Optimizing a Parametric Linear Function over a Non-compact Real Algebraic Variety
Feng Guo
School of Mathematical Sciences Dalian University of Technology
Dalian, 116024, China
[email protected]
Mohab Safey EI Din
Sorbonne Universités, UPMC, Univ Paris 06
INRIA Paris-Rocquencourt POLSYS project-team CNRS LIP6 UMR 7606
[email protected]
Chu Wang and Lihong Zhi
Key Lab. of Mathematics Mechanization
AMSS, Beijing 100190, China
{cwang,lzhi}@mmrc.iss.ac.cn
ABSTRACT
We consider the problem of optimizing a parametric linear function over a non-compact real trace of an algebraic setV. Our goal is to compute a representing polynomial which defines a hypersurface containing the graph of the optimal value function. Rostalski and Sturmfels showed that whenVis irreducible and smooth with a com- pact real trace, then the least degree representing polynomial is given by the defining polynomial of the irreducible hypersurface dual to the projective closure of theV.
First, we generalize this approach to non-compact situations. We prove that the graph of the opposite of the optimal value function is still contained in the affine cone over a dual variety similar to the one considered in compact case. In consequence, we present an algo- rithm for solving the considered parametric optimization problem for generic parameters’ values. For some special parameters’ values, the representing polynomials of the dual variety can be identically zero, which give no information on the optimal value. We design a dedi- cated algorithm that identifies those regions of the parameters’ space and computes for each of these regions a new polynomial defining the optimal value over the considered region.
Categories and Subject Descriptors
I.1.2 [Computing Methodologies]: Symbolic and Algebraic Ma- nipulation—Algorithms; G.1.6 [Numerical Analysis]: Global opti- mization
Keywords
Dual variety; polynomial optimization; recession pointed cone
1. INTRODUCTION
Parametric optimization problems widely arise in both theoretical problems and practical applications, like the maximum likelihood estimation and the model predictive control [5]. It is worthwhile to express the optimal value as an explicit or implicit function of the parameters in the region of interest.
∗Feng Guo is supported by the Fundamental Research Funds for the Central Universi- ties, the Chinese National Natural Science Foundation under grants 11401074. Lihong Zhi and Chu Wang are supported by NKBRPC 2011CB302400, the Chinese National Natural Science Foundation under grants 91118001. Mohab Safey El Din is supported by the GEOLMI grant (ANR 2011 BS03 011 06) of the French National Research Agency. The authors are supported by CNRS and INRIA through theECCAproject at LIAMA.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full cita- tion on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or re- publish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].
ISSAC’15,July 6–9, 2015, Bath, United Kingdom.
Copyrightc 2015 ACM 978-1-4503-3435-8/15/07 ...$15.00.
DOI: http://dx.doi.org/10.1145/2755996.2756666.
In this paper, we consider the problem of optimizing a parametric linear function over a real algebraic variety
c∗0:= sup
x∈Rn
cTx=c1x1+· · ·+cnxn
s.t. h1(x) =· · ·=hp(x) = 0,
(1)
whereh1, . . . , hp∈R[X1, . . . , Xn]are polynomials in the decision variables(X1, . . . , Xn)andc = (c1, . . . ,cn)denotes unspecified parameters.
The optimal valuec∗0can be regarded as a function of the parame- tersc, i.e. theoptimal value function. Our goal is to compute a poly- nomial that defines a hypersurface in the parameters’ space which contains the graph of this function.
Typically, thecylindrical algebraic decomposition(CAD) [8] can be applied to solve (1).
More precisely, by introducing the Boolean operators∧(and), we associate (1) with a Boolean expression
(h1(X) = 0)∧ · · · ∧(hp(X) = 0)∧(c0−cTX ≥0) (2) withX= (X1, . . . , Xn).
Indeed, recall that a CAD can be used to describe the projection of a semi-algebraic set (which is equivalent to eliminating one block of quantifiers).
By computing the CAD of the semi-algebraic set inR2n+1 de- fined by (2) with an ordering where theX-variables are larger than thec-variables, theprojection phaseprovides us a set of polynomi- als inR[c0,c, X], calledprojection level factors, which defines the boundaries ofcellsin the parameters’ spaceRn. However, the com- plexity of CAD algorithms is doubly exponential in the number of variables which limits its practical application to nontrivial problems involving4variables at most and this general approach may return numerous irrelevant polynomials.
In the last decade, several approaches have been developed to de- sign dedicated algebraic techniques for polynomial optimization (see [31,17,16,18,2,34] and references therein). In the non-parametric case, they allow to compute polynomials defining the optimum of a polynomial optimization problem whose degrees are singly expo- nential in the number of decision variables.
Here, our goal is to extend these techniques to the parametric case.
We denote byΦ ∈ Q[c0,c1, . . . ,cn]a polynomial defining a hy- persurface in the parameters’ space which contains the graph of the optimal value function.
The smallest possible degree forΦ in the variablec0 is called thealgebraic degreeof the optimization problem (1). This number measures the complexity of (1). Therefore, a lot of interest has been attracted on finding the polynomialΦand the algebraic degree [6, 15,20,24,25,27,30].
We denoteh = {h1, . . . , hp}the sequence of polynomials ap- pearing in (1) and let
V={v∈Cn|h1(v) =· · ·=hp(v) = 0}. (3)
We assume below thathgenerates a radical and equidimensional ideal. Theregular pointsofVare those points at which the rank of Jacobian matrix associated tohis the codimension ofV.
We letV∗ be the dual variety associated withV, which is the Zariski closure of the vectors in the projective space tangent to the projective closure ofVat its regular points. Its defining polynomials can be seen as polynomials with coefficients inQ[c0,c1, . . . ,cn].
Rostalski and Sturmfels in [30, Theorem 5.23] show that, whenVis irreducible, compact inRnand smooth, the optimal value functionΦ is represented by the defining polynomial ofV∗. Therefore, whenV is compact inRn, the defining polynomial ofV∗can fulfill our goal mentioned above. The compactness in the assumption is included to ensure that the optimumc∗0is well-defined and achieved which are essential in the proof of [30, Theorem 5.23].
However, whenV ∩Rn isnon-compact, the optimal valuec∗0 for some(γ1, . . . , γn) ∈ Rncould be infinite or finite but can not be attained, i.e. c∗0 is anasymptotic critical valueat infinity [21, 22, 26]. Hence, the proof of [30, Theorem 5.23] is not valid in this case. Another issue with the defining polynomials of V∗ is that they might vanish on a Zariski closed set of parameters’ values (γ1, . . . , γn)∈Rn. In other words, they give no information about the optimal values for these parameters’ values. We aim to explore the treatment of the above difficulties.
