Matrix Methods for the Bernstein Form and Their Application in Global Optimization

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1 Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 Jihad Titi Jürgen Garloff University of Konstanz Department of Mathematics and Statistics and University of Applied Sciences / HTWG Konstanz Faculty of Computer Science Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 1 / 26

2 Outline The Bernstein expansion for polynomials over a box and a simplex New method for the computation of the Bernstein coefficients of multivariate Bernstein polynomials over a simplex Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 2 / 26

3 Outline The Bernstein expansion for polynomials over a box and a simplex New method for the computation of the Bernstein coefficients of multivariate Bernstein polynomials over a simplex Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 2 / 26

4 Outline The Bernstein expansion for polynomials over a box and a simplex New method for the computation of the Bernstein coefficients of multivariate Bernstein polynomials over a simplex New method for the calculation of the Bernstein coefficients over sub-boxes generated by subdivision Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 2 / 26

5 Outline The Bernstein expansion for polynomials over a box and a simplex New method for the computation of the Bernstein coefficients of multivariate Bernstein polynomials over a simplex New method for the calculation of the Bernstein coefficients over sub-boxes generated by subdivision Test for the convexity of a polynomial Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 2 / 26

6 Outline The Bernstein expansion for polynomials over a box and a simplex New method for the computation of the Bernstein coefficients of multivariate Bernstein polynomials over a simplex New method for the calculation of the Bernstein coefficients over sub-boxes generated by subdivision Test for the convexity of a polynomial Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 2 / 26

7 Notations We will consider the unit box u := [0, 1] n, since any compact nonempty box x of R n can be mapped affinely upon u. The multi-index (i 1,..., i n ) is abbreviated by i, where n is the number of variables. The multi-index k is defined as k = (k 1, k 2,..., k n ). Comparison between and arithmetic operations with multi-indices are defined entry-wise. For x = (x 1, x 2,..., x n ) R n, its monomials are defined as x i := n j=1 x i j j. The abbreviations k i=0 := k 1 i 1 =0... k n i and ( ) k n=0 i := n α=1 are used. i s,q := (i 1, i 2,..., i s 1, i s + q, i s+1,..., i n ) where 0 i s + q k s. ( kα iα ) Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 3 / 26

8 Bernstein Polynomials Let p be an n-variate polynomial of degree l l p(x) = a i x i. (1) i=0 The i-th Bernstein polynomial of degree k, k l, is the polynomial (0 i k) ( ) B (k) k i (x) = x i (1 x) k i. (2) i Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 4 / 26

9 The Bernstein polynomials of degree k form a basis of the vector space of the polynomials of degree at most k. Therefore, p can represented by p(x) = k i=0 b (k) i B (k) i (x), k l. (3) The coefficients of this expansion are given by (a j := 0 for j k and j k) ( i i j) b (k) i = ( k )a j, 0 i k. (4) j (Bernstein coefficients). j=0 The Bernstein coefficients can be organized in a multi-dimensional array B(u) = (b (k) i ) 0 i k, the so-called Bernstein patch. Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 5 / 26

10 Properties of Bernstein Coefficients Endpoint interpolation property: b 0,0,...,0 = a 0,0,...,0 = p(0, 0,..., 0), b k = k a i = p(1, 1,..., 1). (5) i=0 The first partial derivative of the polynomial p with respect to x s is given by where p x s = i k s, 1 b ib ks, 1,i (x), (6) b i = k s [b is,1 b i ], 1 s n, x u. (7) Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 6 / 26

11 convex hull property:the graph of p over u is contained in the convex hull of the control points. {( ) } {( ) } x i/k : x u conv : 0 i k. (8) p(x) b i Figure 1:The graph of a degree 5 polynomial and the convex hull (shaded) of its control points (marked by squares). Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 7 / 26

12 range enclosing property: For all x u min b (k) i p(x) max b (k) i. (9) Equality holds in the left or right inequality in (9) if and only if the minimum or the maximum, respectively, is attained at a vertex of u, i.e., if i j {0, k j }, j = 1,..., n. Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 8 / 26

13 Simplex Let v 0,..., v n be n + 1 points of R n. The ordered list V = [v 0,..., v n ] is called simplex of vertices v 0,..., v n. The realization V of the simplex V is the set of convex hull of the points v 0,..., v n. R n defined as the Any vector x R n can be written as an affine combination of the vertices v 0,..., v n with weights λ 0,..., λ n called barycentric coordinates. If x = (x 1,..., x n ), then λ = (λ 0,..., λ n ) = (1 n i=1 x i, x 1,..., x n ). Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, 10 9 / 26

14 For every multi-index α = (α 0,..., α n ) N n+1 and λ = (λ 0,..., λ n ) R n+1 we write α := α α n and λ α := n i=0 λα i i. Let k be a natural number. The Bernstein polynomials of degree k with respect to V are the polynomials B k α := ( ) k λ α, α = k. (10) α Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

