Localization of minimax points

J Glob Optim (2008) 40:489–494 DOI 10.1007/s10898-007-9250-1 Localization of minimax points Pando G. Georgiev · Panos M. Pardalos · Altannar Chinchuluun Received: 22 September 2007 / Accepted: 28 September 2007 / Published online: 20 November 2007 © Springer Science+Business Media, LLC. 2007 Abstract In their proof of Gilbert–Pollak conjecture on Steiner ratio, Du and Hwang (Proceedings 31th FOCS, pp. 76–85 (1990); Algorithmica 7:121–135, 1992) used a result about localization of the minimum points of functions of the type max y∈Y f (·, y). In this paper, we present a generalization of such a localization in terms of generalized vertices, when we minimize over a compact polyhedron, and Y is a compact set. This is also a strengthening of a result of Du and Pardalos (J. Global Optim. 5:127–129, 1994). We give also a random version of our generalization. Keywords Minimax · Gilbert–Pollak conjecture · Steiner ratio · g-Vertex · Steiner tree · Spanning tree 1 Introduction We consider a classical minimax problem min max f (x, y), (1) x∈X y∈Y where X is an appropriate set and Y is a compact set. We are interested in determination of a subset B ⊂ X (usually finite) such that min max f (x, y) = min max f (x, y). x∈B y∈Y (2) x∈X y∈Y P. G. Georgiev (B) Computer Science Department, University of Cincinnati, 2600 Clifton Ave., Cincinnati, OH 45221, USA e-mail: pgeorgie@ececs.uc.edu P. M. Pardalos · A. Chinchuluun Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, Gainesville, FL 32611, USA P. M. Pardalos e-mail: pardalos@ufl.edu A. Chinchuluun e-mail: altannar@ufl.edu 123 490 J Glob Optim (2008) 40:489–494 This problem, for specific f (defined by interpolation problems), goes back to Chebyshev, who contributed by it to the foundations of best approximation theory. For more general f , appropriate determination of B in (1) is given by Du and Hwang [1,5, pp. 339–367], as follows: Theorem 1 (Du–Hwang) Suppose that f i , i = 1, . . . , m, are continuous and concave functions on the compact polyhedron X = {x ∈ Rn : x T a j ≤ b j , j = 1, . . . , k}. Then the minimum value of the function g(x) = max i=1,...,m f i (x) over X is achieved at a g-vertex, i.e., at a point x ∗ with the property that there is no y ∈ X such that at least one of the sets M(x ∗ ) and J (x ∗ ) is a proper subset of M(y) and J (y) respectively, where
 J (x) := j ∈ {1, . . . , k} : x T a j = b j , and M(x) := {i ∈ {1, . . . , m} : g(x) = f i (x)}. With the above theorem, Du and Hwang [1,2]√proved the Gilbert–Pollak conjecture (1966) that the Steiner ratio in the Euclidean plane is 3/2 (see also [2,5]). Let G(V, E) be a network with a set of edges E and a set of vertices V . Let M be a subset of V . Let us recall some important notions. • Steiner minimum tree (SMT) on the finite point set M—the shortest network (tree) interconnecting the points in this set. It may contain vertices not in M. • Minimum spanning tree (MST) on M—a shortest spanning tree with vertex set M. • Gilbert–Pollak conjecture: inf M⊂R2 L s (M) L m (M) = √ 3/2, where L s (M) and L m (M) are the lengths of S M T (M) and M ST (M) respectively. An interior