Main contributions.We consider the problem of optimizing a para- metric linear function over a non-compact real trace of an algebraic setV. SupposingVis smooth, we show that the graph of the oppo- site of the optimal value function is contained in the affine cone over a dual varietyV∗, i.e.(−c∗0 :γ1:· · ·:γn)∈ V∗whenever the opti- mal valuec∗0is bounded at(γ1, . . . , γn). We design an algorithm for solving the optimization problem (1) for generic parameters’ values.
It returns a set of two polynomials(Φ, Z)such that
• Φ∈Q[c0,c]andZ∈Q[c];
• for anyγ= (γ1, . . . , γn)6∈V(Z), if the associated optimum c∗0 of (1) is bounded, thenΦ(c0, γ)is not zero and its set of roots contains the optimumc∗0of (1).
IfVis irreducible, smooth and the closure of the convex hull ofV∩
Rncontains no lines, then similar to [30, Theorem 5.23], we show thatV∗ is an irreducible hypersurface and its defining polynomial represents the optimal value function of (1).
WhenVis not smooth but its real trace is compact, we construct recursively a finite number of dual varieties such that(−c∗0 :γ1 :
· · · : γn)lies in the union of these dual varieties. We design an algorithm which returns a finite sequence of(Φi, Zi)with the prop- erty that for anyγ= (γ1, . . . , γn)whose associated optimumc∗0is bounded, there exists ani, ifZi(γ)6= 0, thenc∗0is contained in the roots ofΦi(c0, γ).
It may happen that for some special parameters’ values, the poly- nomials obtained with the above approach are identically0. Then, they provide no information on the optimization problem when the parameters are instantiated to these values.
We design a parametric variant of [18] that solves this problem.
Under the assumption thatVis smooth and when the parameters are instantiated, the algorithm in [18] allows to obtain a polynomial of degree singly exponential in the number of decision variablesX whose set of roots contains the global optimum of the instantiated polynomial optimization problem.
We use the algebraic nature of the algorithm in [18] to design a parametric variant that returns a list of triples
(Φ1, Z1,P1), . . . ,(Φk, Zk,Pk) such that
• Φi ∈Q[c0,c],Zi∈ Q[c]andPi ⊂Q[c]generates a prime ideal for1≤i≤k;
• ∪ki=1V(Pi)is the whole parameters’ space andV(Pi)−V(Zi) is not empty for1≤i≤k;
• for anyγ = (γ1, . . . , γn) ∈ Rn such thatγ ∈ V(Pi)− V(Zi), the set of roots ofΦ(c0, γ1, . . . , γn)contains the global
optimum of the polynomial optimization problem (1) when c1, . . . ,cnare instantiated toγ1, . . . , γn.
This paper is organized as follows. In Section2, we recall some background in convex analysis, algebraic geometry and dual vari- eties needed in this paper. In Section3, we investigate the relation between the graph of the optimal value function and the dual variety V∗when the algebraic varietyVis not compact inRnor not smooth.
In Section4, we present the parametrized variant of [18].
2. PRELIMINARIES 2.1 Convex sets and cones
We first present some ingredients from convex analysis [28]. A non-empty subsetC⊆Rnis said to beconvexif(1−λ)x+λy∈C wheneverx∈ C,y ∈ Cand0< λ <1. We denotecl(C)and int(C)as theclosureandinteriorofC, respectively. Theaffine hull of a convex setC, denoted byaff(C), is the unique smallest affine set containingC. Therelative interiorof a convex setC ⊆ Rn, denoted byri(C), is defined as the interior ofCregarded as a subset ofaff(C). For an arbitrary setC⊆Rn, denoteco(C)as its convex hall.
Thepolarof a non-empty convex setC ⊆Rnis a closed convex set defined as
Co={x∈Rn| ∀y∈C,hx, yi ≤1}.
We haveCoo=cl(co(C∪ {0})).
A subsetK ⊆ Rnis called aconeif it is closed under positive scalar multiplication, i.e. λx ∈ K for allx ∈ K andλ > 0. A convex coneKispointedif it is closed andK∩ −K={0}. The polarof a non-empty convex coneKis defined as
Ko={x∈Rn| ∀y∈K,hx, yi ≤0}.
Therecession cone0+Cof a non-empty convex setCis the set including all vectorsy satisfyingx+λy ∈ C for everyλ > 0 andx∈ C. Importantly, a closed convex setC ⊆ Rnis bounded if and only if0+Cconsists of the zero vector alone. A closed and unbounded convex set C containsno lines if and only if0+C is pointed.
Letfbe a function whose domain is a subsetS⊆Rnand values are real or±∞. Theepigraphoffis defined as
epi(f) ={(x, µ)∈Rn+1|x∈S, µ∈R, µ≥f(x)}.
We say thatf is aconvex functionon S ifepi(f)is convex as a subset ofRn+1. The effective domain of a convex functionfonS is the projection ofepi(f)onRn:
dom(f) ={x∈Rn| ∃µ∈R s.t.(x, µ)∈epi(f)}
={x∈Rn|f(x)<+∞}.
THEOREM 2.1. [19, Theorem 1.2]LetC⊆Rnbe a closed and unbounded convex set, then
1. 0+Co
is ann-dimensional convex set;
2. int 0+Co
⊆ dom(c∗0(c | C)) ⊆ 0+Co
. Moreover, we have f(x) = aTxattains its supremum onCfor every a∈int 0+Co
.
2.2 Dual varieties
DenoteX = (X1, . . . , Xn). For any ideal (homogeneous ideal) I inR[X](R[X0, X]), denote V(I) as the affine (projective) va- riety defined byI inCn (Pn(C)). Now let us review some back- ground about dual varieties inPn(C) [29, 30]. In the following, we abbreviatePn(C)asPnfor convenience. LetI =hf1, . . . , fpi be a homogeneous radical ideal in the polynomial ringR[X0, X]
and V = V(I) ⊆ Pn. The singular locussing(V) is defined by the vanishing of thec×cminors of the p×(n+ 1)Jaco- bian matrixJac(I) = (∂fi/∂Xj), where c = codim(V). Let Vreg =V\sing(V)denote the set of regular points inV. The pro- jective varietyV is smooth ifV =Vreg.