15 Bernstein Polynomials The Bernstein polynomials of degree k form a basis of the vector space R k [X] of polynomials of degree at most k. Therefore, p can be uniquely represented as p(x) = α =k The coefficients of this expansion are given by (a j := 0 for j k and j k) b α (p, k, V )B k α, l k. (11) b α (p, k, ) = β α ( α β) ( k β)a β (12) (Bernstein coefficients). Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

16 Bivariate Case over a simplex A bivariate polynomial of degree l in power form can be expressed as p(x) = a β x β β l = X 1 AX 2, (13) [ ] where X 1 = 1 x 1 x x l 1 1, (14) [ ] X 2 = 1 x 2 x x l 2 2, (15) a 00 a a 0l1 a 10 a a 1l2 A = (16) a l1 0 a l a l1 l 2 Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

17 A bivariate polynomial in the simplicial Bernstein form can be expressed as p(x) = α =k x α 1 1 x α 2 2 (1 x 1 x 2 ) k α 1 α 2 b α1,α 2 α 1! α 2! (k α 1 α 2 )! = X 1 MX 2, (17) [ ] x where X 1 = 1 1 x 2 x 1 2!... α 1 1, (18) α 1! [ ] x X 2 = 2 1 x 2 x 2 2!... α 2 2, (19) α 2! b 00 (1 x 1 x 2 ) k b 01 (1 x 1 x 2 ) k 1 b k! (k 1)!... 0(k 1) (1 x 1 x 2 ) 1! b 0k b 10 (1 x 1 x 2 ) k 1 b 11 (1 x 1 x 2 ) k 2 M = (k 1)! (k 2)!... b 1k (1 x 1 x 2 ) 0 (20)..... b k Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

18 The 2-dimensional array for the Bernstein coefficients can be obtained as b 00 b b 0l1 B = 1 k! (U x 2 (U x1 W ) T ) T b 10 b =..., (21) b l where ( U x1 = U x2 = 1 2 ) 1 2! , (22) (. 1 k ) ( 1 1! k ) 2 2!... 1 W = a 00 k! a 01 (k 1)!... a 0(k 1) 1! a 0k a 10 (k 1)! a 11 (k 2)!... a 1(k 1) 1! (23) a k Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

19 Multidimensional case The polynomial coefficients given by The Bernstein coefficients given by a a a l a a a l a 0l a 1l a l1l a 00l3...0 a 10l a l10l a 01l3...0 a 11l a l11l a 0l2l a 1l2l a l1l 2l a 0l2l 3...l n a 1l2l 3...l n... a l1l 2l 3...l n B = 1 k! (U x n... (U xi... (U x3 (U x2 (U x1 W ) T ) T ) T... ) T... ) T, (24) where W can be obtained by multiplying the entries a i1 i 2...in of A by (k n r=1 i r )! and U xi = U x1 for all i = 2, 3,..., n (they are given in equation (22) ). Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

20 The partial derivative with respect to x s of p in the simplicial Bernstein form is p r (x) = b α(p, k 1, V )B α (k 1) (x) = k (b α b αs, 1 )B α (k 1 s, 1 (x)(25 α =k 1 α =k 1 In the two-dimensional case, the Bernstein coefficients of p over the standard simplex are given as b 00 b b 0l1 b 10 b (26) b l The Bernstein coefficients of p x 1 over are given as b 10 b 00 b 11 b b 1(l2 1) b 0(l2 1) b 00 b b 0l 2 b 20 b 10 b 21 b =... = b 10 b b l10 b (l1 1) b (l 1 1) Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

21 Subdivision From the Bernstein coefficients b (k) i of p over u, we can compute by the de Casteljau algorithm the Bernstein coefficients over sub-boxes u 1 and u 2 resulting from subdividing u in the s-th direction, i.e., u 1 := [0, 1]... [0, λ]... [0, 1], u 2 := [0, 1]... [λ, 1]... [0, 1], (27) for some λ (0, 1). Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

22 De Casteljau algorithm By starting with B 0 (u) = B(u) we set for k = 1, 2,..., n s, { b (k) b (k 1) i, i s k, i = (1 λ)b (k 1) i s, 1 + λb (k 1) i, k i s. (28) To obtain the new coefficients, the above formula is applied for i j = 0, 1,..., j = 1, 2,..., s 1, s + 1,..., l. Then, The Bernstein patch over u 1 is given by b i (u 1 ) = b (ns) i, (29) b i (u 2 ) = b (ns is) i 1,i 2,...,l s,...,i n (30) B(u 1 ) = B (ns) (u), and Bernstein patche B(u 2 ) over the sub-box u 2 are obtained as intermediate values in this computation. Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

