SUM OF SQUARES DECOMPOSITION FOR
SYMMETRIC POLYNOMIAL INEQUALITIES
A thesis presented to the faculty of San Francisco State University '- ifi In partial fulfillment of > The Requirements for The Degree MATH
* £ 6 ^ Master of Arts In Mathematics
by
Logan Coe
San Francisco, California
August 2017 Copyright by Logan Coe 2017 CERTIFICATION OF APPROVAL
I certify that I have read SUM OF SQUARES DECOMPOSITION FOR
SYMMETRIC POLYNOMIAL INEQUALITIES by Logan Coe and that
in my opinion this work meets the criteria for approving a thesis sub mitted in partial fulfillment of the requirements for the degree: Master of Arts in Mathematics at San Francisco State University.
Serkan Hosten Professor of Mathematics
Pierre Langlois Professor of Mathematics
Lipika Deka Associate Professor of Mathematics SUM OF SQUARES DECOMPOSITION FOR
SYMMETRIC POLYNOMIAL INEQUALITIES
Logan Coe San Francisco State University 2017
We develop a new approach to verifying inequalities between symmetric functions.
By reorganizing desired inequalities between these symmetric functions we use non negativity to verify the inequality. Our approach uses symmetry adapted semidefi- nite programming to show that a given symmetric function is not only nonnegative, but in fact a sum of squares. By designing an invariant semidefinite program we re duce our problem size using the natural symmetries in SOS semidefinite programs.
Using these methods we validate known inequalities corresponding to elementary and power sum symmetric functions. We also expand on the conjecture that Schur and complete homogeneous symmetric functions have similar inequalities. We test our approach on all four symmetric functions in cases where hand computation would be impossible. Based on our results we conjecture that a Sum of Squares de composition exists for every inequality including the two conjectured inequalities.
I certify that the Abstract is a correct representation of the content of this thesis. ACKNOWLEDGMENTS
The author thanks Professor Federico Ardila for helping inspire this re search as well as Gina Karunaratne for her contribution to it in the early stages of this work. The author also thanks San Francisco State
University for the funding needed to run optimization problems through
Matlab. Lastly the author thanks his advisor Serkan Hosten for all his patience, explanations, and inspirations. TABLE OF CONTENTS
1 Introduction...... 1
2 Background...... 7
2.1 Symmetric Functions...... 7
2.2 Sum of S q u a re s...... 18
2.3 Semidefinite Programming M ethods...... 26
3 SOS D ecom positions...... 33
4 Reduction ...... 42
4.1 Representation T h e o ry ...... 42
4.2 Symmetry Adaptation ...... 57
5 Conclusion...... 81
6 A p p e n d ix ...... 83
Bibliography ...... 84 LIST OF TABLES
Table
vii LIST OF FIGURES
Figure
2.1 Graph of a Convex Function 1
Chapter 1
Introduction
Applications of symmetric functions are found in many branches of mathematics such as enumerative combinatorics, group theory, Lie algebras, algebraic geome try, and optimization. Inequalities between symmetric functions is a classic topic.
Perhaps the most famous example is the inequality between the Arithmetic and
Geometric means which is shown and proved in [2],
In order to fully understand the goal of this thesis, we must first have a funda mental understanding of symmetric functions. A function of multiple variables is called symmetric if its value remains the same under all permutations of the variables
, i.e., a function f(xu ...,xn) is symmetric if f(x^ i),. .., xa(n)) = f(xu for all a € Sn where Sn is the symmetric group on n elements. For example the function
f(xi, x2, x3) = XiX2 + X1.X3 + x2x3 is symmetric because addition and multiplication are commutative. Or in other 2
words
[xix2 + 0:1X3 + x2x3) = (x2xi + x2x3 + X1X3) = (rr2a;3 + x2xi + x3xi)
and so on. A symmetric function is homogeneous if each term has the same degree, which is true for the example above.
The symmetric functions that we work with are defined by partitions of nonneg ative integers. A partition of a nonnegative integer n is a decreasing sequence of nonnegative integers less than or equal to n where the sum of the terms is equal to n. For instance the set of all partitions of 3 is {(3,0,...), (2,1,0,...), (1,1,1,0,...)} which for simplicity will be denoted {3; 2, 1; 1, 1, 1}. A partial ordering on partitions of the same integer can be used to define a partial order on special homogeneous symmetric functions of the same degree. The focus of our research is to show that methods in Optimization can be used to verify this relationship.
The set of homogeneous symmetric functions of degree r in n variables forms a vector space and has several commonly used bases. In particular, we are interested in the elementary symmetric functions, power sum symmetric functions, (complete) homogeneous symmetric functions, and Schur functions defined in Section 2.1. These are defined by partitions of r. For instance, if the number of variables n — 3 and the degree of the polynomial r — 3, then the elementary symmetric function 3
corresponding to the partition 1,1,1 of 3 is
ei,i,i(xi, x2, x3) = (xi + x 2 + x3)3
For each basis listed above, the term-normalized symmetric function G\{x) as sociated to the element g\(x) is defined to be
G\{x)r r„\ —- 9x{x) 9\(!>•••> 1)’ where A is a partition of r. So, the term-normalized symmetric function for the elementary symmetric function eititi(xi, x2, x3) is
t-t , x ew (x1,x2,x3) fxi + x2 + x3\ 3 E«(*•■*».*»> = ^(IXTT = I 5 J •
Inequalities between two term-normalized symmetric functions F(x) and G(x) have been studied in [3], where for a fixed n F(x) < G(x) or G(x) — F(x) > 0 if
F(a) < G(a) for all a € R>0. 4
T heorem 1.1 ([3]).