A pointu = (u0 : u1 : · · · : un)in the dual projective space (Pn)∗ represents the hyperplane {x ∈ Pn | Pn
i=0uixi = 0}.
We say that u istangent to V at a regular pointx ∈ Vreg if x lies in the hyperplanePn
i=0uixi = 0and its representing vector (u0, u1, . . . , un)lies in the row space of the Jacobian matrixJac(I) at the pointx. Theconormal varietyCN(V)is the closure of the set
{(x, u)∈Pn×(Pn)∗|x∈Vreganduis tangent toV atx}.
Thedual varietyV∗ is the projection ofCN(V) onto the second factor. More precisely, the dual variety is the closure of the set
{u∈(Pn)∗| uis tangent toV at some regular point}.
2.3 Generalized critical values
For a vectorv ∈ Rn,kvkdenotes the standard Euclidean norm ofv. LetV be a smooth affine variety, and letf : V → Rbe a polynomial dominant mapping. DenoteK0(f, V)as the critical values off onV. The set ofasymptotic critical valuesat infinity [21,22,26] offonV is defined as
K∞(f, V) = (
y∈R ∃x(k)∈V,s.t.x(k)→ ∞, f(x(k))→y, kx(k)kν(dx(k)f)→0
) ,
wheredx(k)fstands for the differential off evaluated atx(k)and νstands for the distance ofdx(k)fto the space of degenerate linear maps on the tangent space toV atx(k). The set of generalized critical values offis defined as
K(f, V) =K0(f, V)∪K∞(f, V).
It has been shown in [22, Theorem 3.1] and [21, Theorem 3.3, Corol- lary 4.1] thatK(f, V)is a finite set.
THEOREM 2.2. [21,26]Iffis bounded above, i.e.
f∗= supx∈Vf(x)is finite, thenf∗∈K(f, V).
3. DUALITY IN NON-COMPACT CASE
For the algebraic varietyVdefined in (3), let Ch=cl(co(V ∩Rn)),
i.e. the closure of the convex hull ofV ∩Rn. Then, the problem (1) is equivalent to
c∗0 = sup cTx s.t.x∈Ch. (4) LetV∗be the dual variety to the projective closure ofV. WhenCh
contains no lines, i.e. 0+Ch is pointed, the relation between the optimal value function of (1) and the defining polynomial ofV∗is investigated in [19]. Now we prove the correctness of some results therein without the assumption of pointedness. It will yield an algo- rithm for solving parametric optimization problem (1) with generic parameters.
3.1 Smooth Case
In this subsection, we assume the algebraic variety V in (3) is smooth. Recall thatdom(c∗0(c|Ch))denotes the collection of the parameters’ valuesγ ∈ Rnsuch that the supremum ofγTxonCh
is finite. We generalize Rostalski and Sturmfels’ result [30, Theorem 5.23] to the non-compact case as follows.
THEOREM 3.1. Suppose thatVin(3)is smooth, then
(−c∗0 :γ1:· · ·:γn)∈ V∗ (5) for everyγ∈dom(c∗0(c|Ch)).
PROOF. Fix aγ ∈ dom(c∗0(c | Ch)). By the definition, the supremumc∗0 off(X) = γTX onV ∩Rnis finite. For the case whenc∗0 is a critical value which can be attained, see the proof of [30, Theorem 5.23]. Now by Theorem2.2, we suppose thatc∗0 is an asymptotic critical value off overV ∩Rn. Then, there exists a sequence{x(k)} ⊆ V ∩Rnsuch thatkx(k)k → ∞,f(x(k))→c∗0 andkx(k)kν(dx(k)f)→0. By [36, Lemma 2.1], for eachx(k), we
can find a vectorγ(k)in the normal space ofVat{x(k)}such that kγ(k)−γk = ν(dx(k)f). Then,kx(k)kkγ(k)−γk → 0, which implieskγ(k)−γk →0and(γ(k))Tx(k) →c∗0. It can be checked that
(−(γ(k))Tx(k):γ1(k):· · ·:γn(k))∈ V∗. SinceV∗is closed, we have(−c∗0 :γ1:· · ·:γn)∈ V∗.
COROLLARY 3.2. IfVis irreducible, smooth andChcontains no lines, thenV∗is an irreducible hypersurface and its defining poly- nomial represents the optimal value function of(1).
PROOF. SinceChcontains no lines,0+(Ch)is pointed. Accord- ing to Theorem2.1, (0+Ch)o is a n-dimensional convex set and int (0+Ch)o
is contained indom(c∗0(c | Ch)). Therefore, the affine cone of the Zariski closure of
(−c∗0 :γ1 :· · ·:γn)∈(Pn)∗|γ∈int (0+Ch)o (6) has dimension≥n. By [9, Theorem 12 (i), §3, Chpt. 9], the Zariski closure of (6) is of dimension≥n−1. By Theorem3.1and [30, Proposition 5.10], we havedim(V∗) =n−1. AsVis irreducible, V∗is an irreducible hypersurface [13, Proposition 1.3], and coincides with the Zariski closure of (6) according to [9, Proposition 10 (ii), §4, Chpt. 9]. Then, the conclusion follows.
In the sequel, we say that a property depending on some indeter- minates isgenericif there exists a non-empty Zariski open subset of the space endowed by these indeterminates over which the property holds (we will also say that the property holds for generic values of these indeterminates).
An algorithm can be derived from Theorem3.1for solving the parametric optimization (1) for generic parameter as described be- low. Denoteµ= (µ1, . . . , µp). LetJ⊆Q[c0,c, µ, X]be the ideal generated by
cTX−c0, h1, . . . , hp,ci−
p
X
j=1
µj
∂hj
∂Xi
, i= 1, . . . , n.
Sincehgenerates a radical ideal, for any(c∗0, γ,µ,¯ x)¯ ∈V(J),c∗0is a critical value of the functionγTXonVat a critical pointx.¯
ALGORITHM 3.1. GenericParametricOptimization(h) Input:h1, . . . , hp∈Q[X]which generate a radical ideal Output:(Φ, Z)such that
• Φ∈Q[c0,c]andZ∈Q[c]
• For anyγ∈dom(c∗0(c|Ch))such thatZ(γ)6= 0,Φ(c0, γ) is not zero and its set of roots contains the optimumc∗0of(1).
Step 1Compute the reduced Gröbner basisGofJ∩Q[c0,c]with block lex orderXµcc0.
