A note on convexity of two signomial functions

Journal of Nonlinear and Convex Analysis, vol. 10, pp. 429, 435, 2009

A note on convexity of two signomial functions

Jein-Shan Chen ¹ Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan Chia-Hui Huang ²

Department of Information Management Kainan University

Taoyuan 33857, Taiwan March 24, 2008

(revised on January 12, 2009)

Abstract. In this note, we provide correct proofs for showing the convexity of two signomial functions which are frequently used in some recent papers [4, 6, 7, 8, 9] by Tsai et al.. Their arguments contain repeated flaws that motivate our work of this note.

Key words. Convexity, positive definite, Hessian matrix, signomial function.

1 Motivation and Basic Concepts

In this note, we consider two signomial functions whose convexity play important roles in some recent papers [4, 6, 7, 8, 9] dealing with geometric programming problems. How- ever, the verifications therein contain some certain flaws and those incorrect arguments are repeatedly appeared and cited. From point of scientific research’s view, we hereby provide correct proofs for them.

First, we recall what signomial function is. A function f : IRⁿ₊₊ → IR defined as f (x) = cx^α₁¹x^α₂²· · · x^αnⁿ,

where c > 0 and α_i ∈ IR for all i, is called a monomial function or simply a monomial, see [2]. Note that the exponents αi of a monomial can be any real numbers, but the

1Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. E-mail:

jschen@math.ntnu.edu.tw.

2E-mail:leohuang@mail.knn.edu.tw

coefficient c must be nonnegative. A sum of monomials, namely, a function of the form

f (x) =

∑N k=1

c_kx^α₁^1kx^α₂^2k· · · x^αn^nk,

where c_k > 0 and c_ik ∈ IR, is called a posynomial function with N terms or simply a posynomial. A signomial is a linear combination of monomials of some positive variables x₁, . . . , x_n. Generally speaking, signomials are more general than posynomials.

Next, we review some basic concepts and properties of symmetric matrices which will be used in subsequent analysis. These materials can be found in regular textbooks regarding matrix analysis and convex functions, e.g., [1, 3]. Let f be defined on an open convex set D ⊆ IRⁿ and be twice differentiable, it is known that (i) f is convex on D if and only if the Hessian matrix ∇²f (x) is positive semidefinite (p.s.d. for short) at each x ∈ D; (ii) if ∇²f (x) is positive definite (p.d. for short) at each x ∈ D, then f is strictly convex. The converse of (ii) is false, see the counterexample f (x) = x⁴. Another important criterion for positive definiteness of a symmetric matrix A is via its leading principal minors as below. For convenience, we denote△kas the leading principal minors of A.

Lemma 1.1 Let A be an n× n nonzero symmetric matrix.

(a) If A is positive semidefinite, then all its leading principal minors are nonnegative with not all of them being zero, i.e., △k ≥ 0, k = 1, 2, . . . , n and not all △k = 0.

(b) A is positive definite if and only if all its leading principal minors are positive, i.e.,

△k> 0, for all k = 1, 2, . . . , n.

The positive definiteness of a symmetric matrix can be described not only by its leading principal minors, but also by all principal minors. More specifically, the positivity of any nested sequence of n principal minors of A (not just the leading principal minors) is necessary and sufficient for A to be positive definite (see [3, Theorem 7.2.5]). On the other hand, if all principal minors of A are nonnegative, then A is positive semidefinite (see [3, page 405]).

The converse of Lemma 1.1(a) is false. For example, let A =



 1 0 0 0 0 0 0 0 −1



, we have

⟨x, Ax⟩ = x²1 − x²3 which is not always nonnegative for all x ∈ IR³. But △1 = 1 ≥ 0,

△2 = 0≥ 0, △3 = 0≥ 0. In fact, the converse of Lemma 1.1(a) is true only for n = 2, see [1, page 112]. From the aforementioned discussion, we know that we can not tell the positive semidefiniteness of a symmetric matrix by its leading principal minors whereas we can do it for positive definiteness. Nonetheless, we still can reach the conclusion of the

positive semidefiniteness of a symmetric matrix by the nonnegativeness of its eigenvalues.