E\(x) > E^x), x > 0 <= > A ■< n
P\(x) < Pn(x), x > 0 <*=>■ A ■< /j,
Hx(x) < H,(x), x > 0 A ■< n
S\(x) < Sv(x), x > 0 =► A d: H
where x G R>0, E, P, H , and S are the term normalized Elementary, Power Sum,
Homogeneous, and Schur functions and A and fi are partitions of the same nonnega
tive integer r. The symbol ^ refers to a partial order which we will define in Section
2. 1.
The similarities between these theorems led to Conjecture ?? in [3] that the converses of the last two partial orders are also true.
Here, we take a new approach to verifying the inequalities using semidefinite pro gramming. A semidefinite program is a specific form of optimization problem that we can solve using [8],[7]. One way to show that a polynomial with real coefficients is nonnegative is to show that it is a sum of squares (SOS) of other polynomials.
This is just a sufficient condition as there are some nonnegative polynomials that are not SOS (for instance the Motzkin Poynomial x\x\ + x\x\ — 3x\x\ + 1). Using methods shown in [1] we discover whether a given polynomial can be decomposed as an SOS by solving a semidefinite optimization problem. The problem lies in finding 5
a positive semidefinite matrix that can be used to build our polynomial. Finding such a matrix is achievable through solving a semidefinite program. The positive definiteness of this matrix allows us to construct the polynomial as a sum of squares.
We use other work found in [5], to take advantage of the symmetries within our optimization problem. This involves using representation theory to simplify our semidefinite programs for faster computation. Even in small cases we see massive reductions of problem size. For instance in Section 4.2, we see the ultimate reduction of a 210 variable problem to just 11 variables.
Using all of these methods we verify the results from Theorem 1.1 for n = 2 and 3 and r = 2 and 3. This shows that our approach to verifying theorem 1.1 and testing the conjecture is viable. W ith this evidence, we formed the following conjecture of our own:
Conjecture 1 .1 . Let F{xi,... ,xn) and G(xi,... ,xn) be symmetric functions from the families used in Theorem 1.1. Then F{xi,... ,x„) > G(xi,... ,xn), for x > 0 for the listed cases if and only if F(y\,..., y%) — G(yf,..., y„) is an SOS.
In Chapter 2 we define the tools needed to understand the problem we are trying to solve, as well as the tools we use to solve it. Here we discuss topics such as sym metric functions, integer partitions, partial ordering, and optimization. In Chapter
3, using concepts from [1] to approach the problem of determining whether a given multivariate polynomial is a sum of squares using semidefinite programming, we show how particular inequalities stated in Theorem 1.1 can be solved. In Chapter 4 6
we present some representation theory and describe how it can be used to drastically reduce our problem size. This involves research from [5] that separates our decision variables into independent sets that can be used for smaller sub problems. This process involves a change of basis matrix that depends on n and r. Due to time constraints we were only able to find this matrix for the values 2 and 3. It is our hope that this process can be automated. 7
Chapter 2
Background
We begin by presenting everything needed to fully understand Theorem 1.1. This
Theorem uses terms such as integer partitions, symmetric functions, dominance, and term normalization. These along with definitions of the specific symmetric functions used will be described in this chapter. We also present some optimization, the research of [1], and how it can be used with our method to verify inequalities from
Theorem 1.1. With some small examples we conclude the chapter by discussing further symmetries in our solutions that lead us to the reductions described in
Chapter 4.
2.1 Symmetric Functions
Definition 2.1 . A polynomial in ..., £„] is said to be symmetric if it is invari ant under all permutations of the variables xi,..., xn by elements of the symmetric group Sn. 8
In other words, if a is an element of Sn, and p is a symmetric polynomial, then
X2i • • • i Xn) • • • > •^'o’(n))
Example 2.1.
a = (1, 2)
, X2) = {X!X2)2 + Xi + X2 = (x2xi) 2 + X2 + Xi =p(x2,xi}
Particular symmetric polynomials are defined using partitions of non-negative integers. These partitions are defined in [10] as shown below.
Definition 2.2. A partition A of a nonnegative integer n is a sequence (Ai,..., A*,) £
Nfe satisfying Ai > ... > A* and A* = n.
Here any A; = 0 is considered irrelevant in the full sequence A = (Ai,..., A*,, 0,...).
An even further simplified notation of a partition, which will be used in this thesis, is A = Ai,...,Afc. The set of all partitions of a nonnegative integer n, denoted
”Par(n)” is shown for some small examples below. 9
E xam ple 2.2.
P aril) = {1}
Par( 2) = {2 ; 1, 1}
Par(3) = {3; 2,1,1; 1,1,1}
Par(4) = {4; 3,1; 2,2; 2 ,1, 1 ; 1, 1, 1, 1}
Par(5) = {5; 4,1; 3,2; 3,1,1; 2,2,1; 2,1,1,1 ; 1,1,1,1,1}
The four symmetric functions of particular interest here are all defined by integer partitions. We will start by defining the elementary symmetric function as seen in
[10].