point x of X is a g-vertex iff M(x) is maximal. In general, for any g-vertex, there exists an extreme subset Y of X such that M(x) is maximal over Y . A point x of X is called a critical point, if there exists an extreme set Y such that M(x) is maximal over Y . Thus, every g-vertex is a critical point. However, the inverse is not true. Du and Pardalos [4] generalized the above theorem for the case when the index set is infinite—a compact set, as follows. Theorem 2 (Du, Pardalos) Let f : X × I → R be a continuous function, where X is a compact polyhedron in Rm and I is a compact set in Rn . Let g(x) = max y∈Y f (x, y). If f (x, y) is a concave function with respect to x, then the minimum value of g over X is achieved at some critical point. In this paper we strengthen the results of Du–Pardalos and Du–Hwang, showing that the minimum point in (1) is achieved at a generalized vertex (not only at a critical point), when Y is compact, and X is a compact polyhedron. We give a random version of this strengthening. 123 J Glob Optim (2008) 40:489–494 491 2 Compact polyhedrons Here we give necessary and sufficient conditions for compactness of a polyhedron P = {x ∈ Rn : x T a j ≤ b j , j = 1, . . . , k}. Compact polyhedrons are used in the next theorems, so such a characterization is useful. Proposition 3 The polyhedron P is compact if and only if 0 belongs to the interior of co{a j }kj=1 . (3) Proof Note first that (3) implies that k ≥ n + 1 (otherwise co{a j }kj=1 will have no interior points) and that the span of {a j }kj=1 is all Rn . Without loss of generality, we may assume that b j ≥ 0 (replacing P with P − p0 and b j with b j − p0T a j if necessary, where p0 is any element of P). (a) Assume that (3) is satisfied and P is unbounded. Then there is a sequence xi ∈ P such that xi  → ∞. By compactness of the unit sphere, there is a subsequence {xir } ⊂ {xi } x such that xiir  converges to some x0 with x0  = 1. Since P is closed, x0 ∈ P and r x0T a j ≤ 0, j = 1, . . . , k. (4) By (3) and Carathéodory’s theorem there are positive αr , r = 1, . . . , n + 1, and indices n j1 , . . . , jn+1 such that the span of {a jr }rn+1 =1 is R and n+1  r =1 αr a jr = 0. (5) There is an index r0 such that x0T a jr < 0; otherwise x0 should be orthogonal to all a jr , r = 1, . . . , n + 1, which is a contradiction with the fact that the span of {a jr }rn+1 =1 is all Rn . Multiplying (4) with α jr , r = 1, . . . , n + 1, and summing, we obtain a contradiction. (b) Assume that P is compact and (3) is not satisfied. By separation theorem, we can separate 0 and co{a j }kj=1 , so there is a 0 = y ∈ Rn such that yT a j ≤ 0 ≤ b j , j = 1, . . . , k. So, the ray {βy : β > 0} will belong to P, a contradiction with the compactness of P. ⊔ ⊓ 3 Localization of minimax points Define 
