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

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 June, 10 2 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

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 June, 10 2 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

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 June, 10 2 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

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 June, 10 2 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

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 June, 10 2 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

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 � k := � n � k α � � i n =0 and α =1 i i α 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 . June, 10 3 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

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

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 k b ( k ) B ( k ) � p ( x ) = ( x ) , k ≥ l . (3) i i i =0 The coefficients of this expansion are given by ( a j := 0 for j ≥ k and j � = k ) � i i � b ( k ) j � = � a j , 0 ≤ i ≤ k . (4) i � k j =0 j ( Bernstein coefficients). The Bernstein coefficients can be organized in a multi-dimensional array B ( u ) = ( b ( k ) ) 0 ≤ i ≤ k , the so-called Bernstein patch . i June, 10 5 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

Properties of Bernstein Coefficients Endpoint interpolation property : k � b 0 , 0 ,..., 0 = a 0 , 0 ,..., 0 = p (0 , 0 , . . . , 0) , b 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 ∂ p � b ′ = i B k s , − 1 , i ( x ) , (6) ∂ x s i ≤ k s , − 1 where b ′ i = k s [ b i s , 1 − b i ] , 1 ≤ s ≤ n , x ∈ u . (7) June, 10 6 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

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) . June, 10 7 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

range enclosing property : For all x ∈ u min b ( k ) ≤ p ( x ) ≤ max b ( k ) . (9) i i 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 . June, 10 8 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

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 . R n defined as the The realization | V | of the simplex V is the set of convex hull of the points v 0 , . . . , v n . 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 ) . June, 10 9 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

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 0 + . . . + α n and λ α := � n i =0 λ α i i . Let k be a natural number. The Bernstein polynomials of degree k with respect to V are the polynomials � k � B k λ α , | α | = k . α := (10) α June, 10 10 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

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 � b α ( p , k , V ) B k p ( x ) = α , l ≤ k . (11) | α | = k The coefficients of this expansion are given by ( a j := 0 for j ≥ k and j � = k ) � α � β � b α ( p , k , ∆) = � a β (12) � k β ≤ α β ( Bernstein coefficients). June, 10 11 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

Bivariate Case over a simplex A bivariate polynomial of degree l in power form can be expressed as � a β x β p ( x ) = | β |≤ l = X 1 AX 2 , (13) � � x l 1 x 2 X 1 = , (14) 1 x 1 . . . where 1 1 � � x l 2 x 2 X 2 = 1 x 2 . . . , (15) 2 2   a 00 a 01 . . . a 0 l 1 a 10 a 11 . . . a 1 l 2   A =  . (16)  . . .  . . .   . . . . . .  a l 1 0 a l 1 1 . . . a l 1 l 2 June, 10 12 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

A bivariate polynomial in the simplicial Bernstein form can be expressed as x α 1 x α 2 (1 − x 1 − x 2 ) k − α 1 − α 2 � 1 2 p ( x ) = b α 1 ,α 2 α 1 ! α 2 ! ( k − α 1 − α 2 )! | α | = k = X 1 MX 2 , (17) � � x 2 x α 1 X 1 = , (18) 1 x 1 1 . . . 1 where 2! α 1 ! � � x 2 x α 2 X 2 = , (19) 1 x 2 2 . . . 2 2! α 2 ! b 01 (1 − x 1 − x 2 ) k − 1 b 00 (1 − x 1 − x 2 ) k b 0( k − 1) (1 − x 1 − x 2 )   . . . b 0 k k ! ( k − 1)! 1! b 10 (1 − x 1 − x 2 ) k − 1 b 11 (1 − x 1 − x 2 ) k − 2   . . . b 1 k (1 − x 1 − x 2 ) 0   ( k − 1)! ( k − 2)! M = (20) .   . . . . ... . . . .   . . . .   b k 0 0 0 . . . 0 June, 10 13 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization

The 2-dimensional array for the Bernstein coefficients can be obtained as   b 00 b 01 . . . b 0 l 1 b 10 b 11 . . . 0 B = 1 k !( U x 2 ( U x 1 W ) T ) T =    , (21) . . .  ...  . . .   . . .  b l 1 0 0 . . . 0 where  1 0 0 . . . 0  1 1 0 . . . 0   � 2  �  1 2! 1 . . . 0 U x 1 = U x 2 = , (22)  1  . . . .  ...  . . . .   . . . .   � k � k � � 1 1! 2! . . . 1 1 2   a 00 k ! a 01 ( k − 1)! . . . a 0( k − 1) 1! a 0 k a 10 ( k − 1)! a 11 ( k − 2)! . . . a 1( k − 1) 1! 0   W =  . (23)  . . .  ... . . .   . . . . . .  a k 0 0 0 0 0 June, 10 14 / 26 Matrix Methods for the Bernstein Form and Their Application in Global Optimization