A NOTE on OPTIMIZATION with MORSE POLYNOMIALS 3 with Distinct Critical Values

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∗ ∗ ∗ Therefore fsos ≤ f , and the equality occurs if and only if f − f is a SOS. ∗ ∗ In the case where fsos is strictly less than f , Lasserre proposed finding a sequence of lower bounds for f(x) in some large ball {x ∈ Rn| kxk ≤ R}. This sequence converges to f ∗ when the degrees of the polynomials introduced in the algorithm tend to infinity. However, it may not converge in finitely many steps, and moreover the degrees of the required auxiliary polynomials can be very large. J. Nie, J. Demmel and B. Sturmfels (2006, [11]) have given a method to compute the global infimum f ∗ which terminates in finitely many steps in ∂f ∂f terms of the gradient ideal Igrad(f) = , · · · , . The finite conver- D ∂x1 ∂xn E gence of their SOS relaxation depends mainly on the SOS representation of ∗ f − f modulo the gradient ideal Igrad(f). More explicitly, for each natural number N, let us consider the problem ∗ fN,grad := maximize γ (1.3) m ∂f subject to f(x)−γ− φ (x) is a SOS in R[X] and φ (x) ∈ R[X]2 − +1. i=1 i ∂xi i N d R P n+m Here [X]m denotes the m -dimensional vector space of polynomials of degree at most m. Nie, Demmel and Sturmfels proved the following Theorem 1.1 ([11, Theorem 10]). If f attains its infimum f ∗ over Rn, ∗ ∗ then the sequence {fN,grad}N converges increasing to f . Moreover, if the gradient ideal Igrad(f) is radical, then there exists an integer N such that ∗ ∗ fN,grad = f . It follows from Theorem 1.1 that the radicality of the gradient ideal Igrad(f) is a sufficient condition for the finite convergence of the sequence ∗ R {fN,grad}. Moreover, for almost all polynomials f in [X]d the gradient ideal Igrad(f) is radical [11, Proposition 1], i.e. the ideal Igrad(f) is generi- cally radical. In this paper we introduce a generic class of polynomials whose gradient ideals are radical - the class of Morse polynomials. Recall that a point x0 ∈ Rn is called a critical point of a polynomial f ∈ R[X] if ∂f ∂f (x0)= · · · = (x0) = 0. ∂x1 ∂xn A critical point x0 of the polynomial f is said to be non-degenerate if the Hessian matrix2 ∇2f(x0) of f at x0 is invertible. A polynomial f ∈ R[X] is called a Morse polynomial if all of its critical points are non-degenerate

2 0 n For a polynomial f ∈ R[x1, · · · ,xn], its Hessian matrix at a point x ∈ R is defined by 2 2 ∂ f 0 ∂ f 0 2 (x ) · · · (x )  ∂x1 ∂x1∂xn  2 0 . . ∇ f(x ) :=  . .  .  2 2   ∂ f 0 ∂ f 0  (x ) · · · 2 (x )  ∂xn∂x1 ∂xn  A NOTE ON OPTIMIZATION WITH MORSE POLYNOMIALS 3 with distinct critical values. It is well-known from differential topology and singularity theory that almost all polynomial functions on Rn are Morse (cf. [1]). Moreover, it was shown recently by Hà and Phạm [5, Theorem 5.1] that the set of Morse polynomials in R[X] whose Newton polyhedra at infinity3 N (f) contained in a convenient 4 Newton polyhedron at infinity N is open and dense in N . Hence the class of Morse polynomials in R[X] with a given convenient Newton polyhedron at infinity is large enough to consider for optimization. The main result in this paper is the following Theorem 1.2. Let f ∈ R[X] be a Morse polynomial. Then its gradient ideal Igrad(f) is radical. As a direct consequence of this theorem and Theorem 1.1, we get the following result which was obtained by the author in [8]. Corollary 1.3. Let f ∈ R[X] be a Morse polynomial. Assume that f attains ∗ Rn ∗ its infimum f over . Then there exists an integer N such that fN,grad = f ∗. Remark 1.4. In Corollary 1.3, the assumption that f achieves a minimum on Rn cannot be removed. In fact, let us consider the polynomial f(x,y)= x2 + (xy − 1)2 ∈ R[x,y]. It is easy to check that f has a single critical point at (0, 0) which is non-degenerate. Hence f is a Morse polynomial. Moreover, 1 since lim f( ,y) = 0, we have f ∗ = 0. But f does not attain the value 0 at y→∞ y any points in R2. On the other hand, we have 1 ∂f 1 ∂f f(x,y) − 1= x(1 − y2) + y(1 + y2) . 2 ∂x 2 ∂y ∗ Therefore, it follows from the definition of fN,grad in (1.3), with φ1(x,y) = 1 2 1 2 ∗ ∗ x(1 − y ) and φ2(x,y) = y(1 + y ), that fN,grad = 1 > 0 = f for every 2 2 ∗ ∗ N ≥ 3. This means that the sequence {fN,grad}N never converges to f in this case.