Step 2SetΓto be the set of polynomials inGcontaining the variable c0.
Step 3SetΦto be the polynomial inΓwith the lowest degree inc0. Step 4SetZto be the sum of squares of all coefficients ofΦin view
ofQ[c1, . . . ,cn][c0].
THEOREM 3.3. In Algorithm3.1, we haveΓ6=∅,V∗=V(Γ) and the algorithm is correct.
PROOF. By the definition of dual varieties and Theorem3.1, it suffices to show thatΓ6=∅. Letπn+1(V(J))be the projection of points inV(J)on their firstn+ 1coordinates, thenGis the corre- sponding elimination ideal. By the Closure Theorem [9],πn+1(V(J))
⊆V(G)and there exists a subvarietyW $V(G)such thatV(G)\W
⊆ πn+1(V(J)). Fix a point (c∗0, γ) ∈ V(G). Suppose to the contrary thatΓ = ∅, thenC×γ ⊆ V(G). By Sard’s Theorem, C×γ ∩πn+1(V(J))is an empty set or a finite set. Therefore, C×γ⊆V(G)\πn+1(V(J))⊆Wexcept for at most finitely many points inC×γ. SinceW is closed, C×γ ⊆ W. In particular, (c∗0, γ)∈Wwhich meansV(G) =W, a contradiction.
REMARK 3.1. As proved in Corollary3.2, whenVis irreducible, smooth andChcontains no lines, there is only one polynomial in the setΓin Algorithm3.1. IfChcontains lines,Γmight consist of more than one polynomial andV∗may not be the Zariski closure of the set (6), see Example3.1.
Similar to [31, Theorem 6], with Algorithm3.1and procedures of deciding the emptiness of real algebraic varieties, we can determine whether a genericγ(γ6∈V(Z)) belongs todom(c∗0(c|Ch))and the associated optimumc∗0if it does.
EXAMPLE 3.1. Consider the algebraic varietyVdefined by h(X1, X2) =X12X2−1
which is irreducible, smooth, non-compact inR2andChcontains lines. Letγ= (0,−1), then clearlyc∗0= 0. Running Algorithm3.1, we getΓ ={4c30+ 27c21c2}and hence
Φ = 4c30+ 27c21c2, V(Z) =∅.
Hence,V∗=V(Γ)⊆P2and(0 : 0 :−1)∈ V∗.
Sincedom(c∗0(c | Ch)) ={γ ∈ R2 | γ1 = 0, γ2 < 0}and c∗0= 0for anyγ∈dom(c∗0(c|Ch)), the Zariski closure of
{(−c∗0:γ1:γ2)∈(P2)∗|γ∈dom(c∗0(c|Ch))} (7) is{(0 : 0 :γ2) ∈(P2)∗|γ2∈C}which is of dimension0. Since we havedimV∗= 1,V∗is not the Zariski closure of the set (7).
3.2 Singular case
Now we suppose thatVis irreducible, compact inRnbut isnot smooth. We point out that the inclusion (5) might not hold in this case.
EXAMPLE 3.2. Consider the astroid which is a real locus of a plane algebraic curveVdefined by
h(X1, X2) = X12+X22−13
+ 27X12X22.
It is obvious that for any linear function onV ∩R2, its optimizer is one of the four singular points{(±1,0),(0,±1)}. We haveV∗ = V(Γ)where
Γ ={−c21c22+c20c21+c22c20}.
For a givenγ∈R2, we havec∗0 = max{|γ1|,|γ2|}>0. It is easy to check(−c∗0 :γ1 :γ2)6∈ V∗whenγ1 6= 0orγ26= 0, i.e. (5) does not hold.
Next we recursively construct a finite number of dual varieties such that (5) holds for the union of these varieties. For similar treat- ment, see [35]. The following algorithm has the same input as Algo- rithm3.1and returns a finite sequence of(Φk, Zk)with the property that for anyγ ∈ dom(c∗0(c | Ch)), there exists ak, such that if Zk(γ)6= 0, thenc∗0is contained in the roots ofΦk(c0, γ).
ALGORITHM 3.2. SingularParametricOptimization(h) Step 1Letk= 1andVk=V.
Step 2Compute an equidimensional decompositionVk = ∪iVk,i
withVk,i=V(Ik,i)and eachIk,iis a radical ideal.
Step 3RunGenericParametricOptimization(Ik,i)for eachiand set (V(k))∗=∪iVk,i∗.
Step 4Compute the setΓk⊆Q[c0,c]such thatV(Γk) = (V(k))∗. Step 5SetΦkto be the polynomial inΓkwith the lowest degree in
c0.
Step 6SetZkto be the sum of squares of all coefficients inΦkin view ofQ[c1, . . . ,cn][c0].
Step 7Compute the singular locusVek,iof eachVk,iand setVk+1=
∪iVek,i. IfVk+16=∅, then letk=k+ 1and go to Step 2.
The next theorem shows the correctness of Algorithm3.2.
THEOREM 3.4. The algorithm terminates in a finite number of steps and for everyγ∈Rn, we have
(−c∗0:γ1:· · ·:γn)⊆ ∪lk=1(V(k))∗.
PROOF. Since Vek,iis the singular locus of Vk,i, dim(Vek,i) <
dim(Vk,i)and then the algorithm terminates in a finite number of steps. SinceVis compact inRn, for every parameterγ, the optimum c∗0is finite and attainable. If the optimizerx∗is a smooth point, by Theorem3.1,(−c∗0 :γ1 :. . .:γn)∈ V∗. Ifx∗is a singular point ofV. Then there existk andisuch thatx∗is regular inVk,iand (−c0:γ1:. . .:γn)∈Vk,i∗ .
EXAMPLE 3.2(CONTINUED) The singular locus ofVis defined by h, X15−X1, X13X2+X1X2,3X14−X12+ 2X22−2 ,
and has four real points{(±1,0),(0,±1)}. Running Algorithm3.2, we getΓ1consisting of
(c0−c2)(c0+c2)(c0−c1)(c0+c1)(c20+c21−2c1c2+c22) (c20+c21+ 2c1c2+c22).
By the discussion in Example3.2, it is easy to check that(−c∗0:γ1: γ2)∈V(Γ1) = (V(1))∗for everyγ∈R2.
3.3 “Bad” parameters
It is clear that the polynomialΦin Algorithm3.1gives no infor- mation about the optimal value of (1) with parameters belonging to V(Z). In particular, it might happen for the problem (1) reformu- lated from a general polynomial optimization problem by introduc- ing a new variable.
