . , am + u a : (uj1 , . . . , ujk ) ∈ 0 i=1 i i Z j , j = 1, . . . , m}, where Z j , j = 1, . . . , m is a compact convex set in Rk , aji ∈ Rn , i = 0, 1, . . . , k, j = 1, . . . , m Then, min max max xT My = max max minn xT My. m
x∈S n y∈S m M ∈U 3
M ∈U 3 y∈S
x∈S
(3.11)
Proof. The conclusion follows by the preceding proposition by letting A = S n (which is convex compact and so the infimum is attained on A), Uj = Z j and letting aj , j = 1, . . . , m, be an affine mapping defined by aj (uj ) =
aj0
+
qj X
uji aji ,
uj = (uj1, . . . , ujqj ) ∈ Rqj .
i=1
As a simple application of Proposition 3.1, we derive a robust theorem of the alternative for a parameterized linear inequality system. Corollary 3.1. (Robust Gordan Alternative Theorem) For each j = 1, · · · , m, let aj : Rqj → Rn be an affine function. Let Uj be a convex compact subset of Rqj , j = 1, · · · , m. Then exactly one of the following two statements holds: (i) (∃x ∈ Rn ) (∀uj ∈ Uj ) aj (uj )T x < 0,
j = 1, · · · , m Pm ¯ ¯ ∈ Rm ) (∃¯ (ii) (∃0 6= λ uj ∈ Uj , j = 1, · · · , m), uj ) = 0. + j=1 λj aj (¯
11
Proof. As both (i) and (ii) can not have a solution simultaneously, we only need to show that [Not(i) ⇒ (ii)]. To see this, let M(u) = (a1 (u1 ), . . . , am (um )) ∈ Rn×m and let uj ∈ Rqj , j = 1, 2, · · · , m. Then, T
x M(u)y =
m X
yj aj (uj )T x.
j=1
Let A = Rn , B = {(y1 , . . . , ym ) : Not(i) implies that inf max max xT M(u)y =
x∈A y∈B u∈U
Pm
j=1 yj
= 1, yj ≥ 0} and let U =
infn Pm max
x∈R
max
u ∈Uj j=1 yj =1,yj ≥0 j
Pm
m X
Qm
j=1
Uj . Then
yj aj (uj )T x ≥ 0.
j=1
(Otherwise, inf x∈A maxPm maxuj ∈Uj j=1 yj aj (uj )T x < 0, and so, there exists P Pm j=1 yj =1,yTj ≥0 x0 ∈ A such that j=1 yj aj (uj ) x0 < 0 for all yj ≥ 0 with m j=1 yj = 1 and for all uj ∈ Uj . This means that the statement (i) is true which contradicts our assumption.) Hence, by Proposition 3.1, we have Pm
max
max inf
u ∈Uj x∈Rn j=1 yj =1,yj ≥0 j
m X
yj aj (uj )T x = max max inf xT M(u)y y∈B u∈U x∈A
j=1
= inf max max xT M(u)y ≥ 0. x∈A y∈B u∈U
¯ j ≧ 0, j = 1, · · · , m, not all zero, and u¯j ∈ Uj , j = 1, · · · , m, such Thus, there exist λ P ¯ uj )T x ≥ 0. So, the conclusion follows. that, for each x ∈ Rn , m j=1 λj aj (¯
Remark 3.2. If Uj , j = 1, · · · , m, are singletons, then Corollary 3.1 collapses to the classical Gordan’s alternative theorem [7].
3.4
Isotone Matrix Data Uncertainty
Now, we obtain a form of robust minimax theorem in the case where the matrix data is uncertain and the uncertain matrix is isotone on U in the sense that the mapping u 7→ M(u) satisfies the condition that, for any u1 , u2 ∈ U, max{u1 , u2} ∈ U and u1 , u2 ∈ U, u1 ≥ u2 ⇒ M(u1 ) ≥ M(u2 ). Note that max{u1 , u2 } is the vector whose ith coordinate is the maximum of the ith coordinate of u1 and u2 , and that C1 ≥ C2 means each entry in the matrix C1 − C2 is nonnegative. For a simple example of an isotone matrix data P uncertainty, let U0 = q b {(u1, . . . , uq ) ∈ R : 0 ≤ ui ≤ 1, i = 1, . . . , q}, U0 = {M0 + qi=1 ui Mi : Mi ∈ Rn × Rn , Mi ≥ 0, i = 1, . . . , uq , u = (u1 , . . . , uq ) ∈ U0 }.
Theorem 3.4. (Robust Minimax Theorem IV) Let S n = {(x1 , · · · , xn ) ∈ Rn | xi ≧ Pn m 0, i = 1,P · · · , n, = {(x1 , . . . , xm ) ∈ Rm | xi ≧ 0, i = i=1 xi = 1} and let S m q 1, . . . , m, i=1 xi = 1}. Suppose that U is a convex compact set in R and u 7→ M(u) is an isotone mapping on U. Then, minn xT M(u)y. min max max xT M(u)y = max max m
x∈S n y∈S m u∈U
u∈U y∈S
12
x∈S
(3.12)
Proof. Let A = S n , B = S m . Then, the conclusion will follow from Theorem 2.1 if we show that for any λ ∈ [0, 1], y 1, y 2 ∈ B and u1 , u2 ∈ U, there exists (y0 , u0) ∈ B × U such that xT M(u0 )y0 ≧ λxT M(u1 )y 1 + (1 − λ)xT M(u2 )y 2
∀ x ∈ A.
