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Optimization of the Determinant of the Vandermonde Matrix and Related Matrices

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Optimization of the Determinant of the Vandermonde Matrix and Related Matrices Methodol Comput Appl Probab (2018) 20:1417–1428 https://doi.org/10.1007/s11009-017-9595-y Optimization of the Determinant of the Vandermonde Matrix and Related Matrices Karl Lundengard˚ 1 · Jonas Osterberg¨ 1 · Sergei Silvestrov1 Received: 30 October 2015 / Revised: 1 September 2017 / Accepted: 28 September 2017 / Published online: 14 November 2017 © The Author(s) 2017. This article is an open access publication Abstract The value of the Vandermonde determinant is optimized over various surfaces, including the sphere, ellipsoid and torus. Lagrange multipliers are used to find a system of polynomial equations which give the local extreme points in its solutions. Using Grobner¨ basis and other techniques the extreme points are given either explicitly or as roots of poly- nomials in one variable. The behavior of the Vandermonde determinant is also presented visually in some interesting cases. Keywords Vandermonde determinant · Optimization · Grobner¨ basis · Orthogonal polynomials · Ellipsoid · Optimal experiment design · Homogeneous polynomials Mathematics Subject Classification (2010) 33C45 · 11C20 · 15B99 · 08B99 1 Introduction In this paper we will consider the extreme points of the Vandermonde determinant on various surfaces. The examination is primarily motivated by mathematical curiosity but the techniques used here are likely to be extensible to some problems related to optimal experiment design for polynomial regression, see Section 3. Karl Lundengard˚ [email protected] Jonas Osterberg¨ [email protected] Sergei Silvestrov [email protected] 1 Division of Applied Mathematics, UKK, Malardalen¨ University, Hogskoleplan¨ 1, Box 883, 721 23 Vaster¨ as,˚ Sweden 1418 Methodol Comput Appl Probab (2018) 20:1417–1428 To the authors knowledge this problem has previously been examined on cubes (see Sec- tion 3) and spheres, see Szego(˝ 1939). Here we will consider some techniques to extend some of these results to other surfaces such as ellipsoids, cylinders, p-norm spheres and other surfaces defined by homogeneous polynomials. Our examination will mostly be restricted to three dimensions but many of the techniques can in principle be extended to higher dimensions. 2 The Vandermonde Matrix A rectangular Vandermonde matrix of size m×n is determined by n values x = (x1, ··· ,xn) and is defined by ⎡ ⎤ 11··· 1 ⎢ ··· ⎥ ⎢ x1 x2 xn ⎥ = i−1 = ⎢ ⎥ Vmn(x) xj . (1) mn ⎣ . .. ⎦ m−1 m−1 ··· m−1 x1 x2 xn Note that some authors use the transpose of this as the definition and possibly also let indices run from 0. All entries in the first row of Vandermonde matrices are ones and by considering 0 0 = 1 this is true even when some xj is zero. In this paper we will primarily consider square Vandermonde matrices and for convenience we will use the notation Vn(x) = Vnn(x). The determinant of the Vandermonde matrix is well known. Theorem 1 The determinant of square Vandermonde matrices has the form det Vn(x) ≡ vn(x) = (xj − xi). (2) 1≤i<j≤n This determinant is also simply referred to as the Vandermonde determinant or Vander- monde polynomial or Vandermondian (Vein and Dale 1999). In this paper we will use the method of Lagrange multipliers to optimize the Vandermonde determinant over a surface. For this purpose the following properties will be useful. n(n−1) Lemma 1 The Vandermonde determinant is a homogeneous polynomial of degree 2 . n n(n − 1) Proof Considering (2) the numbers of factor of v (x) is i − 1 = . Thus n 2 i=1 n(n−1) vn(cx) = c 2 vn(x). (3) 3 Application to D-optimal Experiment Designs for Polynomial Regression with a Cost-function Suppose an experiment is conducted where m data points from some compact interval, X ⊂ R, i = 1, 2,...,m, are used to create a polynomial regression model of degree Methodol Comput Appl Probab (2018) 20:1417–1428 1419 m n − 1. A vector containing the data points, xm = (x1,x2,...,xm) ∈ X , is called a design m and a design is said to be D-optimal if det(Mn(xm)) ≥ det(Mn(ym)) for all y ∈ X where ⎡ ⎤ m m − ⎢ m x ... xn 1 ⎥ ⎢ i i ⎥ ⎢ i=1 i=1 ⎥ ⎢ m m m ⎥ ⎢ ⎥ ⎢ x x2 ... xn ⎥ ⎢ i i i ⎥ Mm(x) = ⎢ = = = ⎥ ⎢ i 1 i 1 i 1 ⎥ ⎢ . ⎥ ⎢ . .. ⎥ ⎢ m m m ⎥ ⎣ n−1 n 2n−2 ⎦ xi xi ... xi i=1 i=1 i=1 is the Fischer information matrix, see Kiefer (1959) and Gaffke and Krafft (1982). Optimal experiment design is often used to find the minimum number of points needed for a certain model. If we let m = n we get a interpolation problem defined by a square Van- dermonde matrix and the Fischer information matrix is Mn(x) = Vn(x) Vn(x) and since 