Consider the polynomial optimization problem f∗:= max
x∈Rn
f(x) s.t. h1(x) =· · ·=hp(x) = 0, wheref ∈R[X]. Iffis bounded from above onV ∩Rn, then we havef∗∈K(f,V ∩Rn). Let
Vh,f ={(x, xn+1)∈Cn+1|x∈ V, xn+1−f(x) = 0}.
By Theorem3.1, we have(−f∗: 0 :· · ·: 0 : 1)∈ Vh,f∗ .
EXAMPLE 3.3. [22, Example 2.1] Letf = (X1 +X12X2 + X14X2X3)2. Running Algorithm3.1forVh,f withp = 0, we get Γ ={Φ}where
Φ =1073741824c120 c42c24+ 268435456c110 c21c42c4−
+ 9865003008c100 c42c3c34+· · ·+ 520093696c90c1c32c23c34. We haveΦ(c0,0,0,0,1)≡0which gives no information aboutf∗. In fact, we haveV(Z) =V(c2c4,c3c4,c3c2c1). Hence, forc2= 0,c3 = 0, we always haveΦ(c0,c1,0,0,c4) ≡ 0, i.e. c0can be arbitrary values.
In next section, we aim to design a complete algorithm to solve this problem.
4. COMPLETE ALGORITHM 4.1 Overview
As above, leth = (h1, . . . , hp) ⊂ Q[X1, . . . , Xn]that gener- ates a radical and equidimensional ideal andVbe the algebraic set defined byh1 = · · · =hp = 0; we assume thatVis smooth and denote byrits codimension.
It might happen that for some “bad” parameters’ valuesγ= (γ1, . . . , γn), the defining polynomials of the dual varietyV∗ become identically zero. For such values, this gives no information about the optimumc∗0of the mapx→γTxonV ∩Rn. For instance, in Example3.3, the polynomialΦis a zero polynomial for anyγ = (γ1,0,0, γ4),γ1, γ4 ∈R. In this section, we describe an algorithm that allows to avoid this problem. It can be seen as a parametric version of [18] that provides a complete algorithm for polynomial optimization.
Our algorithm starts by computing a couple of the form(Φ, Z,P) whereΦ∈Q[c0,c],P⊂Q[c]andZ∈Q[c]−hPisuch that for any (γ1, . . . , γn)∈V(P)−V(Z),Φ(c0, γ1, . . . , γn)is not identically 0and the optimum of the restriction of the mapx→γTxtoV∩Rnis a root ofΦ(c0, γ1, . . . , γn). Next, the algorithm is called recursively to study the parametric optimization problem under each constraint Piwhich is a prime component of p
hPi+hZi. Hence, we are led to run our algorithm over an integral domainQ[c1, . . . ,cn]/P whereP ⊂ Q[c1, . . . ,cn]is a prime ideal. Since the domain on which the computations are performed is integral, all operations that we need to manipulate polynomial ideals are available; the only dif- ference is that we need to compute pseudo-inverse of polynomials moduloP, hence simulating computations over the fraction field of Q[c1, . . . ,cn]/P.
The routine that handles these computations over these integral rings is calledBasicParametricOptimization. It is a parametric vari- ant of AlgorithmSetContainingLocalExtremain [18, Section 3].
One of its advantages is that in the fraction field of the integral do- mainsQ[c1, . . . ,cn]/P, it performs a number of operations that is singly exponential inn(see [18, Section 6, Lemma 6.8]) to be com- pared with the doubly exponential complexity innthat is needed by Cylindrical Algebraic Decomposition.
Note that one can also use lazy representations of ideals and dy- namic evaluation techniques (see e.g. [10,23]) to work with these parameters as well as comprehensive Gröbner bases or comprehen- sive triangular sets (see e.g. [7, 37] and references therein). The description below is done assuming that our domain is integral for simplicity; this allows us to focus more on objects and properties related to polynomial optimization and introduced in [18].
Before describing in detail the recursive procedure that is sketched above, let us describe the objects and subroutines that we need and which are extracted from [18].
4.2 Basic objects and properties
We start with polar varieties (see e.g. [1,3] and references therein) and their Noether position properties (see [32]). LetP⊂Q[c]be a finite polynomial sequence generating a prime idealPand letA= Q[c]/P. HenceAis an integral ring.
Forγ∈V(P), we consider the canonical projectionsπi: (x1, . . . , xn)→(x1, . . . , xi)for1≤i≤nand the following projections
πγ:x= (x1, . . . , xn)→γTx=γ1x1+· · ·+γnxn
and for1≤i≤dim(V) =n−r,
πγ,i:x= (x1, . . . , xn)→(γTx, x1, x2, . . . , xi).
Forϑ ∈ C, we denote byVγ,ϑthe algebraic set defined by V ∩ π−1γ (ϑ). We consider
• the set of all (r+ 1)-minors of the truncated Jacobian ma- trixJac([hi1, . . . , hir, γTX],X>i) (columns corresponding to partial derivatives w.r.tX1, . . . , Xiare omitted) for all sub- sets{i1, . . . , ir} ⊂ {1, . . . , p}for1≤i≤n−r−1; we de- note it byM(h, γ, i). For convenience, letM(h, γ, n−r) =∅.
When the entries ofγare parametersc = (c1, . . . ,cn), the set of minors is denoted byM(h,c, i).
• the set of all(r+ 1)-minors of the Jacobian matrix Jac([hi1, . . . , hir, γTX])for all subsets
{i1, . . . , ir} ⊂ {1, . . . , p}; we denote it byS(h, γ).
When the entries ofγare parametersc = (c1, . . . ,cn), the set of minors is denoted byS(h,c).
Letγ∈Cn− {0}andϑ∈C. Assume thatVγ,ϑis smooth and that the idealhh, γTX −ϑi is radical and equidimensional. The polar varietyW(h, γ, ϑ, i)associated toVγ,ϑandπγ,iis the critical locus of the restriction toVγ,ϑofπi. It is defined by the vanishing of the polynomials inhandM(h, γ, i)and the polynomialγTX−ϑ.
We will denote byW(h, γ, i)the algebraic set defined by the van- ishing of the polynomials inhandM(h, γ, i), for1≤i≤n−r.
The polar varietyC(h, γ)associated toVandπγ is the critical locus of the restriction toVofπγ. It is defined by the vanishing of the polynomials inhandS(h, γ).