Definition 2.3. Let n E N, A = Ai,..., A& be a partition of n and
em= ^ 2 Xii'' 'Ximi rn>l i\ <••• ^im with e0 = 1. Then the elementary symmetric function corresponding to the integer partition A is
e A i e A2 * * *
Some small examples of elementary symmetric functions of partitions of 3 in 3 10
variables are shown below to clarify this definition:
Example 2.3.
Ci,i,i(xi, x2, £3) = ejeiex = (#1 + x2 + x3)3
e2,i(a:i, ^2,2:3) = e2ei = (x-lX2 + Xix3 + x2x3)(xt + x2 + x3)
e3 (xux2,x3) =
The majorizations we study are not of the symmetric functions themselves but of the term-normalized versions of them. The definition of which can be found in
[3] as shown below:
Definition 2.4. To each symmetric function g\(x) we will associate a term-normalized symmetric function G\(x) by
<*(.)- »<*> 5a(1, •••,!)
With this definition we can see the term-normalized elementary symmetric func tions from our examples are 11
Example 2.4.
+ £2 + ^ 3 3
{XIX2 + 3-1^3 + X2X3)(x i + X2 + X3) 9
E 3{x u x 2, x3) = ^1X2X3
As stated in Theorem 1.1, a partial order can be defined on these term normalized symmetric functions. Each inequality is determined based on the partial ordering that is already in place on the integer partitions. This ordering is sometimes referred to as dominance and is defined here:
Definition 2.5. A partial order defined on Par(n) for each n e N called the dom inance order or majorization order and denoted ■< where two partitions A and // of the same integer satisfy
/x ^ A •<=>• [i\ + H2 + • • ■ + fit ^ Ai + A2 + • • • + Aj, Vi > 1.
E xam ple 2.5. We can see that in Par( 4), 2 ,1,1 -< 4 because 2<4, 2 + l < 4 + 0, and 2 + 1 + 1 ^ 4 + 0 + 0
So in Par(4),
1,1,1,1 ^ 2, 1,1 ^ 2,2 ^ 3 ,1 < 4 12
and in Par (3),
1,1,1 d 2,1 < 3.
As stated in Theorem 1.1 the partial order on the term-normalized elementary symmetric functions is already complete and is described as in [3] as follows:
E\(x) > Ep(x), x > 0 <=$> A -< n
Example 2.6.
Ei,i,i(x) > E2,i(x) > Ez{x), \/x > 0
Example 2.7. For a less trivial example we consider the partitions of 6. The ordering on these partitions more clearly shows why this is a partial order and not a total order. The Hasse diagrams for the partitions of 6 and corresponding term normalized elementary symmetric functions are shown here, with the most dominant element on top: 13
6 i,i, 1,1,1 | 1 5 ,1 ^2,1,1,1,1 I 1 4 ,2 ^2,2,1,1 \ / \ 4,1,1 3,3 Es, 1,1,1 £*2,2,2 \ r \ / 1,2,1 ^3,2,1 /\ / \ 3,1,1,1 2,2,2 £ 4,1,1 £ 3,3 \ \ / 2, 2,1,1 £ 4,2 | i 2,1,1,1,1 E51 i i’ 1,1,1,1,1,1 E6
Notice that 4, 2 y 4 ,1,1 and 3,3 as described in definition 2.5. However, 4 ,1,1 and 3,3 are not comparable. This is because 4 > 3 but 4 + 1 < 3 + 3.
In Section 3 these results are verified for some examples using semidefinite pro gramming. Before we get to that however we must define the rest of our symmetric functions of interest. Again our definitions come from [10].
Definition 2.6. Let n £ N, A = Ax,..., A*, be a partition of n and
n pm = Y ^ x?i r n > l i with po — n. Then the power sum symmetric function corresponding to the integer 14
partition A is
P \= P \iP \2 - --PXk-
Some small examples of power sum symmetric functions of partitions of 3 in 3 variables are shown below to clarify this definition:
E xam ple 2.8.
Pl.l,lfclj x2i xa) = PlPlPl = (®1 +X 2+ X3)3
P2,l(xi,X2t X3) = P2P1 = (x\ + x\ + xj)(xi + X2 + X3)
p3(x I,x2,x3) = xl + xl + xl
The corresponding term-normalized symmetric functions are thus
(x\ + xl + xl)(xi + x2 + x3) P2, i(xi,x2,x3) = 9 x \+ x \ + xl P3(xux2,x3) 3
The partial order for the term-normalized power sum symmetric functions was also fully proven in [3] as shown below. 15
P\(x) < Pn(x), x>0 4=$ A -< n
Example 2.9. -Pi,i,i(x) < P2,\{x) < Pz(x), Va; > 0
Remark. Notice here that the inequality is reversed compared to the partial order on the term-normalized elementary symmetric functions. As stated in Theorem 1.1, there are two more symmetric functions that have incomplete results which we will define as they are in [10] below.
Definition 2.7. Let n € N, A = Ai,..., A*, be a partition of n and
^1 with ho = 1. Then the complete homogeneous symmetric function corresponding to the integer partition A is
hx = h\1 h\2 ■ ■ ■ hXk.
