i0µi gi,β has only positive terms. If furthermore all gi,αp0q for i 1, . . . , s are greater than or equal to zero, then (4.1.5) is a geometric program.
4.2 Constrained Polynomial Optimization via Signomial and Geometric Programming
In this section, we provide relaxations of the program (4.1.5) following the ideas of Ghasemi and Marshall in [GM13]. The goal is to weaken the assumptions which are needed to obtain a geometric program or at least a signomial program. We provide such relaxations in the programs (4.2.2) and (4.2.3) and provide the desired properties in the Theorems 4.2.1 and 4.2.4. Moreover, we show that under certain extra assumptions the bound obtained by the new program (4.2.2) equals the optimal boundspf,gq from the previous section; see Theorem 4.2.5. Furthermore, we demonstrate in the following Sections 4.3 and 4.4 that the resulting programs can be an alternative for SDP in cases where Lasserre's relaxation has issues.
Let all notation regarding Gpµq be given as in Section 4.1.4. Assume that we have for each i0, . . . , s
gi ¸
βPAi
gi,β xβ with gi,β P R. We have∆pAiq ∆pAq and hence write
gi
¸r j0
gi,αpjqxαpjq ¸
βP∆pAq
gi,βxβ
and set gi,αpjq0for all αpjq PVpAqzAi and gi,β 0 for allβ P∆pAqzAi. We remark that three cases can occur for β P∆pAq XAi:
(1) gi,βxβ is not a monomial square. Then we have βP∆pGq.
(2) gi,βxβ is a monomial square, but there exists another gl such that gl,βxβ is not a monomial square. Then we have βP∆pGq.
(3) gi,βxβ is a monomial square, and there exists no other gl such that gl,βxβ is not a monomial square. Then we have βR∆pGq.
Sums of monomial squares as described in case (3) are ignored in our program (4.1.5).
Thus, we can also ignore this case here. We now investigate the other two cases in detail.
As already mentioned in Section 4.1.4 we can interpret the coecients Gpµqαpjq and Gpµqβ as linear forms in µsince we have for all j 0, . . . , r
We decompose every coecient Gpµqβ into a positive and a negative part such that Gpµqβ Gpµqβ Gpµqβ, where
This decomposition is independent of the choice of µ in the sense that no gi,β can be a summand of both Gpµqβ and Gpµqβ for dierent choices of µ since µ P Rs¥0.
We consider the following optimization problem in the variables µ1, . . . , µs and aβ,1, . . . , aβ,r, bβ for every βP∆pGq:
$' cases. Indeed, with some additional assumptions the program (4.2.2) is a geometric program. Moreover, it is a relaxation of the program (4.1.5).
Theorem 4.2.1. Assume that for all j1, . . . , r the formGpµqαpjq°s
i0µigi,αpjq has exactly one strictly positive term, i.e., there exists exactly one strictly negative gi,αpjq. Then the optimization problem (4.2.2) restricted to µ P p0,8qs is a geometric program. Assume that γsonc denotes the optimal value of (4.2.2) and γ denotes the optimal value of (4.1.5). Then we have
fαp0qγsonc ¤ fαp0qγ ¤ spf,gq.
The typical choice for αp0q is the origin which yields a lower bound for f to be nonnegative on K with the inequality (4.1.4):
Corollary 4.2.2. Let all assumptions be as in Theorem 4.2.1. If αp0q is the origin, then we have
f0γsonc ¤ f0γ ¤ spf,gq ¤ fK.
Proof. (Theorem 4.2.1) If we restrict ourselves toµP p0,8qs, then all functions involved in (4.2.2) depend on variables in R¡0. By assumption every Gpµqαpjq has exactly one strictly positive term. Thus, we can express constraint (1) in (4.2.2) as
°
βP∆pGq
aβ,j Gpµqαpjq
Gpµqαpjq ¤ 1,
with Gpµqαpjq and Gpµqαpjq dened analogously as in (4.2.1). Since Gpµqαpjq is a monomial the left hand side is a posynomial in µand x. The constraints (2) (4) are posynomial constraints in the sense of Denition 4.1.3 of a geometric program. The functionp is also a posynomial since all terms are nonnegative by construction and all exponents are rational. Moreover, everybβ in (4.2.2) has to be greater or equal than the correspondingbβ in (4.1.5) becausemaxta, bu ¥ |ab| for alla, bPRzt0u. Furthermore, since gi,αp0q ¥gi,αp0q holds, the inequality γsonc ¤γ follows by the denitions of (4.2.2) and (4.1.5). The last inequality follows from Theorem 4.1.7.