This can be seen as below.

Lemma 1.2 Let A be an n× n nonzero symmetric matrix. Then, the followings hold.

(a) A is p.s.d. if and only if all of its eigenvalues are nonnegative with at least one eigenvalue being zero.

(b) A is p.d. if and only if all of its eigenvalues are positive.

To close this section, we state another important relation between ln f (x) and f (x) on their convexity that will be needed for proving our main results, i.e., suppose f is defined on a convex set D⊆ IRⁿand f (x) > 0 for all x∈ D, then the convexity of ln f(x) implies f (x) being convex. Note that the converse is false, for instance, f (x) = x² is convex but ln f (x) = 2 ln|x| is not convex.

2 Main Results

Now we are ready to present our main results which show that the following two signo- mial functions are convex functions. As mentioned earlier, signomial functions play an important role in geometric programming. In particular, the convexity of such functions will help in designing solution methods for it which is the main motivation for this note.

Proposition 2.1 Let f1 : IRⁿ₊₊ → IR be defined as f1(x) = c1

∏n i=1

x^α_iⁱ, where c1 > 0 and α_i ≤ 0 for all i = 1, 2, . . . , n. Then f1 is a convex function.

Proof. Since c₁ > 0, it is enough to show that ef₁(x) =

∏n i=1

x^α_iⁱ is convex.

Let g(x)=ln ef₁(x)=

∑n i=1

ln x^α_iⁱ=

∑n i=1

α_iln x_i. Then, we have

∇g(x) = [α₁

x₁ α₂

x₂ · · · α_n x_n

]T

and ∇²g(x) =













−α1

x²₁ 0 · · · 0 0 −α2

x²₂ · · · 0 ... ... . .. ... 0 0 · · · −αn

x²_n













Due to α_i ≤ 0 for all i = 1, 2, . . . , n, we know that all eigenvalues of ∇²g(x) are non-

negative which implies (by Lemma 1.2(a)) that ∇²g(x) is positive semidefinite. Thus, g(x)=ln ef (x) is a convex function which yields ef₁(x) being a convex function. 2

Proposition 2.2 Let f2 : IRⁿ₊₊ → IR be defined as f2(x) = c2

∏n i=1

x^α_iⁱ, where c2 < 0 and

α_i > 0 for all i = 1, 2, . . . , n with 1−

∑n i=1

α_i ≥ 0. Then f2 is a convex function.

Proof. It is not hard to compute that [∇f2(x)]_i = c2αix^α_iⁱ⁻¹

∏n j=1,j̸=i

x^α_j^j. In other words,

∇f2(x) =







c₂α₁x^α₁¹⁻¹x^α₂²· · · x^α_nⁿ c₂α₂x^α₁¹x^α₂²⁻¹· · · x^αnⁿ

...

c₂α_nx^α₁¹x^α₂²· · · x^α_nⁿ⁻¹





.

In addition, it can be verified that [∇²f₂(x)]

ij = ∂²f₂(x)

∂xi∂xj

=





 αiαj

x_ix_jf₂(x), if i ̸= j, αi(αi− 1)

x²_i f2(x), if i = j, namely,

∇²f₂(x)

=















c₂α₁(α₁− 1)x⁻²1

∏n i=1

x^α_iⁱ c₂α₁α₂x⁻¹₁ x⁻¹₂

∏n i=1

x^α_iⁱ · · · c₂α₁α_nx⁻¹₁ x⁻¹_n

∏n i=1

x^α_iⁱ c₂α₂α₁x⁻¹₂ x⁻¹₁

∏n i=1

x^α_iⁱ c₂α₂(α₂− 1)x⁻²2

∏n i=1

x^α_iⁱ · · · c₂α₂α_nx⁻¹₂ x⁻¹_n

∏n i=1

x^α_iⁱ

... ... . .. ...

c2αnα1x⁻¹_n x⁻¹₁

∏n i=1

x^α_iⁱ c2αnα2x⁻¹_n x⁻¹₂

∏n i=1

x^α_iⁱ · · · c2αn(αn− 1)x⁻²n

∏n i=1

x^α_iⁱ















Moreover, the determinant of ∇²f₂(x) can be computed and be shown by induction as det[

∇²f₂(x)]

= (−c2)ⁿ ( _n

∏

i=1

α_ix^nα_i ⁱ⁻² ) (

1−

∑n i=1

α_i )

. (1)

Now, we will complete the proof by discussing the following two cases.