(3.13)
To see this, fix λ ∈ [0, 1], y 1 , y 2 ∈ B, u1 = (u11 , . . . , u1m) ∈ U and u2 = (u21, . . . , u2m ) ∈ U. Let u0 = max{u1 , u2} and y0 = λy 1 + (1 − λ)y 2. As u 7→ M(u) is isotone on U, it follows that u0 ∈ U, M(u0 ) ≥ M(u1 ) and M(u0 ) ≥ M(u2 ). Now, for each x ∈ A, noting that x ∈ Rn+ and y 1, y 2 ∈ Rm + , we obtain that xT M(u1 )y 1 − xT M(u0 )y 1 = xT (M(u1 ) − M(u0 ))y 1 ≤ 0 and xT M(u2 )y 2 − xT M(u0 )y 2 = xT (M(u1 ) − M(u0 ))y 2 ≤ 0. This gives us that λxT M(u1 )y 1 + (1 − λ)xT M(u2 )y 2 ≤ λxT M(u0 )y 1 + (1 − λ)xT M(u0 )y 2 = xT M(u0 )y0 .
References [1] M. Aghassi and D. Bertsimas, Robust game theory, Mathematical Programming 107(1-2) (2006), 231-273. [2] A. Beck and A. Ben-Tal, Duality in robust optimization: Primal worst equals dual best, Operations Research Letters, 37(2009), 1–6. [3] A. Ben-Tal, L.E. Ghaoui and A. Nemirovski, Robust Optimization, Princeton Series in Applied Mathematics, 2009. [4] A. Ben-Tal and A. Nemirovski, Robust optimization—methodology and applications, Mathematical Programming, Ser B, 92 (2002), 453-480. [5] D. Bertsimas and D. Brown, Constructing uncertainty sets for robust linear optimization, Operations Research, 57 (2009), 1483-1495. [6] D. Bertsimas, D. Pachamanova and M. Sim, Robust linear optimization under general norms, Operations Research Letters, 32 (2004), 510-516. [7] B. D. Craven and V. Jeyakumar, Equivalence of a Ky Fan type minimax theorem and a Gordan type alternative theorem, Operations Research Letters, 5(2) (1986), 99–102. 13
[8] K. Fan, Minimax Theorems, Proceedings of the National Academy of Sciences, USA, 39 (1953), 42-47. [9] J.B.G. Frenk, P. Kas and G. Kassay,. On linear programming duality and necessary and sufficient conditions in minimax theory. Journal of Optimization Theory and Applications, 132(3) (2007), 423–439. [10] J.B.G. Frenk and G. Kassay, On noncooperative games, minimax theorems, and equilibrium problems. Pareto optimality, game theory and equilibria, 53–94, Springer Optim. Appl., 17, Springer, New York, 2008. [11] V. Jeyakumar, A generalization of a minimax theorem of Ky Fan via a theorem of the alternative, Journal of Optimization Theory and Applications, 48 (1986), 525–533. [12] V. Jeyakumar, G. M. Lee and N. Dinh, New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs, SIAM Journal on Optimization 14 (2003), 534-547. [13] V. Jeyakumar and G. Li, Characterizing robust set containments and solutions of uncertain linear programs without qualifications, Operations Research Letters, 38 (2010), 188–194. [14] V. Jeyakumar and G. Li, Robust Farkas’ lemma for uncertain linear systems with applications, Positivity, DOI 10. 1007/s11117-010-0078-4. [15] V. Jeyakumar and G. Li, Strong duality in robust convex programming: complete characterizations, SIAM Journal on Optimization, 20 (2010), 3384-3407. [16] G. Li, V. Jeyakumar, G. M. Lee, Robust conjugate duality for convex optimization under uncertainty with application to data classification, Nonlinear Analysis Series A: Theory, Methods and Applications, DOI: 10.1016/j.na.2010.11.036 (2011). [17] G. Li, and K.F. Ng, On extension of Fenchel duality and its application, SIAM Journal on Optimization, 19 (2008), 1489-1509. [18] S.J. Li, G.Y. Chen and G.M. Lee, Minimax theorems for set-valued mappings. Journal of Optimization Theory and Applications 106(1) (2000), 183–199. [19] T. Parthasarathy and T. E. Raghavan, Some Topics in Two-Persons Games, Elsevier, New York, New York, 1971. [20] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, N. J, 1970. [21] J. Von Neumann, Zur Theorie der Gesellschaftsspiele, Mathematische Annalen, 100 (1928), 295-320.
14
Abstract The celebrated von Neumann minimax theorem is a fundamental theorem in two-person zero-sum games. In this paper, we present a generalization of the von Neumann minimax theorem, called robust von Neumann minimax theorem, in the face of data uncertainty in the payoff matrix via robust optimization approach. We establish that the robust von Neumann minimax theorem is guaranteed for various classes of bounded uncertainties, including the matrix 1-norm uncertainty, the rank-1 uncertainty and the column-wise affine parameter uncertainty. Key words. Robust von Neumann minimax theorem, minimax theorems under payoff uncertainty, robust optimization, conjugate functions.
