Complete Hierarchy of Relaxation for Constrained Signomial Positivity

03/08/2020 ∙ by Allen Houze Wang, et al. ∙ University of Waterloo 0

In this article, we prove that the Sums-of-AM/GM Exponential (SAGE) relaxation generalized to signomial over a constrained set is complete, with a compactness assumption. The high-level structure of the proof is as follows. We first apply variable change to convert a set of rational exponents to polynomial equations. In addition, we make the observation that linear constraints of the variables may also be converted to polynomial equations after variable change. Note that any convex set may be expressed as a set of linear constraints. Further, we use redundant constraints to find reduction to Positivstellensatz. We rely on Positivstellensatz results from algebraic geometry to obtain a decomposition of positive polynomials. Lastly, we explicitly show that the decomposition is of a form certifiable by SAGE.

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1 Introduction

A signomial function is one of the form , where are fixed. Optimization of such function subject to signomial inequality and equality is called signomial programming (SP). SPs are non-convex in general and are NP-Hard in special cases [4].

Geometric programming (GP) constitutes a subclass of SPs in which the objective function to be minimized is a posynomial, with , subject to upper constraints on posynomials. GPs have wide applications in many areas such as control in communication systems [4], circuit design [5], approximations to the matrix permanent [14], and the computation of capacities of point-to-point communication channels [3]. However, the modeling power of SPs on arbitrary signomials are useful in many additional applications in chemical engineering [7], aeronautics [15], and communications network optimization [12].

In this article, we are concerned with a broader class of problems, in which one minimizes an arbitrary signomial over any arbitrary convex set characterized by a set of constraints. Such optimization problem may be reduced to the verification of signomial positivity over a constrained set.

2 Outline and Contribution

We first describe the problem of interest. In particular, we describe the relaxation for verifying signomial positivity based on a certification of signomial positivity with at most one negative term. While the certificate has been previously considered, we independently arrived an equivalent formulation in the constrained set and will present additional results. We also show how such verification of positivity can be used for constrained signomial optimization. The verification is a relaxation because although it ensures that a signomial is positive over a constrained set, not all such signomials may be verified as so.

The main contribution of the article is to describe the hiearchy of relaxation and a completeness theorem. While the verification is a relaxation, it has a hiearchy such that with increasing computational complexity, increasing subset of signomials that are positive over a constrained set may be verified as so. The completeness theorem shows that any signomial positive over a constrained set can be certified under the same framework, at finite level on the hierarchy.

3 Background

The problem of interest is verifying the positivity of a signomial over a convex set. That is, given a signomial function and a convex set , whether

(1)

In 2016, Chandrasekaran and Shah proposed the Sums-of-AM/GM Exponential (SAGE) certificates of signomial positivity, which provided a new convex relaxation framework for signomial programs akin to sum-of-square (SOS) methods for polynomial optimization [2]. This work concerns the case where and provides an efficiently computable certificate for a signomial with at most one negative term. The certificate is based on the AM/GM inequality and is exact, and the relaxation for an arbitrary signomial is based on finding a decomposition of a signomial such that each term has at most one negative term and is positive. The relaxation was shown to have a hierarchy that is complete. That is, any positive signomial may be certified as such at some finite (although not bounded or known) level of the relaxation hierarchy.

In 2019, Murray, Chandrasekaran and Wierman extended the Sums-of-AM/GM Exponential (SAGE) relaxation for signomial positivity in the constrained case [10]. We have independently produced the same results, and will describe it in the following sections. However, while the previous work discusses a hiearchy, it does not extend the completeness theorem to the constrained case.

More broadly, the notion of hiearchy of relaxation and completeness is well known in the SOS method for polynomial optimization [6, 8]. The hierarchy and completeness results proved in this article is analogous to the ones in polynomial optimization, and the Positivestellensatz used in this article is the constrained analogue of Reznick’s Positivestellensatz supporting the SOS hiearchy [11].

4 Notations

Given a matrix defined by a finite collection of vectors , let , and let be a matrix having th column removed from . i.e. . Given , use to denote a vector with the th entry removed from

5 Constrained-AGE Certificate

We want to certify the positivity of signomial with at most one negative term over a constrained set. Recall that for any function , its conjugate is defined as:

(2)

With the Fenchel conjugate we recall the celebrated Fenchel-Rockafellar duality theorem. [e.g., [1, Theorem 3.3.5]] Let and be two functions, be a linear map, then

(3)

If, furthermore, and are lower semicontinuous convex and (for instance) , then equality holds and the second infimum is attained if finite. Of interest here is the special case where

(4)

and its conjugate is the support function of :

(5)


The goal is to provide a convex certificate of the following AM/GM-signomial inequality:

(6)

Here for but we allow to be negative (otherwise the answer would be trivially yes). The important observation made in [2] is that we can divide both sides of (6) by and arrive at the equivalent problem:

(7)

Unlike the left-hand side of (6), the left-hand side of (7) is a convex problem (assuming is a convex set).