This definition is very similar to that of the elementary symmetric functions with the slight difference in the summation of hm. Some small examples of complete homogeneous symmetric functions of partitions of 3 in 3 variables are shown below to clarify this definition: 16
E xam ple 2.10.
hi,i,i(xu x2, xz) = hihihi = (xi + x2 + x3)3
h2, i(xi,x2,x3) = h2hi = (xl + xix2 + x1x3 + xl + x2x3 + xl)(xi+x2 + x3)
h3(x i,x2, x3) — xf + x\x2 + x\x3 + xixl + X\X2X3 + xxx\ + x\ + x\x3 + x2x\ + X
The corresponding term-normalized symmetric functions are thus
t t / \ i x i + x 2 + x 3 ' 3 Hl,l,l(Zl,X2,X3) 3 (x\ + XiX2 + X\X3 + x\ + X2X3 + X3)(xi + x 2+ x3) H2,l(xUX2,X3) = 18 U (~ ~ _ \ _ X1 + xix2 + xix3 + xxx\ + XiX2X3 + nxl + x\ + x\x3 + x2x\ + xjj ^ 3 { X i ,X 2,X 3) = — ------
In [3] the following is proven,
H\{x) < H^x), x > 0 4== X < fj, with the conjecture that the converse is also true.
The fourth and final symmetric function listed in Theorem 1.1 is the Schur function defined here:
Definition 2.8. Let n € N, A = Ai,..., A„ be a partition of r and x G Mn, the 17
Schur polynomial S\(xi, • • • > xn) is the ratio between
A i+ n —1 ^ A i + n —1 A i+ n — 1 x 1 n A2+T1—2 ^ 2+n—2 A2+7T'— 2 X1 x 2 det n
rr^n An X1 X2 X and the Vandermonde determinant
n ~ **)■ l<7 It is important to notice that the number of components in the integer partition must be equal to the number of variables in these polynomials. Some small examples of Schur symmetric functions of partitions of 3 in 3 variables are shown below to clarify this definition: Example 2.11. «1,1, \{XI,X2 ,XZ) = XXX2 X3 S2,\{Xi,X2 ,Xz) = ( x i + x 2 )(x1 + x 3 )(x2 + X 3 ) s 3( x i, x2, x3) = x\ + x\x2 + x\x3 + X\X22 + XiX2 X3 + X1.X3 + x 2 + x\x3 + x 2 x\ + xl The corresponding term-normalized symmetric functions are thus 18 s l,l,l(xi,x2,x3) XlX2X3 (Xi + x2)(xx + x3)(x2 + .-r3) S2< i{xi,x2,x3) 8 xf + x\x2 + x\x3 + XiX 2 + XiX2X3 + X\X3 + x2 + x2x3 + x2x\ + £3 S3{x i,x 2,x3) 10 2.2 Sum of Squares Consider the multivariate polynomials f(x\,..., xn) and g{x\,.... xn) in the polynomial ring ]R[xi,..., x„]. In order to show that g(x 1,..., xn) < f(x 1,..., xn) we need to show that f(x 1,..., xn) — ^(xi,..., xn) > 0. In other words, we need to show that this multivariate polynomial is non-negative. Definition 2.9. A multivariate polynomial /(x 1,... ,x„) is said to be nonnegative if /(xi,...,xn) > 0 V(xi,...,x„) eR n One way to do this is by showing that /(x 1,..., xn) can be decomposed into a sum of squares. Definition 2.10. A polynomial /(x) € M[x]n>2d with real coefficients is a sum of squares (SOS) if there exist <71, € K[x]„)(i such that 19 f (x) = qk(x) > °> € R fc=i Note, here R[.x]ni2fi represents the polynomials in n variables of degree 2d. Example 2.12. f (x ) = ^1 + 2X iX 2 + .Tj + x lx 2 = (X1 + ;c2 )2 + ( ^ l - ^ ) 2 As shown in [1], there are methods in semidefinite programming to determine if a given polynomial can be decomposed into a sum of squares. The inequalities that we look at are of the form G\{x) < G^x), Vx > 0 So checking that G;i(x) — G\{x) is an SOS would confirm the inequality for all x in M". This could cause issues if the inequalities do not hold for negative values in x so we must adjust our problem slightly. We do this by simply changing each Xi to a new yf. For instance the function f(x) = x\ + XiX2 + x\ would become f ( y 2) = y$ + v \ y \ + y i Proposition 2.1 . Checking that f(y2) is nonnegative for all y € Rn would show that f (x) is nonnegative for all x G R >0 since every nonnegative Xi can be written 20 as y\ where yt — y/xi. To clarify, our method is outlined as follows, using f{x) and g(x) as arbitrary multivariate polynomials in R[o;]n: f(xi, x2,..., xn) > g{xi, x2, • •., xn) Mxi > 0 f{xi, x2,.. •, x3) - g(xi, ®2, • • • i x3) > 0 Vxj > 0 f(yi,yl,•• • > - £(2/i>• • •>y«) > 0 % e r F(yuV2,---,yn)> o Vj/jSR where F(y) = p(y) - g(y). Example 2.13. If we wish to check the inequality #1,1,1 (*11 x2, x3) > E3(xux2, x3), x > 0 , given by [3], we adjust our inequality as shown below: 21 .