Optimization and Positivity Certificates of Rational Functions using Bernstein Form
Tareq Hamadneh, Hassan Al-Zoubi, Hamza Alzaareer, and Rafael, Wisniewski

TL;DR
This paper explores the representation of rational functions in Bernstein form over a simplex, providing bounds for their range, algebraic positivity certificates, and dimension-independent bounds.
Contribution
It introduces new bounds for rational functions in Bernstein form that do not depend on the dimension, along with algebraic positivity certificates.
Findings
Bounds converge to the true range of the rational function.
Positivity certificates are derived using algebraic identities.
Dimension-independent bounds are established.
Abstract
Rational functions of total degree in n variables have a representation in the Bernstein form defined over dimensional simplex. The range of a rational function is bounded by the smallest and the largest rational Bernstein coefficients over a simplex. Convergence properties of the bounds to the range are reviewed. Algebraic identities certifying the positivity of a given rational function over a simplex are given. Subsequently, a bound established in this work does not depend on the given dimension.
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Polynomial and algebraic computation · Iterative Methods for Nonlinear Equations
Optimization and Positivity Certificates of Rational Functions using Bernstein Form
Tareq Hamadneh1, Hassan Al-Zoubi1,
Hamza Alzaareer1, and Rafael Wisniewski2
Al Zaytoonah University of Jordan, Amman, Jordan1
Aalborg University, 9220 Aalborg East, Denmark2
{t.hamadneh, dr.hassanz, h.alzaareer}@zuj.edu.jo1
Abstract
Rational functions of total degree in variables have a representation in the Bernstein form defined over dimensional simplex. The range of a rational function is bounded by the smallest and the largest rational Bernstein coefficients over a simplex. Convergence properties of the bounds to the range are reviewed. Algebraic identities certifying the positivity of a given rational function over a simplex are given. Subsequently, a bound established in this work does not depend on the given dimension.
Keywords: Bernstein polynomials, rational function, simplex, range bounding, certificates of positivity.
1 Introduction
The problem to decide whether a given rational function in variables is positive, in the sense that all its Bernstein coefficients are positive, goes back to [23, 24]. The same problem was addressed over different domains by other authors in [2] and [21]. The subject of certificates of positivity for polynomials over intervals was considered in [19, 20]. The same subject in the polynomial Bernstein basis was studied in [4], [14] and [21]. The problem of optimizing and approximating the minimum value of a polynomial over simplices was also extensively studied in [3], [5], [9, 13], [16]. The expansion of a given (multivariate) polynomial into Bernstein polynomials is used over a simplex, the so-called simplicial Bernstein form, [3], [5], [7], [15, 16]. The approach was extended to the rational case in [8], [10] and [25], however, without investigating certificates of positivity for rational functions over simplices. Rational functions play an important role in stability analysis of dynamic systems, since the system can be stable with a rational control function [26]. Many researchers were focused on the topic of stability analysis for nonlinear systems, which continues to be a challenging problem [1, 18]. Recently, stability of nonlinear polynomial systems has been translated to certificates of positivity, [11, 12]. In this paper, we extend certificates of positivity to the multivariate rational Bernstein functions over simplices. This ensures the stability of nonlinear dynamic systems with a control in the rational Bernstein form [26]. Specifically, we will optimize the minimum and maximum values of the rational Bernstein function. Subsequently, we investigate certificates of positivity for rational functions in the Bernstein basis with respect to degree elevation and with respect to the maximum diameter of subsimplices. To this end, we will numerically provide valid bounds that approximate the number of subdivision steps and the degree of a rational function. Furthermore, we will provide a new technique for minimizing the range of a rational function over subsimplices. Finally, global and local certificates of positivity are satisfied with these bounds for rational functions in the Bernstein form.
The organization of our paper is as follows. In the next section, we recall the most important background of the simplicial Bernstein expansion. In Section 3, we present polynomials in the Bernstein form. In Section 4, we extend optimization results to the rational case. Rational certificates of positivity are given in Section 5. Finally, Section 6 comprises conclusions.