In the sequel, we will use some properties of polar varieties that hold under generic changes of coordinates.
ForA∈GLn(C)andS⊂Cn, we denote bySAthe image ofS by the mapx→A−1x.
LetA ∈ GLn(Q), we are interested in the parameters’ values γ∈V(P)such that there exists a non-empty Zariski open setO ⊂ Csuch that for anyϑ∈ O, the following holds:
P1:(h, γTX−ϑ)is radical and equidimensional andVγ,ϑis smooth; note that by the Jacobian criterion, this is equivalent to saying that at any point ofVγ,ϑ, the rank of(h, γTX−ϑ) isr+ 1.
P2(A): for1≤i≤n−r, the polar varietyW(hA, γ, ϑ, i) is in Noether position with respect to the projectionπi−1. For those parameters’ values for whichP1 and P2(A) hold, we simply say thatP(A)holds.
We recall now the statement of [18, Proposition 4.2]. It empha- sizes the interest of these properties for polynomial optimization.
PROPOSITION 4.1. [18, Proposition 4.2] LetA∈GLn(C)and letγ∈Cn− {0}such thatP(A)holds. Then the following holds:
• the algebraic setCγAdefined as the Zarsiki closure of
∪n−ri=1
W(hA, γ, i)−C(hA, γ)
∩πi−1−1(0)
has dimension at most1;
• the union ofπγ(C(hA, γ))and the set of non-properness of the restriction ofπγtoCAγ is finite and contains the extremum of the restriction of the mapπγtoV ∩Rn.
From [18, Proposition 4.3], for anyγ∈V(P)there exists a non- empty Zariski open setA ⊂GLn(C)such that for anyA∈A ∩ GLn(Q),P(A)holds.
However, note that in order to use Proposition4.1for parametric optimization, we need to prove a stronger statement: there exists a non-empty Zariski open subsetA ⊂GLn(C)such that the follow- ing holds. For anyA∈A ∩GLn(Q), there exists a Zariski dense subsetU ⊂V(P)such that forγ∈U,P(A)holds.
Basically, our algorithmBasicParametricOptimizationidentifies a polynomialZ such thatV(P)−V(Z)is non-empty and com- putes a polynomialΦ(c0,c)such that for anyγ∈V(P)−V(Z), Φ(c0, γ)defines the union of the finite algebraic sets in the second item of Proposition4.1.
This is what we prove below but before doing that we introduce the data-structure and subroutines used by our algorithm.
4.3 Data-structures and subroutines
From now on,P⊂Q[c]is a polynomial sequence that generates a prime idealP. We denote byAthe integral ringQ[c]/P, byKthe fraction field ofAand byK¯the algebraic closure ofK.
Data-structures.LetF⊂Q[X1, . . . , Xn]that defines a finite alge- braic setV inCn. Then,V can be encoded with a zero-dimensional rational parametrization which is a sequence of polynomialsQ = (q, q0, q1, . . . , qn)⊂Q[U], i.e.V is defined by
q(U) = 0, Xi=qi(U)/q0(U), q0(U)6= 0 for1≤i≤n, withgcd(q, q0) = 1andqis unitary and its degree is the cardinality ofV.
When F defines an algebraic curve V ⊂ Cn, then a rational parametrization forV is a sequence of polynomialsQ= (q, q0, q1, . . . , qn) ⊂Q[U, T]such thatV is the algebraic closure of the set defined by
q(U, T) = 0, Xi= qi(U, T)
q0(U, T), q0(U, T)6= 0 for1≤i≤n
withqunitary inU andT, its degree is the degree of the algebraic curveV andgcd(q, q0) = 1.
Below, we will also consider polynomial systems in the ringA[X1, . . . , Xn]whereAis an integral ring. We denote byKthe fraction field ofAand byK¯ the algebraic closure ofK. Hence, the algebraic sets defined by these polynomial systems lie inK¯n; whenever they define finite algebraic sets or algebraic curves, they can be encoded with rational parametrizations with coefficients inK, which up to normalization can be turned into rational parameterizations with co- efficients inA.
Basic routines.We need to introduce the following routines.
The routineSingularMinorstakes as inputhandPand it returns G˜ = (h,S(h,c)).
The routineSpecialCurvetakes as inputhandPand returnsF˜= ( ˜F1, . . . ,F˜n−r)such that for1≤i≤n−r,F˜iish,M(h,c, i), X1, . . . , Xi−1.
These systems will allow us to compute parametrized representa- tions of the setsCγforγlying in a Zariski dense subset.
The routinePointsPerComponentstakes as inputh∈Q[X1, . . . , Xn]and it returns a zero-dimensional rational parametrization that encodes a finite set of points contained inV=V(h)and meet- ing all the connected components ofV ∩Rn.
The routineValuesTakenByPolytakes as input a zero-dimensional rational parametrizationQ⊂Q[U]that encodes a finite set of points V inCn, the sequence of polynomialsP⊂ Q[c]. It returnsΦ ⊂ Q[c0,c]and a polynomial Z ∈ Q[c]− hPi such that for γ ∈ V(P)−V(Z),Φ(c0, γ)defines the set{γTx|x∈V}. It essen- tially consists of substituting the parametrization in the polynomial cTX−c0, clearing the denominators and eliminating the variableU with a resultant computation to getΦ. Note that these computations are done moduloP(hence inA). Keeping track of exact divisions performed during the resultant computation needed to do this com- putation (or using specialization theorems, see e.g. [12]) yields the polynomialZ. Note thatZ does not belong toP(else we wouldn’t use its factors for performing divisions).
As above, the routineParametricValuesTakenByPolytakes as in- put a zero-dimensional rational parametrizationQbut with coeffi- cients inA, the sequence of polynomialsP⊂Q[c]. The parametriza- tionQencodes a finite set of pointsV inK¯n. It returnsΦ⊂Q[c0,c]
and a polynomialZ∈Q[c]− hPisuch that forγ∈V(P)−V(Z), Φ(c0, γ)defines the set{γTx|x ∈V}. AsValuesTakenByPoly does, this routine works using substitutions and resultant computa- tions.
The following lemma is immediate.
LEMMA 4.2. LetQandPbe as above and(Φ, Z)be the output ofParametricValuesTakenByPoly(Q,P). Then,Z /∈ hPi.