(X\,X 2 , X 3 ) > E 3(X1,X2,X3), V x > 0 x 2 + X%\ 3 > X i X2X3 , > 0 3 J/ 3 X1 +X2 + x3 > - 0 , \ / X i > 0 3 y\ + vl + y\ 1 2 2 2 > V j / j € R F(y) = 1 - 2/ i » 0 , To show F(y) > 0, it is sufficient to show that F(y) is an SOS. To do this we will be using methods outlined in [1]. Consider the multivariate polynomial F(y) in M[y]n,2d- We wish to decompose F(y) into a sum of squares, i.e. F(y) = Qk(y)2- Notice that the leading coefficient of each q\ must be positive, and thus there can be no cancellation of the highest degree term. Therefore each q^ must be at most degree d. Let [y]n,d represent the vector containing all possible monomials of qk without coefficients. Example 2.14. If each qkiy) is in then [y]l 3 = 1, yi, V2, y3 , yfi y m , y m , y'i, v m , yh v\y2> 2/12/3, y w l 2/12/22/3, y iy l yf, yfy3, y2yb vl 22 Using this vector we can rewrite the q^s as l ? i (y) 9 2 ( y ) y i = v Qm(y)_ A where V € Erox20 and its /c’th row contains the coefficients of the polynomial qk- Proposition 2.2. The number of monomials in n variables of up to degree d is M o using this new notation for the r/^’s, we can also rewrite F(y). 771 F(y) = J2qk^2 k= 1 = (V[yU)T {V[yU) = [y]Tn,iVTV[yU = [y]n,lQ[y]n,l where V £ Kmxl and Q is the square matrix VTV. This Q matrix is crucial to our work and to understand why, we must first cover the following definitions: Definition 2.1 1 . A square matrix A is called a symmetric matrix if and only if A = AT. The set of symmetric matrices of size n x n is denoted §n. 23 Symmetric matrices have many useful properties. In particular they have a decomposition that is crucial to our research. This decomposition is defined in the following theorem from [6]: Theorem 2.3. Any symmetric matrix A G R n,n can be factored as A = UAUt , where U G R n,n is an orthogonal matrix, and A is a diagonal matrix containing the eigenvalues Ai,..., An of A in the diagonal. All these eigenvalues are real numbers. Thus, eigenvectors can be chosen to be real and to form an orthonormal basis. The columns u\,...,un of U are indeed the eigenvectors of A, and satisfy Aui = AjUj, i = 1,..., n. This factorization is known as the spectral decomposition for symmetric matrices. Definition 2.12 . A symmetric matrix A is said to be positive semidefinite if and only if vTAv > 0 , Vt; 6 Mn. We denote this as A y 0. The set of positive semidefinite matrices of size n x n is denoted S” . Proposition 2.4 ([6]). Let A G §n with eigenvalues Ai(A),..., An(A). Then A h O <^> Xi(A) > 0, i = 1,... ,n. 24 We can see clearly that Q = VTV is a symmetric matrix, as Qt = (VTV)T = VT(VT)T = VTV = Q which leads us to the following theorem from [1] Theorem 2.5. A multivariate polynomial F(y) E M[y]n,2d is a sum of squares if and only if there exists Q E §( ■* ) satisfying F (y) = [y]Jn+djQ[y](n+dy Q y o. Proof. = > Let F(y) be a polynomial with real coefficients in n variables of degree 2d, i.e. F(y) E IR[y]„,2d- Let F(y) have a sum of squares decomposition F(y) = Yli=\ Qiiu) where qt E i = l,...,fc. There exists a vector [q,] E r( ^ ) such that [y}T{n+d) [qi\ = qi for each i. Let V1 be the (nJd) x k matrix with [q7] being column i for i = 1,..., k. Clearly [y]T„+d\ V'7’ is a k dimensional row vector with qt as its ith V d ) component. Thus [y]T^+d^VTV[y\(n+d'j = Y.i=i ^iv) = F(v)- Let Q = VTV and let vTQv = vTVTVv — tTt where t = Vv E Rfc. Since t is just a vector, tTt is the dot product of t which is tTt = t\ -\------1- 1\ > 0. Therefore, if F(y) E M[y]n,2d 25 is an SOS, then there exists a Q € §( <* ) satisfying ^ (y ) = [y\]n+^Q[y\(n+dy Q y o. Let Q € §( ) such that F (y) = M[n+^Q[y](n+d), Q h 0. Using Theorem 2.3, we can decompose Q as UAUT. Thus [y](«+d)^3[y](n+d) A t/[y](n+d) k 2=1 where A; < (n^d) is the number of eigenvalues of Q, A* is an eigenvlaue of Q, and ut is the ith column of U and corresponding eigenvector to A, for i = 1,..., k. Notice that [y]T(n~dd)ui is a polynomial qt{y) G R[y}( n+d^ and. since Q is positive semidefinite, we know by corollary 2.4 that A* > 0 for i = 1,..., k. Therefore F(y) is an SOS; k F(y) = £ ( ^ a;« )2. 2= 1 □ 26 The existence of such a matrix can be determined using semidefinite program ming and thus, so can the existence of a sum of squares decomposition. 2.3 Semidefinite Programming Methods To understand semidefinite programs, we must first know what an optimization problem looks like, as defined in [6]. Definition 2.13. A standard form optimization problem is a problem of the form: p* = minimize fo(x) subject to fi(x) < 0 i = 1,..., m where • the minimization is over x; • x € Rn is the vector of decision variables; • /o : Mn —> M is the objective function or cost; • fi : Rn —> R, i = 1,..., m, represent the constraints', • p* is the optimal value. In this notation ’’subject to” will often be shortened to s.t.. 27 Example 2.15. min .9x1 — Axix2 + -6x1 — 6Axi — ,8x2 : — 1 < x\ < 2,0 < x2 < 3 which can be written in the standard form: p* = min /o(x) = .9x1 ~ Axix2 + — 6.4xi — .8x2 s.t. /i(x) = —X\ - 1 < 0 f2(x) = Xi-2< 0 f3(x) = —x2 < 0 f4(x) = x2- 3 < 0 Next we will cover a few more definitions from [6] in order to fully