2 Bernstein Expansion
We introduce some notation and necessary material about the simplicial Bernstein basis. Let be points of , the ordered list is called simplex of vertices . Throughout the paper, will denote a non-degenerate simplex of ; viz the points are affinely independent. Let be the associated barycentric coordinates of , i.e., the linear polynomials of such that , and . The realization of the simplex is the subset of defined as the convex hull of the points .
We refer to the multi-index and . Without loss of generality, we will often consider the standard simplex where denotes the canonical basis of , and the origin. This is not a restriction since any simplex in can be mapped affinely upon . Subsequently, if then . For with , we define
[TABLE]
If is any natural number such that , we use the notation
[TABLE]
The Bernstein polynomials of degree with respect to are the polynomials where
[TABLE]
For its multipowers are . Let be a polynomial of degree
[TABLE]
can be uniquely expresses for as
[TABLE]
where are called the Bernstein coefficients of of degree with respect to .
We recall the following notations:
- The grid points of degree associated to are the points
[TABLE]
which leads us to the control points associated to of degree with respect to
[TABLE]
The control points of form its control net of degree .
- The discrete graph of of degree with respect to is formed by the points
[TABLE]
Proposition 2.1**.**
*[16, Proposition 2.7] For and , the following properties hold.
(i) Linear precision: If degree , then
[TABLE]
(ii) Interpolation at the vertices: If denotes the canonical
basis of then
[TABLE]
(iii) Convex hull property: The graph of over is contained in
*the convex hull of its associated control points;
(iv) Range enclosing property:
[TABLE]
It follows from in Proposition 2.1, the interval
[TABLE]
encloses the range of of degree over .
Finally, we denote the distance between two intervals , by
[TABLE]
3 Polynomial Bernstein Form
In this section, we present the most important properties of the Bernstein expansion over a simplex we will employ throughout the paper.
In the following remark, we provide the simplicial polynomial Bernstein form of a given on .
Remark 3.1**.**
For , let be a polynomial of degree . The simplicial Bernstein form of of degree on is given by
[TABLE]
where
[TABLE]
and
[TABLE]
The Bernstein coefficients of degree () can be given as linear combinations of Bernstein coefficients of degree , see, e.g., [15, Proposition 1.12].
Let be points of , . By multiplying both sides of (5) with and rearranging the result we obtain, see [6, Lemma 1.1]
[TABLE]
where
[TABLE]
Hence, the range of of degree over can be bounded by
[TABLE]
Remark 3.2**.**
The number of Bernstein coefficients of an variate polynomial of degree is equal .
The following definiton is given in [15].
Definition 3.1**.**
Let be a non-degenerate simplex of For and define the second differences of of degree with respect to as
[TABLE]
with the convention The second differences constitute the collection
[TABLE]
Let denotes the maximum of the second differences, i.e.,
[TABLE]
Theorem 3.1**.**
[15, Theorem 4.2] Let and . Then
[TABLE]
where
[TABLE]
A similar statement holds for the minimum.
4 Rational Bernstein Form
We may assume a rational function where both and have the same degree since otherwise we can elevate the degree of the Bernstein expansion of either polynomial by component where necessary to ensure that their Bernstein coefficients are of the same order Since any simplex can be mapped upon the standard simplex by an affine transformation, we will extend results from polynomials to rational functions over . Let the range of over be The simplicial rational Bernstein coefficients of of degree with respect to are given by
[TABLE]
Without loss of generality, we assume throughout the paper that , .
The range enclosing property for the rational function is given from [17, Theorem 3.1] as
[TABLE]
By application of (6) to (9), the following theorem provides the sharpness property [25, Theorem 4] of with respect to its enclosure bound.
Theorem 4.1**.**
The equality holds in the right hand side of (10)
[TABLE]
if and only if
[TABLE]
and . A similar statement holds for the equality in the left hand side of (10).