1
Introduction
The celebrated von Neumann Minimax Theorem [21] asserts that, for an (n×m) matrix M, minn xT My, xT My = max minn max m m y∈S
x∈S y∈S
x∈S
n
where S is the n-dimensional simplex. It is a fundamental equality in two-person zero-sum games [19]. Due to its importance in mathematics, decision theory, economics and game theory, numerous generalizations have been given in the literature (see [9, 10, 11, 18] and ∗
The authors are grateful to the referee and the editors for their valuable comments and constructive suggestions which have contributed to the final preparation of the paper. The first and second authors were partially supported by a grant from the Australian Research Council. The third author was supported by the Korea Science and Engineering Foundation (KOSEF) NRL program grant funded by the Korea government (MEST)(No. ROA-2008-000-20010-0). † Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia ‡ Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia § Department of Applied Mathematics, Pukyong National University, Busan 608-737, Korea
1
other reference therein). However, these generalizations and their applications have so far been limited mainly to problems without data uncertainty, despite the reality of data uncertainty in many real-world problems due to modeling or prediction errors [2, 3, 5, 4, 6, 13, 14, 15]. For related recent work on incomplete-information games, see [1] and other references therein. The purpose of this paper is to present a new form of the von Neumann minimax theorem, called robust von Neumann minimax theorem, for two-person zero sum games under data uncertainty via robust optimization and to establish that the robust von Neumann minimax theorem always holds under various classes of uncertainties, including the matrix 1-norm uncertainty, the rank-1 uncertainty and the column-wise affine parameter uncertainty. minn xT My xT My and the maxmin value γ2 := max The minimax value γ1 := minn max m m y∈S
x∈S y∈S
can be calculated by the following two optimization problems: γ1 = max xT My ≤ t} and γ2 =
y∈S m
max
(y,t)∈S m ×R
x∈S
min
(x,t)∈S n ×R
{t :
{t : minn xT My ≥ t}. Whenever the cost function x∈S
is affected by data uncertainty, the effect of uncertain data on the cost matrix M can be captured by a new matrix M(u) where u is an uncertain parameter and it belongs to the compact uncertainty set U ⊆Rq . For instance, the effect of uncertain data a1 a2 (a1 , a2 , a3 ) on the cost matrix M = can be captured by the new matrix a2 a3 a1 (u) a2 (u) M(u) = where u ∈ U ⊆ R. So, the minimax value and the maxmin a2 (u) a3 (u) value in the face of cost matrix data uncertainty can be obtained by the following two uncertain optimization problems (UPI )
minn
xT M(u)y ≤ t} {t : max m
max
{t : minn xT M(u)y ≥ t}.
(x,t)∈S ×R
y∈S
and (UPII )
(y,t)∈S m ×R
x∈S
The robust counterpart [3, 15, 16] of the uncertain optimization problem (UPI ) is a deterministic optimization problem defined by (RPI )
min
(x,t)∈S n ×R
xT M(u)y ≤ t, for all u ∈ U}, {t : max m y∈S
(1.1)
and the optimistic counterpart [2, 15, 16] of the uncertain optimization problem (UPII ) is another deterministic optimization problem defined by (OPII )
max m
(y,t)∈S ×R
{t : minn xT M(u)y ≥ t, for some u ∈ U}. x∈S
(1.2)
The robust minimax theorem states that the optimal values of the robust counterpart problem (RPI ) (“worst possible loss of Player I”) and the optimistic counterpart (OPII ) (“best possible gain of Player II”) are equal. Equivalently, it asserts that minn xT M(u)y. min max max xT M(u)y = max max m
x∈S n y∈S m u∈U
u∈U y∈S
2
x∈S
(1.3)
Employing conjugate analysis [20] and Ky Fan’s minimax theorem [8], we derive the robust minimax equality (1.3) under a concave-like condition. We also show that the concave-like condition is also necessary for the robust minimax theorem in the sense that it holds if and only if inf max max{xT M(u)y + xT a} = max max inf {xT M(u)y + xT a}, ∀a ∈ Rn .
x∈A y∈B u∈U
u∈U y∈B x∈A
Importantly, we establish that the robust minimax theorem always holds for various classes of bounded uncertainty sets, including the matrix 1-norm uncertainty set, the rank-1 matrix uncertainty set, the column-wise affine parameter uncertainty set and isotone matrix-data uncertainty set. Consequently, we also derive a robust theorem of the alternative for uncertain linear inequality systems from the robust minimax theorem.
2
A Robust Minimax Theorem under Uncertainty
In this Section, we present a concave-like condition ensuring (1.3). We also show that the condition is also necessary for (1.3) to hold for every linear perturbation. We begin this section by fixing notation and preliminaries of convex analysis. Throughout this paper, Rn denotes the Euclidean space with dimension n. The inner product in Rn is defined by hx, yi := xT y for all x, y ∈ Rn . The nonnegative orthant of Rn is denoted by Rn+ and is defined by Rn+ := {(x1 , . . . , xn ) ∈ Rn : xi ≥ 0}. For a set A in Rn , the convex hull of A is denoted by coA. We say A is convex whenever µa1 + (1 − µ)a2 ∈ A for all µ ∈ [0, 1], a1 , a2 ∈ A. A function f : Rn → R ∪ {+∞} is said to be convex if for all µ ∈ [0, 1] f ((1−µ)x+µy) ≤ (1−µ)f (x)+µf (y) for all x, y ∈ Rn . The function f is said to be concave whenever −f is convex. As usual, for any proper (i.e., domf 6= ∅) convex function f on Rn , its conjugate function f ∗ : Rn → R ∪ {+∞} is defined by f ∗ (x∗ ) = supx∈Rn {hx∗ , xi − f (x)} for all x∗ ∈ Rn . Clearly, f ∗ is a proper lower semicontinuous convex function and for any proper lower semicontinuous convex functions f1 , f2 (cf. [12, 17]), f1 ≤ f2 ⇔ f1∗ ≥ f2∗ ⇔ epif1∗ ⊆ epif2∗ .