Let , , and , then applying the Fenchel-Rockafellar duality we have (note that hence the condition in Section 5 holds):

(8)

Here we have used the fact that for the function ,

(9)

where is the Euler constant. Note that the infimum in (8) is actually attained if the infimum is finite. Thus, w.l.o.g., we can rewrite (8) :

(10)

We remark that the condition in (8) remains to be sufficient to guarantee (7), for any set , while it is also necessary if is closed convex. On the other hand, the left-hand side of (8) is always a convex problem hence can be efficiently computed.

More generally, define the following. Given a matrix defined by a finite collection of vectors and convex set consider signomials defined by with a most one negative coefficient occurring at the th index that are positive over a convex set . i.e. . Then such signomials are defined as:

(11)
(12)
(13)

We also define the coefficients of such signomials Given a matrix defined by a finite collection of vectors and convex set , the set of coefficients defining are described as follows:

(14)
(15)
(16)

We observe that is a closed convex cone. Using the above two definitions, we may define summation of such signomials.

and are both closed under closure Clear from the definition of positivity over a convex set and positivity of coefficients on all but th term, both are closed under addition.

6 Constrained-SAGE Relaxation

We may use the above certificate to provide relaxation for signomial positivity over a constrained set. The key is to find a decomposition of a signomial such that each part has at most one negative term and is positive over a constrained set (i.e. Sums-of-AGE). Given a matrix defined by a finite collection of vectors and convex set , signomials defined by that can be decomposed into are defined as:

(17)

From the definitions the following are clear: We also define the coefficients of such signomials Given a matrix defined by a finite collection of vectors and convex set , the set of coefficients defining are described as follows:

(18)
(19)
(20)

and are both closed under closure Follows from additive closure of and as in Section 5.

7 Convex Relaxation for Constrained Signomial Optimization

We discuss how the above definitions can be used for convex relaxation of constrained signomial optimization.

Given a matrix defined by a finite collection of vectors , and a convex set , let . WLOG assume that . Consider the following problem:

(21)

The minimization problem can be reformulated as a certification of positivity.

(22)
(23)

7.1 Inner Approximation

However, the problem is in general intractable, as certifying signomial positivity is intractable. The definition of provides a tractable lower bound. Consider the following problem and its solution. Let . Recall Section 6. Then it is clear that . A more concrete formulation of the problem is below:

(24)
(25)

where , and is characterized by convex constraints in section 6.

The above can be considered a relaxation through an inner approximation of signomial positivity over a constrained set. That is, we rely on the the property that . It is an approximation because the LHS is a subset of the RHS, but not necessarily equal to it.

8 Hierarchy of Relaxation

Given a matrix defined by a finite collection of vectors , define the following notation. For some

Note that . Also it is clear that . Using the above set of exponentials, we may define a quantity as follows. Let The the significance of the quantity is expressed in the following theorem. For some , and any , . That is, convex relaxation of the constrained optimization problem has a hierarchy that becomes non-decreasingly accurate. Before proving the theorem, we prove the following two theorems for better understanding of the set . For any , if ,
then . Proof of Section 8. Let . Suppose . Then where . By definition, this implies and .

The conditions hold for as well. So

Finally, by additive closure property (section 6) of ,
we have For any , Proof of Section 8. Note that multiplication of a posynomial does not change the positivity of a signomial over a convex set. In particular, for any , if and only if . Thus .

Proof of Section 8. Let . Then

And we have the first inequality Let . Then

And we have the second inequality

9 A Completeness Theorem

Let be a collection of rational vectors. Let be a set of (possibly infinite) halfspaces defined by rationals. where and . Let . Assume that is compact. Let

Consider a signomial defined by such collection of vectors. . Suppose that the signomial is positive over . There exists some s.t. We also note the following classic result from convex analysis [13]. Any closed convex set may be expressed as the intersection of (possibly infinite) halfspaces. Here, we restrict the halfspaces to be defined by rationals.