Remark 4.1**.**
We conclude from [25, Theorem 5] that
[TABLE]
In the following theorem, we review the linear convergence [25] of the range of a rational function to the enclosure bound under degree elevation. We include the positive and negative cases of , since in [25] just the positive case is given.
Theorem 4.2**.**
For it holds that
[TABLE]
where
[TABLE]
and
[TABLE]
Proof. The proof follows by using arguments similar to that given in the proof of Theorem 5.4.
Assume that has been subdivided with respect to a point in , , where the interiors of the simplices are disjoint. Denote the union of the enclosure bounds over , , by . The following theorm reviews the quadratic convergence [25, Theorem 7] of the range of a rational function to the enclosure bound with respect to subdivision. The proof also follows by using arguments similar to that given in the proof of Theorem 5.4.
Theorem 4.3**.**
Let be a subdivision of the standard simplex and be an upper bound on the diameters of the s. Then we have for
[TABLE]
where
[TABLE]
and is the constant (14) independent of .
5 Rational Certificates of Positivity
We study the positivity of rational functions over a non-degenerate simplex . In order to do so, we use the simplicial rational Bernstein form. Certifying the positivity of rational functions is desired in many applications such stability analysis of dynamic systems, optimization and control theory. The enclosure property of shows that if all Bernstein coefficients of over are positive, then the rational function is positive over . The converse need not to be true. There are rational functions which are positive over and some Bernstein coefficients are negative.
Example 5.1**.**
Let
[TABLE]
which is positive over , but the list of Bernstein coefficients has a negative value at .
The (univariate) Bernstein polynomials of of degree on
[TABLE]
take positive values over . Note that is positive at and is positive at . The Bernstein coefficient is the value of at and is the value at . Hence, if all Bernstein coefficients of are positive, the rational Bernstein form of over a given domain provides certificates of positivity for over the same domain. Without loss of generality, we assume that the (multivariate) rational case is studied on the standard simplex . Denote by the list of Bernstein coefficients of a rational function with respect to , we define by:
[TABLE]
The rational Bernstein form of of degree is positive on if In the following subsections, we decide if a rational function is positive and gives certificates of positivity in the rational Bernstein form by sharpness, degree elevation (global certificates), subdivision (local certificates) and minimization of a rational function. At the last, we provide a bound does not depend on the number of variables of .
5.1 Certificates by Sharpness
The sharpness property in Theorem 4.1 satisfies the certificate of positivity of a rational function over . The equality holds in the left hand side of (10) if
[TABLE]
This implies the following proposition.
Proposition 5.1**.**
Given is positive on . If , then satisfies the certificate of positivity.
5.2 Global Certificates
If big enough, the minimum rational Bernstein coefficient of converges linearly to the minimum range over . We show that the positive rational function has a global certificate of positivity at degree over . The Bernstein degree is estimated in the following theorem.
Proposition 5.2**.**
Given is a positive rational function of degree over . If
[TABLE]
where is the constant (13), then satisfies the global certificate of positivity.
Proof. Let so that
[TABLE]
Then are nonnegative. Theorem 4.2 implies that
[TABLE]
the interpolation property shows that , , are positive.
Observing the obtained global certificate of positivity, we give the following corollary.
Corollary 1**.**
If is a rational function of degree positive over , then there exist some such that the minimum rational Bernstein coefficient of of degree is positive.
Example 5.2**.**
Let a rational function
[TABLE]
of degree , which is positive over . Note that is negative. The rational Bernstein form of (17) has a global certificate of positivity at , since , and .
5.3 Local Certificates
In this section, we will not elevate the degree any more. This will lead to local certificates of positivity.
Definition 5.1** (15, Definition 5.4).**
Let be a subdivision of the simplex , and the interiors of the simplices are disjoint. If satisfies the certificate of positivity for all , we say that satisfies the local certificate of positivity associated to the subdivision , which we write
We recall that the subdivision scheme consisting in steps of binary splitting has a shrinking factor .