Case (i): If 1−

∑n i=1

α_i = 0, we will show that y^T∇²f₂(x) y ≥ 0 for any y ∈ IRⁿ which

says ∇²f₂(x) is a positive semidefinite matrix by definition, and hence f₂(x) is a convex function under this case. To see this, we first write out the expression of y^T∇²f₂(x) y as below

y^T∇²f2(x) y

= c₂

∏n i=1

x^α_iⁱ











α₁(α₁− 1)x⁻²₁ y₁²+ α₁α₂x⁻¹₁ x⁻¹₂ y₁y₂ +· · · + α1α_nx⁻¹₁ x⁻¹_n y₁y_n + α₂α₁x⁻¹₂ x⁻¹₁ y₁y₂+ α₂(α₂− 1)x⁻²2 y²₂ +· · · + α2α_nx⁻¹₂ x⁻¹_n y₂y_n

+ ... ... ...

+ α_nα₁x⁻¹_n x⁻¹₁ y₁y_n+ α_nα₂x⁻¹_n x⁻¹₂ y₂y_n+· · · + αn(α_n− 1)x⁻²_n y_n²











= c₂

∏n i=1

x^α_iⁱ











α₁x⁻¹₁ y₁[

(α₁− 1)x⁻¹1 y₁ + α₂x⁻¹₂ y₂+· · · + αnx⁻¹_n y_n] + α₂x⁻¹₂ y₂[

α₁x⁻¹₁ y₁+ (α₂− 1)x⁻¹2 y₂+· · · + αnx⁻¹_n y_n]

+ ... ... ...

+ α_nx⁻¹_n y_n[

α₁x⁻¹₁ y₁+ α₂x⁻¹₂ y₂+· · · + (αn− 1)x⁻¹n y_n]











= c₂

∏n i=1

x^α_iⁱ











α₁x⁻¹₁ y₁[

α₁x⁻¹₁ y₁+ α₂x⁻¹₂ y₂+· · · + αnx⁻¹_n y_n− x⁻¹1 y₁] + α₂x⁻¹₂ y₂[

α₁x⁻¹₁ y₁+ α₂x⁻¹₂ y₂+· · · + αnx⁻¹_n y_n− x⁻¹2 y₂]

+ ... ... ...

+ α_nx⁻¹_n y_n[

α₁x⁻¹₁ y₁+ α₂x⁻¹₂ y₂+· · · + αnx⁻¹_n y_n− x⁻¹n y_n]











= c2

∏n i=1

x^α_iⁱ

{ (

α₁x⁻¹₁ y₁+ α₂x⁻¹₂ y₂+· · · + αnx⁻¹_n y_n)2

− (

α₁x⁻²₁ y²₁ + α₂x⁻²₂ y₂²+· · · + αnx⁻²_n y_n²) }

. (2)

Next, we will argue that the whole thing inside the big parenthesis of (2) is nonpositive by applying Cauchy-Schwarz inequality. In order to apply Cauchy-Schwarz inequality, we make the following arrangement:

[(√α₁x⁻¹₁ y₁)2

+(√

α₂x⁻¹₂ y₂)2

+· · · +(√

α_nx⁻¹_n y_n)2] [ (√

α₁)²+ (√

α₂)²+· · · + (√ α_n)²

]

≥[

α₁x⁻¹₁ y₁+ α₂x⁻¹₂ y₂+· · · + αnx⁻¹_n y_n]2

. (3)

Since [(√α₁)2

+(√

α₂)2

+· · · +(√

α_n)2]

= 1, inequality (3) is equivalent to (α1x⁻¹₁ y1 + α2x⁻¹₂ y2+· · · + αnx⁻¹_n yn

)2

−(

α1x⁻²₁ y₁²+ α2x⁻²₂ y₂²+· · · + αnx⁻²_n y_n²)

≤ 0.