(2.1)
The following special case of Ky Fan minimax theorem [8] plays a key role in deriving our robust von Neumann minimax theorem. Recall from Ky Fan [8] that the function f (., y) is said to be convex-like whenever (∀x1 , x2 ∈ C) (∀λ ∈ (0, 1)) (∃x3 ∈ C) (∀y ∈ D) f (x3 , y) ≤ λf (x1 , y) + (1 − λ)f (x2 , y). The function f (x, .) is said to be concave-like whenever (∀y1 , y2 ∈ D) (∀λ ∈ (0, 1)) (∃y3 ∈ D) (∀x ∈ C) f (x, y3 ) ≥ λf (x, y1) + (1 − λ)f (x, y2 ), where f : C × D → R and C and D are sets. Theorem 2.1. [8, 11] Let C be a compact subset of Rn and let D ⊂ Rm . Let f : C × D → R. Suppose that f (., y) is concave-like and f (x, .) is convex-like and that f (., y) is upper-semi-continuous. Then, max inf f (x, y) = inf max f (x, y). x∈C y∈D
y∈D x∈C
3
Theorem 2.1. (Robust von Neumann Minimax Theorem) Let A be a closed convex subset of Rn and let B be a convex compact subset of Rm . Let U be a convex compact subset of Rq . Assume that (∀λ ∈ [0, 1]) (∀(y1 , u1 ), (y2, u2 ) ∈ B × U) (∃(y, u) ∈ B × U) (∀x ∈ A) xT M(u)y ≧ λxT M(u1 )y1 + (1 − λ)xT M(u2 )y2 .
(2.2)
Then inf max max xT M(u)y = max max inf xT M(u)y.
x∈A y∈B u∈U
u∈U y∈B x∈A
Proof. Let z = (y, u) ∈ Rm × Rq and define F : Rn × Rm × Rq → R by F (x, z) = xT M(u)y. Then, we see that x 7→ F (x, z) is linear for any z ∈ Rn × Rq and z 7→ F (x, z) is concavelike. So, by Theorem 2.1 gives us that inf max F (x, z) = max inf F (x, z).
x∈A z∈B×U
z∈B×U x∈A
Thus, the conclusion follows.
Pn Corollary 2.1. Let A = {(x1 , . . . , xn ) ∈ Rn | xi ≧ 0, i = 1, . . . , n, i=1 xi = 1}; let m BSbe a convex compact subset of R and let U be a convex compact subset of Rq . If ( u∈U , y∈B {M(u)y} − Rn+ ) is a convex set then inf max max xT M(u)y = max max inf xT M(u)y.
x∈A y∈B u∈U
u∈U y∈B x∈A
Proof. The conclusion will follow from Theorem 2.1 if we show that (2.2) S holds. To see this, let λ ∈ [0, 1]) and let (y1 , u1 ), (y2, u2 ) ∈ B×U). Then, M(ui )yi ∈ ( u∈U , y∈B {M(u)y}− Rn+ ), for i = 1, 2. By the convexity hypothesis, we can find (y, u) ∈ B × U such that M(u)y − λM(u1 )y1 − (1 − λ)M(u2 )y2 ∈ Rn+ .
This together with the fact that x ∈ A gives us the required inequality (2.2). n It is worth noting Pn that whenever A is simplex, i.e. A = {(x1 , · · · , xn ) ∈SR | xi ≧ 0, i = 1, · · · , n, i=1 xi = 1}, (2.2) is equivalent to the convexity of the set ( u∈U , y∈B {M(u)y}− Rn+ ). As an illustration, we provide a simple numerical example verifying Corollary 2.1.
Example 2.1. Let A = B = {(x1 , x2 ) : x1 , x2 ≥ 0 and x1 + x2 = 1}. Let U = [0, 1] and let M(u) = M0 + uM1 where −1 0 0 5/6 . and M1 = M0 = 0 1 1 1/2 Then,
[
[
{M(u)y} =
u∈U , y∈B
{
u∈[0,1] (y1 ,y2 )∈co{(0,1),(1,0)}
=
[
u∈[0,1]
co{
4
1 2
−u 1
1 2
5 6
+u
−u , }. +u 1 5 6
y1 y2
}
1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 1 S Clearly, the set u∈[0,1], y∈B {(M0 +uM1 )y} which is shown by the shaded region of figure 1, is not convex; whereas the set [ 5 {(M0 + uM1 )y} − R2+ = {(a1 , a2 ) : a1 ≤ , a2 ≤ 1.5} 6 u∈[0,1], y∈B
is convex. To verify the robust minimax equality (1.3), let x2 = 1 − x1 and y2 = 1 − y1 . Then, 4 1 1 1 xT (M0 + uM1 )y = ( − u − y1 )x1 + + u + y1 − uy1. 3 3 2 2 Calculating extreme values with respect to each variable gives us 5 max max min xT (M0 + uM1 )y = max max min fu (x1 , y1 ) = . u∈U y∈B x∈A u∈[0,1] y1 ∈[0,1] x1 ∈[0,1] 6 where fu (x1 , y1) = ( 31 −u− 43 y1 )x1 + 12 +u+ 12 y1 −uy1 . Also, minx∈A maxu∈U maxy∈B xT (M0 + uM1 )y = 65 . We now show that our jointly concavelike condition of Theorem 2.1 is indeed a characterization for the robust von Neumann minimax theorem in the sense that the condition holds if and only if the robust von Neumann minimax theorem is valid for every linear perturbation, i.e., inf max max{xT M(u)y + xT a} = max max inf {xT M(u)y + xT a}, ∀a ∈ Rn .
x∈A y∈B u∈U
u∈U y∈B x∈A
Theorem 2.2. (Characterization) Let A be a closed convex subset of Rn and let B be a convex compact subset of Rm . Let U be a convex compact subset of Rq . Then, the following statements are equivalent: 5
(1) (∀λ ∈ [0, 1]) (∀(y1 , u1 ), (y2, u2 ) ∈ B × U) (∃(y, u) ∈ B × U) (∀x ∈ A) xT M(u)y ≧ λxT M(u1 )y1 + (1 − λ)xT M(u2 )y2 .