develop the tools we need for semidefinite programming. Definition 2.14. A function from / from R" —)• R is linear if and only if it is preserved under the scaling and addition of input arguments, i.e. Vx E Mn and a € R, fifitx) = a f{x) ; V.Ti, x2 G Rn, f(xi + x2) = f(x i) + fix 2) . A function is affine if it is a linear function plus a constant. As shown in [6], a function is affine if and only if it can be expressed as f(x) = 28 aTx + b. Example 2.16. /(x) = 3xi + 5x2 — 7 is affine as \ Xi f{x) = 3xi + 5x2 - 7 = ^3 5^ \X2J Next we define what it means for a set or a function to be convex: Definition 2.15. A subset C C M" is said to be convex if it contains the line segment between any two points in it. i.e. Xi, X2 £ C, A £ [0, 1] v Axi "I- (1 — A)x 2 £ C. Definition 2.16. A function / : Rn —» R is convex if the domain of / is a convex set, and for all xi,x2 £ dom(f) and all A £ [0,1], /(Axi + (1 - A)x 2) < A /( x i) + (1 - A ) /( x 2) Perhaps the following image of a convex function from [6] will provide some clarity on these definitions 29 Figure 2.1: Graph of a Convex Function With these definitions, we are now able to define a convex optimization problem as in [6]. Definition 2.17. An optimization problem of the form p* = min f0(x) s-t. fi(x) < 0 i = 1, ... ,771 hk(x) = 0, k = l,...,q is called a convex optimization problem if • the objective function /o is convex; # the functions defining the inequality constraints, fi(x) < 0, i = 1, are convex; 30 • the functions defining the equality constraints, hkix) = 0, k = 1,... ,<7, are affine. The last tool that we need to develop, is form of constraints that we will use to define a semidefinite program. Again the following definition is from [6] Definition 2.18. A linear matrix inequality (LMI) in standard form is a constraint on a vector of variables x G Rm of the form m F(x) = F0 reiFi t 0, i=1 where the n x n coefficient matrices F0, Fi,..., Fm are symmetric. Finally we have everything we need in order to define the program we use to determine whether our polynomial F(y) is a sum of squares or not. Below is the definition of a semidefinite program (SDP) as defined by [6]. SDP’s are themselves convex optimization problems and can generalize in particular linear programs and convex quadratic programs. Definition 2.19. A semidefinite program (SDP) is a convex optimization problem, where one minimizes a linear objective function under an LMI constraint, i.e. p* = min cTx s.t. F(x) >z 0 31 where m F(x) = F0 + J2^F i, i= 1 Fi, i = 0, 1,..., m are given nxn matrices, c € is the given objective direction, and x G Rm is the optimization variable. SDPs can also be written in a generic conic-form as mm trace((7X) s.t. X >- 0 trace(AiX) = bt, i = 1,..., m where Ai is a symmetric matrix for i = 1,..., m. Now that we have all the definitions that we need, let us look at the inequality Ei,i(x1 ,x2) > E2(xux2) Example 2.17. In order to show x2) > E2{x\,x2) holds for all x > 0, we will show that Or more simply, F(y) =yt~ 2y\y\ + y\ > 0, Vy, G R 32 By Theorem 2.5, if we can rewrite F(y) as F(y) = [y\l,2Q\y\2,2, QtO, then F(y) is an SOS. 33 Chapter 3 SOS Decompositions In this chapter we describe, in full, the method we use to see if a multivariate polynomial is a sum of squares. As we work through one major example we see where our first reductions can be made to the problem size. The end of this chapter lists our results, showing that our method can be used to verify the inequalities from Theorem 1.1 in several cases. Finally we notice further symmetries that lead us to our next reduction to our problem size. Referring back to example 2.17, let qa,b-,i,j represent the component of Q that will be multiplied with the monomials y^y\ and y\yJ2- Thus Q is written as: 34 <7l,0;0,0 <7l,0;l,0 <7l,0;0,l <7l,l;0,0 9l,l;l,0 9l,l;0,l 9l,l;2,0 9 l,l;l,l 91,1:0,2 ?0,2;0,0 <70,2;1,0 90,2;0,1 ?0,2;2,0 Qo,2-1,1 90,2;0,2 Notice that the equation F(y) = y\ — 2y\y\ + y2 = [y]^2Q[y]2,2 gives us many restrictions on what Q can be. For instance we know that the coefficient of the yf term in F(y) must be 1. The only way we can get this from the right hand side of our equation is if q2,0,2,0 — 1- Similarly we notice that we will have many constraint equations equal to zero. In fact, every coefficient of a monomial less than degree 4 will be zero. This leads us to the dangerous assumption that components of Q such as <7i,i;i,o = 0. This is not necessarily the case however as our constraints state that 2<7i,i;i,o + 2<72,o;o,i = 0 as apposed to simply qi,1,1,0 = 0. Notice that the symmetry