This together with c₂ < 0 implies that y^T∇²f₂(x) y ≥ 0 for any y ∈ IRⁿ. Thus, we complete the proof of case (i).

Case(ii): If 1−

∑n i=1

α_i > 0, then we know from (1) that

△i = (−c2)ⁱ ( _i

∏

j=1

α_j x^iα_j ^j−2 ) (

1−

∑i j=1

α_j )

, (4)

where △i denotes the i-th leading principal minor of the Hessian matrix of f₂(x). Note that c₂ < 0, α_i > 0 for all i = 1, 2,· · · , n, and 1 −

∑n i=1

α_i > 0. Therefore, it can be seen that △i > 0 for all i = 1, 2,· · · , n, which implies (by Lemma 1.1(b)) that ∇²f₂(x) is a positive definite matrix. This says that f₂(x) is strictly convex under this case. 2

For Proposition 2.1, Tsai et al. claimed that (e.g. [4, Prop. 5(i)], [6, Prop. 1] and [9, Prop. 2]) all principal minors△k≥ 0 and concluded directly that f1 is a convex function.

As mentioned earlier, this property holds only for n = 2 and is not satisfied for general n≥ 3. For Proposition 2.2, Tsai et al. made the same mistakes again and did not notice that the case 1−∑n

i=1α_i = 0 will cause the error therein (e.g. [4, Prop. 5(ii)], [6, Prop.

2] and [9, Prop. 3]).

We want to point out that our results also provide an alternative proof for the main result (Theorem 7) of [5]. Indeed, Maranas and Floudas in [5, Theorem 7] further discuss another condition as below

∃j such that αj ≥ 1 −

∑n i̸=j

α_i, and α_i ≤ 0, ∀i ̸= j, i = 1, 2, · · · n. (5) to guarantee that f₁ defined as in Prop. 2.1 is a convex function. Our approach can be also employed to verify this fact. To see this, we arrange all powers α_i in decreasing order. In other words, without loss of generality, we assume

α₁ > α₂ ≥ · · · ≥ αn. (6)

Notice that condition (5) implies that α1 is positive and all the other α2,· · · , αn are nonpositive with α₁ ≥ 1 −∑_n

i=2α_i. As mentioned in Prop. 2.1, we only need to show that the function ef₁(x) =

∏n i=1

x^α_iⁱ is convex. By similar arguments as in the proof of Prop.

2.2, we know that

△bi = (−1)ⁱ ( _i

∏

j=1

α_j x^iα_j ^j−2 ) (

1−

∑i j=1

α_j )

,

where b△i denotes the i-th leading principal minor of the Hessian matrix of ef₁(x). From conditions (5) and (6), it is easily verified that

( 1−

∑i j=1

α_j )

< 0 for each i. It is also

not hard to observe that

∏i j=1

α_j is positive if i is odd, and is negative if i is even. In other words, for each i there holds

(−1)ⁱ ( _i

∏

j=1

αjx^iα_j ^j−2 )

< 0.

In addition, we observe that b△n= 0 when α₁ = 1−∑_n

i=2α_i. Thus, from all the above, we have either

△b1 > 0,· · · , b△n−1 > 0, b△n > 0 if α₁ > 1−

∑n i=2

α_i (7)

or

△b1 > 0,· · · , b△n−1 > 0, b△n= 0 if α₁ = 1−

∑n i=2

α_i. (8)

Then, Lemma 1.1(b) says that ∇²fe₁(x) is positive definite for case (7) whereas following the similar arguments as in Prop. 2.2 implies that ∇²fe1(x) is positive semidefinite for case (8). Thus, we conclude that ef₁ is also a convex function under condition (5).

Acknowledgement. The authors would like to thank anonymous referees for their com- ments and suggestions which help improve this paper a lot.

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