(2.3)
(2) inf max max{xT M(u)y + xT a} = max max inf {xT M(u)y + xT a} ∀a ∈ Rn . x∈A y∈B u∈U
u∈U y∈B x∈A
Proof. [(1) ⇒ (2)] Let z = (y, u) ∈ Rm × Rq and define F˜ : Rn × Rm × Rq → R by F˜ (x, z) = xT M(u)y + aT x. Then, we see that x 7→ F˜ (x, z) is linear for any z ∈ Rm × Rq and z 7→ F˜ (x, z) is concavelike. So, the Ky Fan’s minimax theorem (Theorem 2.1) gives us the statement (2). [(2) ⇒ (1)] We will establish this implication by the method of contradiction and suppose that (1) fails. Then, there exist λ ∈ [0, 1], y1 , y2 ∈ B and u1 , u2 ∈ U such that for all (y, u) ∈ B × U, there exists x ∈ A such that xT M(u)y < λxT M(u1 )y1 + (1 − λ)xT M(u2 )y2 .
(2.4)
Let a0 = λM(u1 )y1 + (1 − λ)M(u2 )y2 and let a = −a0 . Then, by (2.4) and statement (2), inf max max xT M(u)y − a0 = max max inf xT M(u)y − a0 < 0. x∈A y∈B u∈U
u∈U y∈B x∈A
Let h(x) := max max xT M(u)y + δA (x). Then, h is convex and y∈B u∈U
−h∗ (a0 ) = inf max max xT M(u)y − a0 < 0. x∈A y∈B u∈U
Thus, (a0 , 0) ∈ / epih∗ . Let hy,u (x) = xT M(u)y, for (y, u) ∈ B × U. As h ≥ hy,u , for each (y, u) ∈ B × U, we see from (2.1) that epih∗ ⊇ epih∗y,u for each (y, u) ∈ B × U. This together with the convexity of epih∗ gives us that [ [ epih∗ ⊇ co epih∗y,u = co {M(u)y} × [0, +∞). y∈B,u∈U
y∈B,u∈U
Since (a0 , 0) ∈ / epih∗ , it follows that λM(u1 )y1 + (1 − λ)M(u2 )y2 = a0 ∈ / co
[
M(u)y,
y∈B,u∈U
which is impossible. It is easy to see that, the jointly concavelike condition (2.3) holds if the classical condition “(u, y) 7→ xT M(u)y is concave” is satisfied, and so, robust von Neumann minimax theorem holds under the classical condition. However, we shall see, in the following simple example, that this classical condition is hard to satisfy even in the case of a linear perturbation: 6
m1 m2 Example 2.2. Let M0 = , and consider M(∆) = M0 + ∆ where ∆ is a m2 m3 (2 × 2) symmetric matrix (which can be equivalently regarded as a vector in Rq with q = 3). Let n = m = 2. Now, we show that (∆, y) 7→ xT (M0 + ∆)y is not concave for any fixed x ∈ R2+ \{0}. To see this, fix x = (x1 , x2 )T ∈ R2+ \{0} and let a1 a2 ∆= a2 a3 Then, for each fixed x = (x1 , x2 ), the mapping (∆, y) 7→ xT (M0 + ∆)y can be equivalently rewritten (up to an invertible linear transformation) as f (a1 , a2 , a3 , y1, y2 ) = (m1 + a1 )x1 y1 + (m2 + a2 )x1 y2 + (m2 + a2 )x2 y1 + (m3 + a3 )x2 y2 . Note that an invertible linear transformation preserves concavity, we only need to show f is not concave. To see this, note that, for each (a1 , a2 , a3 , y1 , y2 ) ∈ R5 , ∇2 f (a1 , a2 , a3 , y1 , y2) is a constant (5 × 5) matrix 0 0 0 x1 0 0 0 0 x2 x1 0 0 0 0 x C= 2 x1 x2 0 0 0 0 x1 x2 0 0 As x = (x1 , x2 )T ∈ R2+ \{0}, eT5 Ce5 = 4x1 + 4x2 > 0 where e5 = (1, 1, 1, 1, 1)T . So, f is not concave.
From the preceding example, we see that the classical sufficient condition “(∆, y) 7→ xT (M + ∆)y is concave“ is somewhat limited from the application viewpoint. However, we shall see in the next section that our condition (2.2) can be satisfied under various types of simple and commonly used data uncertainty sets, and hence produces various classes of the robust von Neumann minimax theorems in the face of payoff matrix data uncertainty.
3
Classes of Robust Minimax Theorems
In this Section, we establish that robust von Neumann’s minimax theorem always holds under various classes of uncertainty sets by verifying the joint concavelike condition in Theorem 2.1.
3.1
Matrix 1-Norm Uncertainty
In the first case, we assume that the matrix data in the bilinear function of the von Neumann’s minimax theorem is uncertain and the uncertain data matrix belongs to the matrix 1-norm uncertainty set U 1 = {M0 + ∆ ∈ Rn×m | k∆k1 ≤ ρ} where M0 ∈ Rn×m , k∆k1 is the matrix 1-norm defined by k∆k1 = sup k∆xk1 and k∆xk1 is the x∈Rm ,kxk1 =1
l1 -norm of the vector ∆x ∈ Rn .
7
Theorem 3.1. (Robust Minimax Theorem I) Let M0 ∈ Rn×n and let U 1 = {M0 + ∆ ∈ Rn×mP| k∆k1 ≤ ρ}. ρ > 0. Let S n = {(x1 , . . . , xn ) ∈ Rn | xi ≧ n m 0, i = 1,P . . . , n, = {(x1 , . . . , xm ) ∈ Rm | xi ≧ 0, i = i=1 xi = 1} and let S m 1, . . . , m, i=1 xi = 1}. Then, we have minn xT My. min max max xT My = max max m
x∈S n y∈S m M ∈U1
M ∈U1 y∈S
x∈S
(3.5)
Proof. Let A = S n , B = S m . Consider U1 = {∆ : k∆k1 ≤ ρ} ⊆ Rn×m as a subset of Rq with q = mn and let M(∆) = M0 + ∆, ∆ ∈ U1 . Note that (3.5) is equivalent to minn xT M(∆)y max xT M(∆)y = max max minn max m m
x∈S y∈S
∆∈U1
∆∈U1 y∈S
x∈S
Thus, to see the conclusion from Theorem 2.1, it suffices to show that: for any λ ∈ [0, 1], y1 , y2 ∈ B and ∆1 , ∆2 ∈ U1 , there exists (y, ∆) ∈ B × U1 such that xT M(∆)y ≧ λxT M(∆1 )y1 + (1 − λ)xT M(∆2 )y2
∀ x ∈ A.