2. Proof of Theorem 1.2 In this section we give a proof for Theorem 1.2, applying some known results in singularity theory. Let f ∈ R[X] be a Morse polynomial. Let us denote by

n ∂f ∂f Vgrad(f) := {x ∈ C | (x)= · · · = (x) = 0} ∂x1 ∂xn

3 n A subset N of the positive cone R+ is called a Newton polyhedron at infinity if it is the convex hull in Rn of the set A ∪ {0}, where A ⊆ Nn is a finite set. 4 n A polyhedron N ⊆ R+ is called convenient if it intersects each coordinate axis at a α α1 αn point different from the origin. For a polynomial f(x)= α uαx := α uαx1 · · · xn ∈ R[X], its Newton polyhedron at infinity N (f) is definedP by the convexP hull in Rn of the set {α|uα 6= 0} ∪ {0}. 4 LÊ CÔNG-TRÌNH the set of complex critical points of f. Since f is a polynomial, the set n Vgrad(f) is actually an algebraic subset of C defined by the gradient ideal Igrad(f).

Lemma 2.1. If f is a Morse polynomial, then Vgrad(f) is finite. Proof. It is well-known that each complex critical point of the Morse polynomial f is isolated (cf. [10, Corollary 2.3]). Therefore the set Vgrad(f) is n discrete. On the other hand, the set Vgrad(f) is algebraic in C , hence it has only finitely many connected components (cf. [9, p.408]). Note that, each critical point itself is a connected component of the algebraic set Vgrad(f). This implies that Vgrad(f) has only finitely many elements. 1 N n Assume Vgrad(f)= {p , · · · ,p }, and let p ∈ C denote one of the points i n p . If p = (p1, · · · ,pn) ∈ C , denote by C{x1 − p1, · · · ,xn − pn} the ring of convergent power series at the point p. The number

µ(f,p) := dimC C{x1 − p1, · · · ,xn − pn}/Igrad(f) is called the Milnor number of f at p. Lemma 2.2. µ(f,p) is finite. Proof. The result follows from the Hilbert’s Nullstellensatz, since p is an isolated critical point of f (cf. [4, Lemma I. 2.3]). Denote N i µ(f) := µ(f,p ) = dimC C[x1, · · · ,x ]/I (f), X n grad i=1 which is called the total Milnor number of f. It follows from Lemma 2.2 that µ(f) is finite. The following result gives a relation between the number |Vgrad(f)| of complex critical points of f and the total Milnor number of f, which yields also a criterion to check for radicality of the gradient ideal Igrad(f).

Lemma 2.3 ([3, Theorem 2.10]). |Vgrad(f)|≤ dimC C[x1, · · · ,xn]/Igrad(f), with equality if and only if the gradient ideal Igrad(f) is radical. To apply this lemma, we need to prove the following Lemma 2.4. µ(f,pi) = 1 for every i = 1, · · · ,N. Proof. Let us denote by p one of the points pi, i = 1, · · · ,N. By translating the variety accordingly, without loss of generality we may assume that p is the origin 0 ∈ Cn. Consider the polynomial map n n (f1, · · · ,fn) : C → C , ∂f where f := ∈ C[x1, · · · ,x ], i = 1, · · · ,n. We have f (0) = 0 for all i ∂xi n i i = 1, · · · ,n. Moreover, we will show in the following that the determinant of the Jacobian matrix J(f1, · · · ,fn) of this polynomial map at the origin A NOTE ON OPTIMIZATION WITH MORSE POLYNOMIALS 5

0 ∈ Cn is non-zero. In fact, since f is Morse, we see that 0 is a non-degenerate critical point of f. It follows that the determinant of the Hessian matrix ∇2f(0) of f at 0 is non-zero. Observe that ∂f1 (0) · · · ∂f1 (0) ∂x1 ∂xn  . .  det J(f1, · · · ,fn)(0) = det . .    ∂fn (0) · · · ∂fn (0) ∂x1 ∂xn ∂2f ∂2f 2 (0) · · · ∂x ∂x (0)  ∂x1 1 n  = det . .  . .   ∂2f ∂2f  (0) · · · 2 (0)  ∂xn∂x1 ∂xn  = det ∇2f(0).

Hence det J(f1, · · · ,fn)(0) 6= 0. It follows from the implicit function theorem for convergent power series (cf. [4, Theorem I.1.18]) the following isomor- phism of C-vector spaces:

C{x1, · · · ,xn}/ hf1, · · · ,fni =∼ C. This implies that µ(f, 0) = dimC C = 1. Proof of Theorem 1.2. Assume that f is a Morse polynomial. Then it follows from Lemma 2.1 that Vgrad(f) is a finite set whose elements are p1, · · · ,pN . By Lemma 2.4, at each complex critical point pi of f we have µ(f,pi) = 1. Hence N µ(f)= µ(f,pi)= N = |V (f)| . X grad i=1 The theorem now follows from Lemma 2.3. Acknowledgements The author would like to express his gratitude to Professor Hà Huy Vui for his valuable discussions on Morse theory and polynomial optimization. He would also like to thank the anonymous referees for their careful reading and detailed comments with many helpful suggestions.

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Lê Công-Trình, Department of Mathematics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon, Binh Dinh E-mail address: [email protected]