23 Subdivision Direction Selection Subdivision can be performed in any coordinate direction. It may be advantageous to subdivide in a particular direction to increase the probability of finding a sharp range enclosure. The merit function for the subdivision in coordinate direction K = min {j : j {1, 2,..., l}, y(j) = max {y(s), s = 1, 2,..., l}}.(31) Rule A: y(s) = wid(u s ), where wid(u s ) is the width(edge length) of the box in the direction s. Rule B: y(s) = max p x s = max i ks, 1 b is,1 b i. Rule C: y(s) = [max i ks, 1 (b is,1 b i ) min i ks, 1 (b is,1 b i )] wid u s. Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

24 The Bernstein coefficients can be calculated over a sub-box by premultiplying the matrix representing the Bernstein patch B(u) by matrices which depend on the subdivision parameter point λ. E.g., when the subdivision is applied in the first coordinate direction, then the Bernstein patches over u 1 and u 2 are given as B(u 1 ) = L m L m 1... L 1 B(u), B(u 2 ) = L ml m 1... L 1B(u), (32) where for t = 1, 2,..., m [ ] I L t = t 0, L t = (1 λ)e 1,t M m+1 t [ M m+1 t λe m+1 k,1 0 I t ]. (33) where I t is the txt identity matrix, E 1,t, E m+1 k,1 R m 1 t,t with all of their entries are zero except the (1, t) and (m + 1 t, 1) entry is 1, respectively, and M m+1 t = (mij), Mm+1 t = (m ij ) Rm+1 t,m+1 t, λ if i = j, 1 λ, if i = j, m ij := 1 λ, if i = j + 1, mij := λ, if i = j 1, (34) 0, if otherwise, 0, if otherwise. Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

25 The matrix method has the following advantages over the de Casteljau algorithm: Elegant. Easier to handle. The Bernstein coefficients over each sub-box appear directly. The matrix method of computation of the Bernstein coefficients over each sub-box and the matrix method proposed by Ray and Nataraj [6](for computation of the Bernstein coefficients over the entire box) are complement each other. Thus, the Bernstein coefficients can be calculated by using matrix methods only. Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

26 Notations IR: set of the compact, nonempty real intervals [a] = [a, a], a a. IR n : set of n-vectors with components from IR, interval vectors. IR n,n : set of n-by-n matrices with entries from IR, interval matrices. Elements from IR n and IR n,n may be regarded as vector intervals and matrix intervals, respectively, w.r.t. the usual entrywise partial ordering, e.g., [A] = ([a ij ]) n i,j=1 = ( [a ij, a ij ] ) n i,j=1 = [A, Ā], where A = ( a ij ) n i,j=1, A = (a ij) n i,j=1. A vertex matrix of [A] is a matrix A = (a ij ) n i,j=1 with a ij {a ij, a ij }, i, j = 1,..., n. Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

27 An interval matrix [A] R n,n can be represented as [A] = [A c, A c + ] = {A : A c A A c + } (35) with A c, R n,n and symmetric, 0. We introduce an auxiliary index set Y := {z R n ; z j = 1 for j = 1, 2,... n}, with cardinality 2 n. For each z Y define the matrix A z := A c T z T z, (36) where T z is an nxn diagonal matrix with diagonal vector z. A z [A] for each z Y. The number of mutually different matrices A z is at most 2 n 1. Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

28 Test for the convexity of a polynomial p Theorem (Bialas and Garloff, 1984; Rohn, 1994) Let [A] be a square interval matrix. Then, [A] is positive semidefinite if and only if A z is positive semidefinite for each z Y. Second order convexity condition Let the function f : R n R be twice differentiable, that is, its Hessian matrix 2 f exists at each point in the domf. Then f is convex if and only if the domf is convex and its Hessian matrix is positive semidefinite for all x domf, i.e., 2 f 0. Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

29 References [1] G.T. Cargo and O. Shisha, The Bernstein Form of a Polynomial, J. Res. Nat. Bur. Standards 70(B):79 81, [2] J. Garloff, Convergent Bounds for the Range of Multivariate Polynomials, Interval Mathematics 1985, K. Nickel, Ed., Lecture Notes in Computer Science, vol. 212, Springer, Berlin, Heidelberg, New York, 37 56, [3] J. Garloff, The Bernstein Algorithm, Interval Comput. 2: , [4] P.S.V Nataraj and M. Arounassalame, A New Subdivision Algorithm for the Bernstein Polynomial Approach to Global Optimization, Int. J. Automat. Comput. 4(4): , [5] S. Ray and P.S.V. Nataraj, An Efficient Algorithm for Range Computation of Polynomials Using the Bernstein Form, J. Global Optim. 45: , [6] S. Ray and P.S.V. Nataraj, A Matrix Method for Efficient Computation of Bernstein Coefficients, Reliab. Comput. 17(1):40 71, Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26

30 Thank you for your attention! Matrix Methods for the Bernstein Form and Their Application in Global Optimization June, / 26