Properness. We describe now a routine CheckProperness that takes as inputh, a matrixA∈GLn(Q)andP⊂Q[c]as above.
When there are no generic parameters’ values inV(P)for which P(A)holds, the routineCheckPropernesssimply returns(0). Else it returnsZ ∈ Q[c]− hPisuch that for anyγ ∈ V(P)−V(Z), propertyP(A)holds.
Roughly speaking, the above routine identifies those parameters’
valuesγfor whichP(A)holds.
LEMMA 4.3. We use the above notation and assumptions. Then, there exists a non-empty Zariski open setA ⊂GLn(C)such that for anyA∈A ∩GLn(Q)the following holds.
LetZbe the output ofCheckProperness(h,A,P). Then,V(P)−
V(Z)is Zariski dense inV(P)and forγ∈V(P)−V(Z), prop- ertyP(A)holds.
PROOF. Note that by construction, under our assumptions,Z /∈ hPi. Hence, sincehPiis prime,V(P)−V(Z)is Zariski dense in V(P).
It remains to prove that there exists a non-empty Zariski open set A ⊂GLn(C)such that for anyA∈A∩GLn(Q)andγ∈V(P)−
V(Z), propertyP(A)holds.
By [18, Proposition 4.3], for anyγ ∈ V(P), there exists a non- empty Zariski open setA ∈GLn(C)such thatP(A)holds forγ.
We prove below that there existsZ /∈ hPisuch that for anyA∈A andγ0∈V(P)−V(Z),P(A)holds forγ0.
Consider a minimal Gröbner basisG of the ideal generated by hh,cTX−c0iand all(r+ 1)-minors ofJac(h,cTX−c0)with K(c0)as a ground field. We claim thatGis(1). Indeed, if it was not the case, this would imply that for anyγ ∈ V(P)which does not cancel the finitely many denominators that appear in a computa- tion ofG,P1does not hold; hence a contradiction. We deduce that Gis(1)as claimed and letZ0 be the product of all denominators appearing during the computation ofG.
Now letAbe ann×nmatrix with entriesAi,jas indeterminates.
By [33], one can ensure Noether position properties by setting the non-vanishing of some denominators of a minimal reduced Gröbner basis of the ideals generated by
hA,M(hA,c, i),cTAX−c0
withK(Ai,j)as a ground field. The coefficients of these denomina- tors lie inKwhich containsQ. Now, we define the Zariski open set A ⊂GLn(C)by the non-vanishing of the coefficients of the mono- mials inc,c0. This set is non-empty because, as above, it would contradict that for anyγ∈V(P),P(A)holds forAgeneric.
Now, remark that forA∈A, one can defineZ00as the denomina- tors that appear in the computation of the minimal reduced Gröbner basis of the ideals generated by
hA,M(hA,c, i),cTAX−c0
withKas a ground field.
TakingZ=Z0Z00ends the proof.
Rational parametrizations. Consider a sequence of polynomials polynomialsF= (f1, . . . , fs)andG= (g1, . . . , gk)inA[X1, . . . , Xn]andI ⊂K[X1, . . . , Xn]be the saturation ofhFibyhGi; we denote it byhFi:hGi∞. We assume thatIhas dimension1and is equidimensional.
We are interested in studying the complex solutions ofFthat are not solutions ofGwhere the admissible values for the parameters clie in the irreducible algebraic set associated toP. We consider a routineParametricCurveRepresentationthat takes as inputF, GandPand which returns a finite sequence of polynomialsQ= (q, q0, q1, . . . , qn)⊂K[U, T]and a polynomialZinQ[c]− Psuch that for anyγ= (γ1, . . . , γn)∈V(P)−V(Z)the curve associated tohFγi:hGi∞is the Zariski closure of the set defined by
qγ(U, T) = 0, Xi=qi,γ(U, T)/q0,γ(U, T), q0,γ(U, T)6= 0 for1 ≤ i ≤ n(where qγ andqi,γ denote the polynomials ofQ obtained by instantiatingctoγinqandqi,γfori∈ {0, . . . , n}).
LEMMA 4.4. LetFandGbe as above and let(Q, Z)be the out- put ofParametricCurveRepresentation(F,G,P). Then the Krull dimension ofhPi+hZiis less than the Krull dimension ofhPi.
PROOF. Without loss of generality, one can assume that we are in generic coordinates. Using Gröbner bases withKas ground field and linear algebra inK(X1)[X2, . . . , Xn]one can compute a ratio- nal parametrization ofI(see e.g. [11,4]). During this computation, some polynomials (which are not0moduloPby construction) are used to perform divisions. TakingZas the product of these polyno- mials is a valid output and since these polynomials are not0modulo P,Zis not. SincePis prime, we deduce thatP+hZihas dimension less than the dimension ofP.
Reusing the above notations and ParametricCurveRepresen- tation, it is straightforward to obtain a routineUnionParametric- Curvethat takes as input a sequence of sequences of polynomials F = (F1, . . . ,Fl) a sequence of polynomialsGandPsuch that the idealhFki : hGi∞ ⊂ K[X1, . . . , Xn]has dimension1and is equidimensional for1≤k≤l. It returns a finite sequence of poly- nomialsQ= (q, q0, q1, . . . , qn)⊂K[U, T]and a polynomialZin Q[c]− Psuch that for anyγ= (γ1, . . . , γn)∈V(P)−V(Z)the curve associated toTl
k=1hFk γi:hGi∞is the Zariski closure of the set defined by
qγ(U, T) = 0, Xi=qi,γ(U, T)/q0,γ(U, T) q0,γ(U, T)6= 0
for1 ≤ i ≤ n(whereqγandqi,γ denotes the polynomials ofQ obtained by instantiatingctoγinqandqi,γ fori ∈ {0, . . . , n}).
The following lemma is an immediate consequence of Lemma4.4.
LEMMA 4.5. LetFandGbe as above and let(Q, Z)be the out- put ofUnionCurveParametric(F,G,P). Then the Krull dimension ofhPi+hZiis less than the Krull dimension ofhPi.
Intersection of a curve with a variety. We describe now the rou- tineParametricIntersectionwhich takes as input a one-dimensional rational parametrizationQof a curveC ⊂ K¯n, a polynomial se- quenceG ∈ A[X1, . . . , Xn]andP. The sequenceGdefines an algebraic setH inK¯n. Assume that the intersection ofC andH is finite. Then, following [14] one can compute a parametric zero- dimensional rational parametrizationQ0that encodesC∩H. This is done by substituting inGthe parametrizations of theXi’s hence re- ducing the computation to computing the intersection defined by the vanishing of two bivariate polynomials with coefficients inK(using resultant computations). Again, keeping track of the denominators appearing during the computation or using specialization theorems, one can finally return a parametric zero-dimensional parametrization Qand a polynomialZ /∈ hPisuch that for anyγ∈V(P)−V(Z), Q0γencodesCγ∩Hγ.