(3.6)
To see this, fix λ ∈ [0, 1], y1 , y2 ∈ B, ∆1 ∈ U1 and ∆2 ∈ U1 . Let y = λy1 +(1−λ)y2 ∈ S m and a = λ∆1 y1 + (1 − λ)∆2 y2 . Now, consider a matrix ∆ defined by ∆ = aeT , where e ∈ Rm with each coordinate is equal to 1. As y ∈ S m , we have ∆y = a and kyk1 = 1. Moreover, as kyk1 = 1, we have k∆k1 = sup k∆xk1 = kak1 sup |eT x| ≤ kak1 sup kxk1 = kak1 . kxk1 =1
kxk1 =1
kxk1 =1
Note that kak1 = kλ∆1 y1 + (1 − λ)∆2 y2 k1 ≤ λk∆1 k1 ky1 k1 + (1 − λ)k∆2 k1 ky2 k1 ≤ ρ. So, ∆ satisfying ∆y = a = λ∆1 y1 + (1 − λ)∆2 y2 and k∆k1 ≤ ρ. Now, for any x ∈ A, from ∆y = a, we have λxT M(∆1 )y1 + (1 − λ)xT M(∆2 )y2 = = = =
λxT (M0 + ∆1 )y1 + (1 − λ)xT (M0 + ∆2 )y2 xT M0 y + λxT ∆1 y1 + (1 − λ)xT ∆2 y2 xT M0 y + xT a xT (M0 + ∆)y = xT M(∆)y.
So, the conclusion follows from Theorem 2.1.
3.2
Rank-1 Matrix Uncertainty
Secondly, we derive the robust minimax theorem in terms of rank-1 uncertainty sets U 2 = {M0 + ρuv T | u ∈ Rn , v ∈ Rm , kuk∞ ≤ 1 and kvk∞ ≤ 1} where kuk∞ (resp. kvk∞ ) is the l∞ -norm of u = (u1 , . . . , un ) ∈ Rn (resp. v = (v1 , . . . , vm ) ∈ Rm ) defined by kuk∞ = max1≤i≤n |ui | (resp. kvk∞ = max1≤i≤m |vi |).
8
Theorem 3.2. (Robust Minimax Theorem II) Let M0 ∈ Rn×n . Let S n = Pn m n = {(x1 , . . . , xm ) ∈ {(x1 , · · · , xn ) ∈ R | xi ≧ 0, i P = 1, · · · , n, i=1 xi = 1} and let S m m T R | xi ≧ 0, i = 1, . . . , m, | u ∈ Rn , v ∈ i=1 xi = 1}. Let U 2 = {M0 + ρuv m R , kuk∞ ≤ 1 and kvk∞ ≤ 1} where ρ > 0. Then, minn xT My. min max max xT My = max max m
x∈S n y∈S m M ∈U 2
M ∈U 2 y∈S
x∈S
(3.7)
Proof. Let A = S n , B = S m . Consider U2 = {ρuv T : u ∈ Rn , v ∈ Rm , kuk∞ ≤ 1 and kvk∞ ≤ 1} ⊆ Rn×m as a subset of Rq with q = mn and let M(∆) = M0 + ∆, ∆ ∈ U2 . Note that (3.7) is equivalent to minn xT M(∆)y. min max max xT M(∆)y = max max m
x∈S n y∈S m ∆∈U2
∆∈U2 y∈S
x∈S
The conclusion will follow from Theorem 2.1, if we show that for any λ ∈ [0, 1], y1 , y2 ∈ B and ∆1 , ∆2 ∈ U2 , there exists (y, ∆) ∈ B × U2 such that xT M(∆)y ≧ λxT M(∆1 )y1 + (1 − λ)xT M(∆2 )y2
∀ x ∈ A.
(3.8)
To see this, fix λ ∈ [0, 1], y1 , y2 ∈ B and ∆1 , ∆2 ∈ U2 . Then, we can find u1 , u2 ∈ Rn and v1 , v2 ∈ Rm such that ku1 k∞ ≤ 1, ku2 k∞ ≤ 1, kv1 k∞ ≤ 1, kv2 k∞ ≤ 1, ∆1 = ρu1 v1T and ∆2 = ρu2 v2T . Now, consider a matrix ∆ defined by ∆ = ρaeT , where e ∈ Rm with each coordinate is equal to 1 and a = λu1 v1T y1 + (1 − λ)u2 v2T y2 . Letting y = λy1 + (1 − λ)y2 , we see that y ∈ S m and ∆y = ρa. Moreover, as ky1 k1 = 1 and ky2k1 = 1, it follows that kak∞ = ≤ ≤ ≤
kλu1 v1T y1 + (1 − λ)u2 v2T y2 k∞ λku1 k∞ |v1T y1 | + (1 − λ)ku2 k∞ |v2T y2 | λ|v1T y1 | + (1 − λ)|v2T y2 | λkv1 k∞ ky1 k1 + (1 − λ)kv2 k∞ ky2k1 ≤ 1.