J (x) := j ∈ {1, . . . , k} : x T a j = b j and M(x) := {y ∈ Y : g(x) = f (x, y)}. The point x̂ will be called a generalized vertex (in short g-vertex) if there is no z ∈ X such that the set J (x̂) ∪ M(x̂) is a proper subset of J (z) ∪ M(z). 123 492 J Glob Optim (2008) 40:489–494 Theorem 4 (General localization theorem) Suppose that Y is a compact topological space, X ⊂ Rn is a compact polyhedron, f : X × Y → R is a continuous function and f (·, y) is concave for every y ∈ Y . Define g(x) = max y∈Y f (x, y). Then the minimum of g over X is attained at some generalized vertex x̂. Proof Let x ∗ be a minimum point for g. We will prove, applying the Zorn lemma, that there exists a g-vertex x̂ satisfying M(x ∗ ) ⊂ M(x̂) and J (x ∗ ) ⊂ J (x̂). Let us consider the partial ordering  on the set S = {x ∈ X : M(x ∗ ) ⊆ M(x), J (x ∗ ) ⊆ J (x)} defined by x  y ⇔ (M(x) ∪ J (x)) ⊆ (M(y) ∪ J (y)) . ∞ be a linearly ordered subset of (S, ), i.e., Let {xi }i=1 M(x ∗ ) ⊆ M(x1 ) ⊆ M(x2 ) ⊆ · · · ⊆ M(xn ) . . . J (x ∗ ) ⊆ J (x1 ) ⊆ J (x2 ) ⊆ · · · ⊆ J (xn ) . . . Since X is compact, there exists a subsequence {xik } converging to some x ′ . Since Y is compact, the function x  → max f (x, Y ) is continuous. So, if y ∈ M(xn ) for some n, then y ∈ M(xki ) for sufficiently large i, therefore max f (x ′ , Y ) = lim max f (xik , Y ) = lim f (xki , y) = f (x ′ , y), k→∞ M(x ′ ) which means y ∈ or M(xn ) ⊆ M(x ′ ) for every n. Similarly we can prove that ′ J (xn ) ⊆ J (x ) for every n. Thus, x ′ is an upper bound for {xn }∞ n=1 with respect to . By the Zorn lemma, (S, ) has a maximal element x̂, i.e., x̂ is a g-vertex. In particular, M(x ∗ ) ⊆ M(x̂) and J (x ∗ ) ⊆ J (x̂). The latter inclusion implies that x̂ T a j = b j , ∀ j ∈ J (x ∗ ). There is λ0 > 0 such that x(λ) := x ∗ + λ(x ∗ − x̂) ∈ P for every λ ∈ (0, λ0 ). We will prove that x̂ is a minimum point of g. Assume the contrary. Consider the cases: Case 1 There exists λ ∈ (0, λ0 ) such that M(x(λ)) ∩ M(x ∗ ) = ∅. Take y ∈ M(x(λ)) ∩ M(x ∗ ). Then x∗ = By concavity of f (., y) we have f (x ∗ , y) ≥ a contradiction. λ 1 x̂ + x(λ). 1+λ 1+λ λ 1 f (x̂, y) + f (x(λ), y) > f (x ∗ , y), 1+λ 1+λ Case 2 M(x(λ)) ∩ M(x ∗ ) = ∅ for every λ ∈ (0, λ0 ). Let y(λ) ∈ M(x(λ)). By compactness, there exists a sequence {λk } converging to zero such that y(λk ) converges to some y ∗ ∈ Y . Since g(x(λk )) = f (x(λk ), y(λ)), by continuity we obtain g(x ∗ ) = f (x ∗ , y ∗ ), so y ∗ ∈ M(x ∗ ). Since M(x ∗ ) ⊆ M(x̂), it follows that f (x̂, y ∗ ) = g(x̂) > g(x ∗ ). Therefore, for sufficiently large k, f (x̂, y(λk )) > g(x ∗ ). 123 J Glob Optim (2008) 40:489–494 493 Since y(λk ) ∈ M(x(λk )) and y(λk ) ∈ M(x ∗ ), we have f (x(λk ), y(λk )) = g(x(λk )) ≥ g(x ∗ ), and f (x ∗ , y(λk )) < g(x ∗ ). Thus f (x ∗ , y(λk )) < min{ f (x(λk ), y(λk )), f (x̂, y(λk ))}, which is a contradiction with the concavity of f (x ∗ , .). ⊔ ⊓ Remark The above theorem generalizes a theorem of Du and Pardalos [4]. Namely, they prove that the minimum of g is attained at a critical point x̂ characterized by the following: (*) There exists an extreme subset Z of X such that x̂ ∈ Z and the set M(x̂) = {y ∈ Y : g(x̂) = f (x̂, y)} is maximal over Z . It is easy to see that every g-vertex is a critical point, but the inverse is not true (see for instance, [5], Remark 2). 