of Q allowed us to let <7i,i;i,o + Even though our assumption is unjustified, the constraint is, in fact, satisfied if each term is zero. This observation led us to the following idea: If we can show 35 that the block of Q containing only components corresponding to terms of order 4 is positive semidefinite, does that mean all of Q is positive semidefinite assuming the rest of Q contains zeros? This idea is the inspiration for the lemma below. Lemma 3.1. The block matrix Q = with the matrix X in the bottom right hand comer and zeros everywhere else, is positive semidefinite if and only if X is positive semidefinite. Proof. ■ Let X G §" be a positive semidefinite matrix, Q be the m x m matrix with X in the bottom right corner and zeros everywhere else, and v G Mm. Then i 1 ---- o 0 0 -H v t Q v = v 1 II = {v'yxv' __ * o * 1 1 i i—"* where v' is the vector containing only the last n components of v. Since X is positive semidefinite, (y')TXv' > 0, and since (v')T Xv' = vTQv and v was arbitrary, QtO. Hence if X >z 0, then the block matrix >- 0. 36 For the other direction of this proof, let the block matrix Q be positive semidefinite. Then vTQv > 0 for all v in ]Rm. Let w be a vector in Rn and w' be the vector in Rm with the components of w as its last n components and zeros everywhere else. Then (■w')TQw = wTXw and since Q >z 0, (w')TQw = wTXw > 0. Since w was arbitrary, 0 0 X is a positive semidefinite matrix. Therefore if the block matrix h 0, then X >- 0. Hence the lemma holds. □ Using Lemma 3.1, all we need to show is that the matrix containing the compo nents corresponding to monomials of degree 4 is positive semidefinite. Our problem is reduced to just a quarter of the original matrix shown here: <72,0;2,0 Q’ = <7i ,1;2,o 9 i,i;i,i <7i,1;0,2 Q0,2;2,0 <70,2;1,1 90,2;0,2 Now the only constrains remaining are those for the coefficients of the terms of order 4. This new linear space is shown below: Q2,0;2,0 <72,0;1,1 Q2,0;0,2 y \ y \ y m v i Ql,l;2,0 9l,l;l,l 9l,l;0,2 y m = F(y) = y\ - 2y\yl + y\ 90,2;2,0 ?0,2;1,1 90,2;0,2 to to ___ i 1 37 2^20,11 = 0 2^20,02 + <711,11 = —2 2^11,02 = 0 <702,02 = 1 Recall An SDP is of the form min trace(CA) s.t. X y 0 trace(AjX) = &,, i 1 38 So to show that Q is positive semidefinite, we let X = O' 1 0 0 0 1 0 Al = 0 0 0 A2 = 1 0 0 0 0 0 0 0 0 h = lL 62 = 0 0 0 1 0 0 0 A3 = 0 1 0 a 4 = 0 0 1 1 0 0 0 1 0 b 3 = - 2 64 = 0 0 0 0 0 0 0 0 0 1 with the freedom to choose the cost matrix C as we please. For this thesis we chose to use the trivial identity cost matrix. The motivation for this choice is beyond solely the simplicity of the matrix and will be explained further. Finally we have everything we need to solve our semidefinite program. The program we use to run our semidefinite optimization problem is CVX, a package for specifying and solving convex programs in Matlab [8] [7]. Our program yields the 39 following optimal solution: 1 0 -1 Q ' = 0 0 0 b 0 -1 0 1 Using linear algebra, we can decompose this matrix into a form that clearly shows the sum of squares decomposition of F(y). Using the spectral decomposition from Theorem 2.3, we can decompose Q' into 0 - y f l - f i 0 0 0 0 1 0 1 1 Q' = UAUt = 1 0 0 0 0 0 O _ 1 _ 0 0 2 0 0 - s f i a . _ 1 1 Showing that F(y) = vl y m yl Q' y\ v m yl y\ — Zym + 2/2 - 2 \y\ + \J\yl) ^°> vy GR2 Thus we have shown that F(y) = y\) — E^yX- yf) > 0 Vy € R2 and therefore that the inequality £q,i(xi, X2) > ^(xi, X2) Vxi,x2 > 0. We were able to use our method to verify all the inequalities from Theorem 1.1 for n = 2, r = 2 shown here: 40 E.Avlvl) - Mvlvl) = vi - 1 = j (-fly! + ;y22) / , x 2 .2 „,2\ D / 2 „.2\ _ 2/l yiV2 , vi _ 1 I / 1 ,2 , 1..2 W ,y 22)-Pi,i(yU 2) = T " T + 4^4 V "2 yi + 2y2 2 .2 „,2\ rr /„,2 „,2\ _ Vl y\y\ . Vt _ ^ ( I ^ 2 , 1„.2 H2{yW2) ~ H M .vi) = 12 _ (3 12 = 6 \ \J ~2 2 J S2(y?, y|) - 5i,i(yi, yf) = Vl - 2yx2y| + y^ = 2 f J ~ ^ V i + 7^ 2] In these cases, our SOS were each simply a sum of one square term. For the following example, the solution was not as sparse; E xam ple 3.1. We wish to show Ew (yl,y%,yl) > E3(yl,y%y$), Vy e M3 and after running our semidefinite program as we did above, we get the following SOS decomposition: The number of square terms that we have in our decomposition is equivalent to the number of eigenvalues of Q'. This is the idea that lead us to use the triv ial cost matrix C = I. For if we use the identity, our objective function becomes min trace(Q') = Y ^ i= 1 We had hoped that by minimizing the sum of the eigen values of Q we would reduce the number of terms in our decomposition. This was not the case however, as in example 3.1, our solution had more terms. The usefulness of this choice in C was not lost however as it lead to the two symmetric polynomials shown in example 3.1. With this observation, we turned to Representation theory in search of a method to abuse the symmetries of our semidefinite program. It will be shown that the natural symmetries in our semidefinite programs leads us to massive reductions to our problem sizes. We believe that this reduction is necessary to continue to test inequalities of larger size. The process is explained in the following chapter. 