So, ∆ = ρaeT ∈ {ρuv T : u, v ∈ Rn , kuk∞ ≤ 1 and kvk∞ ≤ 1}. Then, we see that ∆ ∈ U2 and it satisfies ∆y = ρa = ρ(λu1 v1T y1 + (1 − λ)u2v2T y2 ). Now, for each x ∈ A, we have xT M(∆1 )y1 + (1 − λ)xT M(∆2 )y2 = = = =
λxT (M0 + ρu1 v1T )y1 + (1 − λ)xT (M0 + ρu2 v2T )y2 xT M0 y + ρxT (λu1 v1T y1 + (1 − λ)u2 v2T y2 ) xT M0 y + ρxT a xT (M0 + ∆)y = xT M(∆)y.
So, the conclusion follows from Theorem 2.1.
9
3.3
Column-wise Affine Parameter Uncertainty
Thirdly, we obtain our robust minimax theorem in the case where the matrix data is uncertain and the uncertain data matrix is columnwise affinely parameterized, i.e., the matrix data M belongs to the uncertainty set U3 = {
a10
+
q1 X
u1i a1i
,...,
am 0
+
qm X i=1
i=1
m um : (uj1 , . . . , ujqj ) ∈ Z j , j = 1, . . . , m}, i ai
where Z j , j = 1, . . . , m is a compact convex set in Rqj , aji ∈ Rn , i = 0, 1, . . . , qj , j = 1, . . . , m. To begin with, we first derive the following proposition as a preparation. Proposition 3.1. Let A be P a closed convex set in Rn and let S m = {(x1 , . . . , xm ) ∈ m m qj n R | xj ≧ 0, j = 1, . . . , m, j=1 xi = 1}. Let aj : R → R , j = 1, · · · , m, be affine functions and let U = {(a1 (u1 ), . . . , am (um )) : uj ∈ Uj } where Uj ⊆ Rqj is a convex compact set, j = 1, . . . , m. Then, inf max max xT My = max max inf xT My. m
x∈A y∈S m M ∈U
M ∈U y∈S
x∈A
(3.9)
Q Pm q Proof. Let B = S m and consider U = m j=1 Uj as a subset of R with q = j=1 qj and define M(u) = (a1 (u1 ), . . . , am (um )) , u = (u1 , . . . , um ) ∈ U. Note that (3.9) is equivalent to inf xT M(u)y. inf max max xT M(u)y = max max m
x∈A y∈S m u∈U
u∈U y∈S
x∈A
As it was seen earlier, the conclusion will follow from Theorem 2.1 if we show that for any λ ∈ [0, 1], y 1, y 2 ∈ B and u1 , u2 ∈ U, there exists (y, u) ∈ B × U such that xT M(u)y ≧ λxT M(u1 )y 1 + (1 − λ)xT M(u2 )y 2
∀ x ∈ A.
(3.10)
To see this, fix λ ∈ [0, 1], y 1 , y 2 ∈ B, u1 = (u11 , . . . , u1m) ∈ U and u2 = (u21, . . . , u2m ) ∈ U. So, for any x ∈ A, we have λxT M(u1 )y 1 + (1 − λ)xT M(u2 )y 2 = λxT a1 (u11 ), . . . , am (u1m ) y 1 + (1 − λ)xT a1 (u21 ), . . . , am (u2m ) y 2 m m X X 1 1 T yj aj (uj ) x + (1 − λ) = λ yj2 aj (u2j )T x =
j=1 m X
j=1
λyj1aj (u1j ) + (1 − λ)yj2aj (u2j )
j=1
T
x.
Let y = λy 1 + (1 − λ)y 2. Then, y = (y1 , . . . , ym ) with yj = λyj1 + (1 − λ)yj2. Let u = (u1 , . . . , um ) where each uj , j = 1, . . . , m, is given by ( 1 1 λyj uj +(1−λ)yj2 u2j if yj 6= 0 yj uj = 1 uj else. 10
So, uj ∈ Uj and yj uj = λyj1u1j + (1 − λ)yj2u2j (this equality is straightforward by the construction of uj when yj 6= 0. On the other hand, if yj = 0, then yj1 = yj2 = 0 so the equality again follows). This implies that u ∈ U and yj a(uj ) = λyj1 aj (u1j ) + (1 − λ)yj2aj (u2j ). Thus, T
1
1
T
2
2
λx M(u )y + (1 − λ)x M(u )y =
m X
yj aj (uj )T x = xT M(u)y.
j=1
Thus, the conclusion follows. Remark 3.1. Using a similar method of proof, if we further assume that A ⊆ Rn+ , then the assumption “each aj is an affine function” can be relaxed to “each aj is a concave function”. Now, we establish the robust minimax theorem for columnwise affine parameterization case. Theorem 3.3. (Robust Minimax Theorem III) Let S n = {(x1 , · · · , xn ) ∈ Rn | xi ≧ Pn S m = {(x1 , . . . , xm ) ∈ Rm | xi ≧ 0, i = 0, i = 1, · · · , n, i=1 xi = 1} and let Pm P Pk m m 1, . . . , m, i=1 xi = 1}. Let U 3 = { a10 + ki=1 u1i a1i , . . . , am + u a : (uj1 , . . . , ujk ) ∈ 0 i=1 i i Z j , j = 1, . . . , m}, where Z j , j = 1, . . . , m is a compact convex set in Rk , aji ∈ Rn , i = 0, 1, . . . , k, j = 1, . . . , m Then, min max max xT My = max max minn xT My. m
x∈S n y∈S m M ∈U 3
M ∈U 3 y∈S
x∈S
(3.11)
Proof. The conclusion follows by the preceding proposition by letting A = S n (which is convex compact and so the infimum is attained on A), Uj = Z j and letting aj , j = 1, . . . , m, be an affine mapping defined by aj (uj ) =
aj0
+
qj X
uji aji ,
uj = (uj1, . . . , ujqj ) ∈ Rqj .