One expects the programs (4.1.5) and (4.2.2) to have a similar optimal value if, for example, gi,αp0q ¥ 0 for most i 1, . . . , s and if either Gpµqβ or Gpµqβ is identi-cally zero for most β P ∆pGq. Note that one of Gpµqβ, Gpµqβ is zero if and only if maxtGpµqβ, Gpµqβu |Gpµqβ Gpµqβ| |Gpµqβ| and the latter holds if and only if gi,β ¥0 orgi,β ¤0for i0, . . . , s.
We give an example to demonstrate how a given constrained polynomial optimization problem can be translated into the geometric program (4.2.2). In Section 4.3, we provide several further examples including actual computations of inma using the GP-solver CVX.
Example 4.2.3. Let f 1 2x2y4 12x3y2 and g1 13 x6y2. From these two polynomials we obtain a function
Gpµq
1 1 3µ
2x2y4 µx6y2 1 2x3y2.
For Gpµq to be an ST-polynomial, we have to choose µ P p0,3q. Here, the vertices of NewpGpµqq are αp0q p0,0q, αp1q p2,4q, and αp2q p6,2q, and we have
∆pGq tβu tp3,2qu. Thus, we introduce 4 variables paβ,1, aβ,2, bβ, µq. First, we compute the barycentric coordinates of β and get
λp0βq 3
10, λp1βq 3
10, λp2βq 2 5. We match the coecients ofGpµq with the verticesαpjq:
• g1,αp0q maxt13,0u 13,
• Gpµqαp1q2, Gpµqαp2qµ,
• Gpµqβ 12, Gpµqβ does not exist.
Hence, program (4.2.2) is of the form: In what follows, we extend Theorem 4.1.7 by reformulating the program (4.2.2) such that it is always applicable. On the one hand, the new program is only a signomial program instead of a geometric program in general. On the other hand, the reformulated program covers the missing cases of Theorem 4.2.1 and also yields better bounds than the corresponding geometric program (4.2.2) in general. We dene
qpµ,tpaβ, cβq : β P∆pGquq
The key dierence between this program and (4.2.2) is that
cβ ¥ maxtGpµqβ Gpµqβ, Gpµqβ Gpµqβu |Gpµqβ|.
We obtain the following statement.
Theorem 4.2.4. The optimization problem (4.2.3) restricted to µP p0,8qs is a signo-mial program. Assume that γsnp denotes the optimal value of (4.2.3) andγsonc, γ denote the optimal values of (4.2.2) and (4.1.5) as before. Then we have
fαp0qγsonc ¤ fαp0qγsnp ¤ fαp0qγ ¤ spf,gq.
Particularly, we have γsnp γ if the program (4.1.5) attains its optimal value for µP p0,8qs.
Proof. The proof is analogous to the proof of Theorem 4.2.1. The only dierence is that now certain terms can have a negative sign and hence posynomials then become signomials. The statement follows with the denition of a signomial program; see Section 4.1.2.
Finally, we show that if we strengthen the assumptions in Theorem 4.2.1, then, the output fαp0qγsonc of the program (4.2.2) equals the output fαp0qγ of the program (4.1.5) and particularly equals the boundspf,gq.
Theorem 4.2.5. Assume that for every1¤j ¤rthe formGpµqαpjq °s
i0µigi,αpjq
has exactly one strictly positive term. Furthermore, assume that gi,αp0q ¥ 0 for all i1, . . . , s, and that ∆pAq XAiXAl H for all 0¤i l ¤s. Let γ be the optimal value of the program (4.1.5). If the optimal value spf,gq suptGpµqsonc :µPRs¥0u is attained for some µP p0,8qs, then fαp0qγsonc fαp0qγ spf,gq, where, as before, γsonc denotes the optimal value of (4.2.2).
Note that the condition∆pAq XAiXAl H is satised if the supports of gi and gl dier in all elements that are not vertices of NewpGpµqq.
Proof. The assumption ∆pAq XAi XAl H for all 0 ¤ i l ¤ s implies for every βP ∆pGq that Gpµqβ
°s
i0µigi,β µkgk,β, for some k 0, . . . , s. Therefore, we have for everyβ P∆pGq that
maxtGpµqβ, Gpµqβu |µkgk,β| |Gpµqβ|.
Moreover, we have gi,αp0q ¥ 0 for all i 1, . . . , s by assumption and thus we obtain
°s
i1µigi,αp0q °s
i1µigi,αp0q. Hence, the two programs (4.1.5) and (4.2.2) coincide.
By assumption, every Gpµqαpjq consists of exactly one positive term. Therefore, (4.2.2) is a GP by Theorem 4.2.1. Considering Theorem 4.2.1 it suces to show that the inequalityfαp0qγsonc ¥spf,gq holds, such that fαp0qγsoncfαp0qγ spf,gq