LEMMA 4.6. LetQ,GandPas above and(Φ, Z)be the out- put ofParametricIntersection(Q, f,P). Then,dim(hPi+hZi)<
dim(P).
Computing a set of non-properness. LetQbe a one dimensional rational parametrization with coefficients inA; it defines an algebraic curveC1 ⊂ K¯n. Then, there exists a Zariski dense subsetU of V(P)such that forγ∈U,V(Qγ)defines an algebraic curveCγ.
The routine ParametricSetOfNonProperness computes(Φ, Z) such thatV(P)−V(Z) ⊂U and is non-empty and such that for γ ∈V(P)−V(Z),Φ(c0, γ)is not0and its set of roots contains the set of non-properness of the restriction of the mapx→γTxto Cγ.
We denote byC2the algebraic curve{(γ0, x)|x∈C1and γ0 =cTx}. We also denote byC1the projective closure ofC1in Pn( ¯K). Forx = (x0 : x1 : · · · : xn) ∈ C1 withx0 6= 0, we denote byx˜the point
x1 x0, . . . ,xxn
0
and byC2the quasi-projective set{(γ0, x)|x∈C1andcTx˜=γ0}.
Following algorithm given in [9] our routine reduces to the fol- lowing steps:
• compute a representation of the projective closureC2; this can be done using Gröbner bases withKas a ground field;
• compute the intersection ofC2with the hyperplane at infinity defined byX0 = 0; this is a finite set of points and again it can be done using Gröbner bases withKas a ground field.
Keeping track of all denominators that appear during the computa- tions yields the polynomialZ /∈ hPias above.
The following lemma is immediate.
LEMMA 4.7. LetQandPbe as above and let(Φ, Z)be the out- put ofParametricSetOfNonProperness(Q,P). Thendim(hPi+ hZi)<dim(P).
4.4 Basic routine for parametric optimization
We describe now our basic subroutineBasicParametricOptimiza- tion. It can be seen as a parametrized version of AlgorithmSetCon- tainingLocalExtremain [18].
This latter algorithm consists in reducing the problem of comput- ing the optimum of a polynomial function restricted to a real alge- braic setV ∩Rnto the problem of computing the optimum of the same polynomial function restricted to a curve. Obviously, this is done in such a way that both optimization problems share the same optimum.
We describe the main steps and refer to the steps of algorithm BasicParametricOptimizationcorresponding to their parametric vari- ants. The algorithm starts by computing sample points inV ∩Rnand
gets(i)the values attained by the polynomial function to optimize at those points (this corresponds to Steps1-2). Next, it computes rep- resentations of linear sections of polar varieties that define algebraic curves (Step5-6). Finally, it computes(ii)the set of non-properness of the restriction of the considered function to the curve (Step7) and gets(iii)the critical values of this function restricted to the curve (Step8-9). All this is done in such a way that the optimum lies in the set of values(i),(ii)and(iii).
Input:h= (h1, . . . , hp)⊂Q[X1, . . . , Xn]andP⊂Q[c]
Properties:Pgenerates a prime ideal andhh1, . . . , hpigenerates a radical equidimensional ideal defining a smooth algebraic set.
Output:(Φ, Z)such that
• Φ∈Q[c0,c]andZ∈Q[c];
• Zis not0modulohPi;
• For anyγ ∈ V(P)−V(Z)such thatZ(γ) 6= 0,Φ(c0, γ) is not zero and its set of roots contains the optimum of the restriction ofπγtoV ∩Rn.
BasicParametricOptimization(h,P) 1. R=PointsPerComponents(h)
2. (Φ0, Z0) =ValuesTakenByPoly(R,cTX,P) 3. Choose randomlyA∈GLn(C)
4. Z00 =CheckProperness(h,A,P)
5. F˜=SpecialCurve(hA,P),G˜ =SingularMinors(hA,P) 6. (R1, Z1) =UnionParametricCurve( ˜F,G,˜ P)
7. (Φ1, Z10) =ParametricSetofNonProperness(R1,P,cTX) 8. R2=ParametricIntersection(R1,G,˜ P)
9. (Φ2, Z2) =ParametricValuesTakesByPoly(R2,cTX,P) 10. TakeZ=Z0Z00Z1Z10Z2andΦ = Φ0Φ1Φ2
11. return(Φ, Z)
THEOREM 4.8. LethandPbe as above and(Φ, Z)be the out- put ofBasicParametricOptimization(h,P). Then,Z /∈ hPiand its output is correct.
PROOF. The fact thatZ /∈ hPiis an immediate consequence of the fact thathPiis prime and that its factorsZ0, Z00, Z1, Z10 andZ2
do not belong tohPifrom Lemmata4.2,4.3,4.5,4.6and4.7.
It remains to prove the correctness of the output. SinceAis cho- sen at random at Step3, one can assume thatAbelongs to the non- empty Zariski open setA defined in Lemma4.3.
Now, remark that for anyγ ∈ V(P)−V(Z). By Lemma4.3, propertyP(A)holds.
Hence, without loss of generality one can assume that algorithm SetContainingLocalExtremain [18] runs by choosing the matrixA selected at Step3ofBasicParametricOptimization. On inputγTX andhthe output ofSetContainingLocalExtrema is a polynomial ϕ∈Q[c0]whose set of roots contains the optimum of the restriction of the mapx→γTxtoV ∩Rn. From Lemmata4.2,4.5,4.6and 4.7this polynomialϕis exactlyΦ(c0, γ). Hence, correctness of al- gorithmSetContainingLocalExtrema[18, Proposition 4.2] implies the one ofBasicParametricOptimization.
4.5 Recursive procedure
We present now our recursive procedure. It uses the routineBasic- ParametricOptimization presented above. It also uses a routine PrimeDecompositionwhich takes as input a polynomial familyP⊂ Q[c]and a polynomialZ ∈ Q[c]. It returns polynomial families P1, . . . ,Pksuch that
phPi+hZi=∩ki=1hPii andhPiiis prime for1≤i≤k.