Remark 5.1**.**
From [15, Definition 5.5] and [16, Definition 2.14], the mesh of , denoted by is its diameter. If is a subdivision scheme, we write the subdivision of obtained after successive subdivision steps. is said to have a shrinking factor if for every simplex , , where is the largest mesh among the subsimplices .
From Theorem 4.3, the following proposition can be similarly shown as proposition 5.2.
Proposition 5.3**.**
Let be a rational function, positive over . Let be an integer and a subdivision scheme with a shrinking factor . Assume that
[TABLE]
where is the constant (16). Then satisfies the local certificate of positivity associated to .
Example 5.3**.**
We consider the rational function (17) over . The coefficients of over subintervals of width are given as follows:
, , and . The rational Bernstein function still has a negative value over . Therefore, halving the interval and finding the coefficients over the new subintervals of
[TABLE]
satisfy the local certificate of positivity at the second subdivision step.
5.4 Optimization of Rational Functions
The enclosure bound leads to a lower bound of on . By repeatedly subdividing , the minimum of over can then be approximated within any desired accuracy. Choosing , the number of subdivision steps is bounded in Theorem 5.4.
Remark 5.2**.**
Let a rational Bernstein form of be on and let the minimum Bernstein coefficient of of degree be . The value is defined as
[TABLE]
where is attained at , . Then by (3) and (10) one can deduce that
[TABLE]
Theorem 5.4**.**
Let a rational function , and a subdivision scheme with a shrinking factor . Let and an integer satisfying
[TABLE]
where is the constant (16), then
[TABLE]
Proof. Assume that
[TABLE]
Let We can conclude from (20) and the corresponding grid point that
[TABLE]
[TABLE]
[TABLE]
Taking absolute values and using (10) we can estimate
[TABLE]
where the last inequality follows by Proposition 5.3 which completes the proof.
Corollary 2**.**
Given is a rational function of degree over . Assume under the assumptions of Theorem 5.4 that . If and are satisfying:
[TABLE]
then satisfies the local certificate of positivity
5.5 Independent Bounds
In this section, we provide a bound does not depend on the number of variables of . Such this bound is the best in high dimensions as explained in [9], [14] and [22].
Powers and Reznick in [22] have proved the following bound:
Theorem 5.5**.**
[21, Theorem 3] Let be a polynomial of degree , positive on the standard simplex . Let be the minimum of on . Then for
[TABLE]
the Bernstein form of of degree has positive coefficients.
In the following corollary, we hold Theorem 5.5, [22, Proposition 4] and results from [21] to the rational case.
Corollary 3**.**
Let be a rational function, positive on . If
[TABLE]
then satisfies the global certificate of positivity.
Proof. Let be large enough and , hence allows from Theorem 5.5 and [22, Proposition 4] the (global) certificate of positivity. Hence, If
[TABLE]
then . It follows that are positive, which completes the proof .
Corollary 4**.**
Let and , where is the minimum of on . Then the positive rational function satisfies the (global) certificate of positivity over if
Remark 5.3**.**
Given is a rational function of degree , negative on . Then satisfies certificates of negativity by applying the same arguments above to the upper bounds.
6 Conclusions
In this paper, we considered the multivariate rational function in the Bernstein form. The expansion of the numerator and denominator polynomials into Bernstein polynomials was applied. We reviewed properties such sharpness and monotonicity of bounds of a multivariate rational function. The linear and the quadratic convergence of the enclosure bound to the range of a rational function improved the bounds of over simplices. By repeatedly subdividing the simplex, the minimum of over a simplex was approximated within a desired accuracy. We addressed an optimization of a (multivariate) rational function and bounded the number of subdivision steps. Subsequently, we investigated certificates of positivity in the simplicial rational Bernstein form by sharpness, degree elevation and subdivision. At the last, we estimated the degree of the Bernstein expansion by a bound which is not depending on the given dimension.
Acknowledgments
The authors gratefully acknowledge support from Al-Zaytoonah University of Jordan under the grant number 2019-2018/585/G12. The first author would like to thank Professor Amjed Zraiqat for his careful reading the manuscript and the constructive comments.
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