10 Proof of Completeness Theorem

The proof structure is as follows. First, we show that the halfspace constraints and rational exponents can be converted into polynomial equations after a change of variable. Then, we make modifications to the polynomials so that they are homogeneous, and add redundant constraints so that the its extension from positive orthant to the non-negative orthant does not increase the feasible region. The goal of variable change operation is to reduce signomial positivity over a convex set to positivity over the intersection semi-algebraic set and the non-negative orthant. Then, we apply Positivestellansatz result from algebraic geometry to decompose the positive polynomial into sum of homogeneous polynomials [11]. Lastly, we show that the decomposition is certifiable as SAGE.

Without loss of generality, we may make the following assumptions about the collection of vectors .

  1. the first n vectors are linearly independent

The assumptions on the exponents are in fact not restrictive. To satisfy the first condition, we may select a set of linearly independent vectors as the first

. The proof is easily generalized to the case when the span of the vectors has dimension less than . The second condition is not restrictive either, since we may insert a zero vector into the set of exponents. However, it is a variable required for satisfying certain conditions in the proof.

10.1 Variable Change

In this section we prove the following. Consider a signomial function and a set of halfspaces satisfying conditions in Section 9. Then, there exists a set at the intersection of the positive orthant and the a set of polynomial equations, such that

10.1.1 Exponents Defined by Rational Vectors

Consider set of exponents defined by rational vectors satisfying conditions in Section 9. Apply a change of variable by letting . First, since the first vectors are linearly independent, they span . Thus the set of first exponents are free, and may take any value. Next, since , . The rest of the vectors may be expressed as linear combinations of the first vectors. Moreover, the linear combinations are defined by rationals since the exponents are rational. Thus, they are constrained with respect to the first vectors. For with , . Then;

The last step is from the fact that . and since ’s are rationals, we may raise both sides by the smallest common denominator to clear the fractions. For example, .

We may apply such operation to for all . Note that the operation is only valid in the positive orthant. Thus, with the assumptions in Section 9, change of variable has converted a set of rational exponents to polynomial equations as follows.

Where ’s are obtained from the procedure as above.

10.1.2 Halfspace Defined by Rationals

We first consider a single rational halfspace constraint on . Let . and .

We note the following known theorem in linear algebra [9] Let be a polyhedron and let

be a linear transformation. Then

be a polyhedron. Further, if P is a rational polyhedron and T is a rational linear transformation (that is, the matrix of T is rational), then T(P) is a rational polyhedron. Thus, given the rational halfspace constraint on , we may find a rational polyhedron constraint on . for some and . The dimension is arbitrary, but is finite by the above theorem. We apply a series of elementary arithmetic operation as below.

The last step moves exponents with negative terms by multiplication on both sides. For example; . Again, since has rational entries, we may raise both sides by a common denominator to clear fraction.

Where is the common denominator of the fractions.

10.1.3 Intersection of Halfspaces Defined by Rationals

It is easy to extend the above to intersection of (possibly infinite) halfspaces. In the above, single halfspace constraint has generated a finite set of polynomial equations. Given a set of halfspaces, we simply take the intersection of the polynomial equations generated from them. For each , Let and . We may write as below.

is a set of (possibly infinite) indices, each correspoding to some polynomial generated from the rational halfspaces.

10.1.4 Positivity of Signomial to Positivity of Polynomial

In the previous sections, we have performed a change of variable transforming rational exponents over the intersection of rational halfspaces to a vector in the positive orthant constrained by polynomial equations. Now we consider the optimization in the new variables. We define a set of feasible values of as below.

And the optimization problem can be recasted as follows

We recall Section 10.1, which can be written as follows. Based on the variable change from section 10.1, we have:

10.2 Positivity to Positivestellensatz

So far we have reduced the positivity of a signomial over a convex set to the positivity of a polynomial over the intersection of the positive orthant and a set defined by (possibly infinite) polynomial equations. Now we note the key Positivestellensatz results [11] Let be and be homogeneous polynomials on such that for all and . Then for some , there exists homogeneous polynomials such that all of their coefficients are nonnegative and The condition of this Posivistellensatz requires positivity of a polynomial over the intersection of nonnegative orthant and a semialgebraic set. is not such set. There are three conditions required by the Posivistellensatz theorem that does not satisfy.

  1. does not include the faces of the nonnegative orthant.

  2. is defined by polynomials that are possibly non-homogeneous.

  3. is defined by possibly infinite polynomials.

The goal of this section is to show the following: There exists a set defined as the intersection of the nonnegative orthant and a semialgebraic set s.t.
Note that implication only needs to be true in one direction. In other words, satisfy the conditions for Posivistellensatz, positivity over implies positivity over . In the next three sections, we show how modifications on to satisfy the three conditions, while positivity condition remains true.

10.2.1 Inclusion of Points on the Faces of Nonnegative Orthant

First consider the following set.

which extends the definition of