4 Random g-vertices In this section (, ) is a measure space, i.e.,  is a set and  is a σ algebra on . A (multi)function F :  → Rn is measurable iff F −1 (B) is measurable for each closed subset B of Rn (i.e., F −1 (B) := {ω ∈  : F(ω) ∩ B = ∅} ∈ ). For a comprehensive description of measurable relations, see for instance [6]. Theorem 5 (Random g-vertices) Suppose that X in Theorem 1 is a compact random polyhedron, i.e., it depends on ω and is given by X (ω) = {x ∈ E : x T a j (ω) ≤ b j (ω) j = 1, . . . , k}, where a j :  → Rn and b j :  → R are measurable functions. Then there exists a measurable function ω  → x̂(ω) with x̂(ω) being a generalized vertex of X (ω) for every ω ∈ . Proof First we prove that there is a measurable minimizer x ∗ (ω) of the minimization problem minimize max f (ω, x, Y ) (6) subject to x ∈ X (ω) (7) The proof of this fact is routine and is based on known measurability theorems, contained, for instance, in [6]. Next, following the proof of Theorem 4, we prove that there is a g-vertex x̂(ω), which is a measurable function on ω. Let us consider the partial ordering  on the set of all measurable selections of the multifunction X (ω) i.e., S = {x :  → X (ω) : M(x ∗ (ω)) ⊆ M(x(ω)), J (x ∗ (ω)) ⊆ J (x(ω)), ∀ω ∈ } defined by x  y ⇔ (M(x(ω)) ∪ J (x(ω))) ⊆ (M(y(ω)) ∪ J (y(ω))) ∀ω ∈ . 123 494 J Glob Optim (2008) 40:489–494 ∞ be a linearly ordered subset of (S, ), i.e., Let {xi }i=1 M(x ∗ (ω)) ⊆ M(x1 (ω)) ⊆ M(x2 (ω)) ⊆ · · · ⊆ M(xn (ω)) . . . ∀ω ∈ , J (x ∗ (ω)) ⊆ J (x1 (ω)) ⊆ J (x2 (ω)) ⊆ · · · ⊆ J (xn (ω)) . . . ∀ω ∈ . Consider the set     1 X n (ω) = x ∈ X (ω) : x + B ∩ {xk (ω)}∞ = ∅ . k=n n Since X n (ω) = ∞   k=n xk (ω) +  1 B , n we have X (ω) \ X n (ω) = ∞ k=n  X (ω)  xk (ω) + 1 B n  . (8) Thus, by Theorem 4.1 of [6], the multifunction ω  → X (ω)\X n (ω) is measurable, and by Theorem 4.5 of [6], the function ω  → X n (ω) is measurable. Putting C(ω) = ∞ n=1 X n (ω) and applying again Theorem 4.5 of [6], we obtain that the multifunction ω  → C(ω) is measurable. Now by the Kuratowski, Ryll–Nardzewski selection theorem (see [6], Theorem 5.1), there is a measurable selection x ′ (ω) ∈ C(ω), ∀ω ∈ . Since C(ω) is the set of all cluster ′ points of the sequence {xn (ω)}∞ n=1 , the point x (ω) is a cluster point too, depending in a measurable way on ω. As in the proof of Theorem 4, we obtain that x ′ (ω) is an upper bound for {xn (ω)}∞ n=1 with respect to . By the Zorn lemma, (S, ) has a maximal element x̂(ω) (measurable on ω), i.e., x̂(ω) is a measurable g-vertex. Further we prove exactly as in Theorem 4 that x̂(ω) is a minimum point of the function max f (ω, ·, Y ) over X (ω), which completes the proof of the theorem. ⊔ ⊓ References 1. Du, D.Z., Hwang, F.K.: An approach for proving lower bounds: solution of Gilbert–Pollak’s conjecture on Steiner ration. Proceedings 31th FOCS, pp. 76–85 (1990) 2. Du, D.Z., Hwang, F.K.: A proof of the Gilbert–Pollak conjecture on the Steiner ratio. Algorithmica 7, 121–135 (1992) 3. Du, D.Z., Pardalos, P.M., Wu, W.: Mathematical Theory of Optimization. Kluwer Academic Publishers (2001) 4. Du, D.Z., Pardalos, P.M.: A continuous version of a result of Du and Hwang. J. Global Optim. 5, 127–129 (1994) 5. Du, D.Z.: Minimax and its applications. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization. Kluwer (1995) 6. Himmelberg, C.J.: Measurable relations. Fund. Math. 87, 53–72 (1975) 123