2 Vn(x) is an n × n matrix det(Mn(x)) = det(Vn(x) ) det(Vn(x)) = det(Vn(x)) . Thus the maximization of the determinant of the Fischer information matrix is equivalent to find- ing the extreme points of the determinant of a square Vandermonde matrix in some volume given by the set of possible designs, see Gaffke and Krafft (1982). Optimal designs for various kinds of polynomial regression models are known, see Dette and Trampisch (2010) for an overview. The typical set of possible design is given by con- straining each parameter of the model to be in a certain interval. Usually these intervals are also normalized such that the vector of parameters x can be found in the n-dimensional cube x ∈[−1, 1]n. When consider certain other sets of possible designs the results presented in this paper might be useful. Suppose there is a cost-function associated with the data such that the total cost of the experiment being below some threshold value, g(x) ≤ 1, defines some compact set, G ={x ∈ Rm|g(x) ≤ 1}, such that G ⊂ X m. Since the Vandermonde determinant is a homogeneous polynomial for any c>1 |vn(x)| > |vn(cx)| the extreme points will be on the surface of the compact set and thus it is enough to consider the set of points defined by g(x) = 1. 4 Optimization using Grobner¨ Bases Grobner¨ bases together with algorithms to find them, and algorithms for solving a polyno- mial equation is an important tool that arises in many applications. One such application is the optimization of polynomials over affine varieties through the method of Lagrange mul- tipliers. We will here give some main points and informal discussion on these methods as an introduction and to fix some notation. Definition 1 (Cox et al. 1997)Letf1, ··· ,fm be polynomials in R[x1, ··· ,xn].Theaffine n variety V(f1, ··· ,fm) defined by f1, ··· ,fm is the set of all points (x1, ··· ,xn) ∈ R such that fi(x1, ··· ,xn) = 0forall1≤ i ≤ m. 1420 Methodol Comput Appl Probab (2018) 20:1417–1428 When n = 3 we will sometimes use the variables x,y,z instead of x1,x2,x3. Affine vari- eties are the common zeros of a set of multivariate polynomials. Such sets of polynomials will generate a greater set of polynomials (Cox et al. 1997)by m f1, ··· ,fm ≡ hifi : h1, ··· ,hm ∈ R[x1, ··· ,xn] , i=1 and this larger set will define the same variety. But it will also define an ideal (a set of polynomials that contains the zero-polynomial and is closed under addition, and absorbs multiplication by any other polynomial) by I(f1, ··· ,fm) =f1, ··· ,fm .AGrobner¨ basis for this ideal is then a finite set of polynomials {g1, ··· ,gk} such that the ideal gener- ated by the leading terms of the polynomials g1, ··· ,gk is the same ideal as that generated by all the leading terms of polynomials in I =f1, ··· ,fm . In this paper we consider the optimization of the Vandermonde determinant vn(x) over surfaces defined by a polynomial equation on the form n p sn(x1, ··· ,xn ; p; a1, ··· ,an) ≡ ai|xi| = 1, (4) i=1 where we will select the constants ai and p to get ellipsoids in three dimensions, cylinders in three dimensions, and spheres under the p-norm in n dimensions. The case of the ellipsoid is suitable for solution by Grobner¨ basis methods, but due to the existing symmetries the spheres are more suitable for other methods, as provided in Section 8. From (3) and the convexity of the interior of the sets defined by (4), under a suitable choice of the constant p and non-negative ai, it is easy to see that the optimal value of n | |p ≤ n | |p = vn on i=1 ai xi 1 will be attained on i=1 ai xi 1. And so, by the method of Lagrange multipliers we have that the minimal/maximal values of vn(x1, ··· ,xn) on sn(x1, ··· ,xn) ≤ 1 will be attained at points such that ∂vn/∂xi − λ∂sn/∂xi = 0for1≤ i ≤ n and some constant λ and sn(x1, ··· ,xn) − 1 = 0, Tyrrell Rockafellar (1993). For p = 2 the resulting set of equations will form a set of polynomials in λ, x1, ··· ,xn. These polynomials will define an ideal over R[λ, x1, ··· ,xn], and by finding a Grbner basis for this ideal we can use the especially nice properties of Grbner bases to find analytical solutions to these problems, that is, to find roots for the polynomials in the computed basis. 5 Extreme Points on the Ellipsoid in Three Dimensions In this section we will find the extreme points of the Vandermonde determinant on the three dimensional ellipsoid given by ax2 + by2 + cz2 = 1, (x,y,z) ∈ R3 (5) where a>0, b>0, c>0. Using the method of Lagrange multipliers together with (5) and some rewriting gives that all stationary points of the Vandermonde determinant lie in the variety V = V ax2 + by2 + cz2 − 1,ax+ by + cz, ax(z − x)(y − x) − by(z − y)(y − x) + cz(z − y)(z − x)) .
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