i=1
As a simple application of Proposition 3.1, we derive a robust theorem of the alternative for a parameterized linear inequality system. Corollary 3.1. (Robust Gordan Alternative Theorem) For each j = 1, · · · , m, let aj : Rqj → Rn be an affine function. Let Uj be a convex compact subset of Rqj , j = 1, · · · , m. Then exactly one of the following two statements holds: (i) (∃x ∈ Rn ) (∀uj ∈ Uj ) aj (uj )T x < 0,
j = 1, · · · , m Pm ¯ ¯ ∈ Rm ) (∃¯ (ii) (∃0 6= λ uj ∈ Uj , j = 1, · · · , m), uj ) = 0. + j=1 λj aj (¯
11
Proof. As both (i) and (ii) can not have a solution simultaneously, we only need to show that [Not(i) ⇒ (ii)]. To see this, let M(u) = (a1 (u1 ), . . . , am (um )) ∈ Rn×m and let uj ∈ Rqj , j = 1, 2, · · · , m. Then, T
x M(u)y =
m X
yj aj (uj )T x.
j=1
Let A = Rn , B = {(y1 , . . . , ym ) : Not(i) implies that inf max max xT M(u)y =
x∈A y∈B u∈U
Pm
j=1 yj
= 1, yj ≥ 0} and let U =
infn Pm max
x∈R
max
u ∈Uj j=1 yj =1,yj ≥0 j
Pm
m X
Qm
j=1
Uj . Then
yj aj (uj )T x ≥ 0.
j=1
(Otherwise, inf x∈A maxPm maxuj ∈Uj j=1 yj aj (uj )T x < 0, and so, there exists P Pm j=1 yj =1,yTj ≥0 x0 ∈ A such that j=1 yj aj (uj ) x0 < 0 for all yj ≥ 0 with m j=1 yj = 1 and for all uj ∈ Uj . This means that the statement (i) is true which contradicts our assumption.) Hence, by Proposition 3.1, we have Pm
max
max inf
u ∈Uj x∈Rn j=1 yj =1,yj ≥0 j
m X
yj aj (uj )T x = max max inf xT M(u)y y∈B u∈U x∈A
j=1
= inf max max xT M(u)y ≥ 0. x∈A y∈B u∈U
¯ j ≧ 0, j = 1, · · · , m, not all zero, and u¯j ∈ Uj , j = 1, · · · , m, such Thus, there exist λ P ¯ uj )T x ≥ 0. So, the conclusion follows. that, for each x ∈ Rn , m j=1 λj aj (¯
Remark 3.2. If Uj , j = 1, · · · , m, are singletons, then Corollary 3.1 collapses to the classical Gordan’s alternative theorem [7].
3.4
Isotone Matrix Data Uncertainty
Now, we obtain a form of robust minimax theorem in the case where the matrix data is uncertain and the uncertain matrix is isotone on U in the sense that the mapping u 7→ M(u) satisfies the condition that, for any u1 , u2 ∈ U, max{u1 , u2} ∈ U and u1 , u2 ∈ U, u1 ≥ u2 ⇒ M(u1 ) ≥ M(u2 ). Note that max{u1 , u2 } is the vector whose ith coordinate is the maximum of the ith coordinate of u1 and u2 , and that C1 ≥ C2 means each entry in the matrix C1 − C2 is nonnegative. For a simple example of an isotone matrix data P uncertainty, let U0 = q b {(u1, . . . , uq ) ∈ R : 0 ≤ ui ≤ 1, i = 1, . . . , q}, U0 = {M0 + qi=1 ui Mi : Mi ∈ Rn × Rn , Mi ≥ 0, i = 1, . . . , uq , u = (u1 , . . . , uq ) ∈ U0 }.
Theorem 3.4. (Robust Minimax Theorem IV) Let S n = {(x1 , · · · , xn ) ∈ Rn | xi ≧ Pn m 0, i = 1,P · · · , n, = {(x1 , . . . , xm ) ∈ Rm | xi ≧ 0, i = i=1 xi = 1} and let S m q 1, . . . , m, i=1 xi = 1}. Suppose that U is a convex compact set in R and u 7→ M(u) is an isotone mapping on U. Then, minn xT M(u)y. min max max xT M(u)y = max max m
x∈S n y∈S m u∈U
u∈U y∈S
12
x∈S
(3.12)
Proof. Let A = S n , B = S m . Then, the conclusion will follow from Theorem 2.1 if we show that for any λ ∈ [0, 1], y 1, y 2 ∈ B and u1 , u2 ∈ U, there exists (y0 , u0) ∈ B × U such that xT M(u0 )y0 ≧ λxT M(u1 )y 1 + (1 − λ)xT M(u2 )y 2
∀ x ∈ A.
(3.13)
To see this, fix λ ∈ [0, 1], y 1 , y 2 ∈ B, u1 = (u11 , . . . , u1m) ∈ U and u2 = (u21, . . . , u2m ) ∈ U. Let u0 = max{u1 , u2} and y0 = λy 1 + (1 − λ)y 2. As u 7→ M(u) is isotone on U, it follows that u0 ∈ U, M(u0 ) ≥ M(u1 ) and M(u0 ) ≥ M(u2 ). Now, for each x ∈ A, noting that x ∈ Rn+ and y 1, y 2 ∈ Rm + , we obtain that xT M(u1 )y 1 − xT M(u0 )y 1 = xT (M(u1 ) − M(u0 ))y 1 ≤ 0 and xT M(u2 )y 2 − xT M(u0 )y 2 = xT (M(u1 ) − M(u0 ))y 2 ≤ 0. This gives us that λxT M(u1 )y 1 + (1 − λ)xT M(u2 )y 2 ≤ λxT M(u0 )y 1 + (1 − λ)xT M(u0 )y 2 = xT M(u0 )y0 .
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