42 Chapter 4 Reduction As stated in Chapter 3, we now utilize Representation theory to manipulate our problems so that they are far simpler. We first review some basics of representation theory of finite groups and then show how we can apply these concepts to our problem. We show that for even small cases, such as n = 3 and r = 3, the problem size can be reduced from 55 variables to just 13. For larger n and r, this reduction may be necessary for computation. 4.1 Representation Theory In order to utilize representation theory to reduce the size of our optimization problems we must first understand the following definitions from [9]. Let Matd be the set of all d x d matrices with complex entries and GLd be the 43 complex general linear group of degree d GLd — {X : X = (xij)dxd G Matd, X is invertible} Definition 4.1. A matrix representation of a group G is a group homomorphism X : G -> GLd. This means X(gh) = X(g)X(h)\f g, h G G. Next we look at some examples of representations on the symmetric group S3. Example 4.1. The Trivial Representation of S3: X : S3 -> GLi X(g) — 1 \/g £ S3 Example 4.2. The Sign Representation of S3: X : S3 -»• GLi X(g) = sign(g) Mg G S3 where sign(y) is the function yielding — 1 if g is an odd permutation, and 1 if g is an even permutation. 44 Example 4.3. The Defining Representation of S3: X : S3 -> GL3 1 0 0 0 1 0 X(e) = 0 1 0 * ( ( 1 ,2 ) ) = 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 * ( ( 1 ,3 ) ) = 0 1 0 * ( ( 2 ,3 ) ) = 0 0 1 1 0 0 0 1 0 0 0 1 0 1 0 *((1,2,3)) = 1 0 0 *((3,2,1)) = 0 0 1 0 1 0 1 0 0 Since we are using matrices, which correspond to linear transformations, we can think of representations in a similar manner. Definition 4.2. Let V be a finite dimensional vector space and G be a group. Then V is a G-module if there is a group homomorphism p \ G GL(V), 45 where GL(V) is the set of all invertible linear transformations of V to itself, called the general linear group of V. This means that V is a G-module if there is a multiplication, gv, of elements of V by elements of G such that 1. gv E V, 2. g(cv + dw) = c{gv) + d(gw), 3. (gh)v = g(hv), and 4. ev = v for all g, h £ G; v, w E V; and scalars c, d E C Example 4.4. Let 1/= spanjyf, yfy2, y\yz, y\y\, y m y 3, Viyb y|> 2/22/3, 2/22/3* 2/3}• We will define a group action by permuting the indices of the basis elements of V. For example (1,2)y^y3 = y^ys- This gives us ten matrices in the basis of V similar to the one shown bellow; 46 0000001000 0001000000 0000000100 0100000000 0000100000 * ((1 ,2 )) = 0000000010 1000000000 0010000000 0000010000 0000000001 Notice the 1 in column 3 row 8 comes from the fact that (1, 2) takes our third basis element y\yi to our eighth basis element y12/3- Just like positive integers, G-modules can be reduced into their irreducible com ponents. Definition 4.3. Let V be a G-module. A submodule of V is a subspace W that is closed under the action of G, i.e., w € W gw e W \/g e G. Example 4.5. Consider the defining representation from example 4.3. The sub- 47 space spanned by is closed under the action of S3 and is thus a one dimensional 1 submodule of the ^-module corresponding to the defining representation. Example 4.6. The subspace of V from example 4.4 spanned by 2/12/22/3 is a submodule of V. If we continue to find submodules, we eventually break down our G-modules into its smallest components Definition 4.4. A nonzero G-module V is reducible if it contains a non-trivial submodule W. Otherwise, V is said to be irreducible. Irreducible representations are difficult to find so we will need to develop a few more tools from [9] to discover them. We will start with the definition of the direct sum of two subspaces in order to understand how G-modules are broken up into their submodules. Definition 4.5. Let V be a vector space with subspaces U and W. Then V is the (internal) direct sum of U and W, written V = U © W, if every v E V can be written uniquely as a sum V = u + w, u E U, w EW. 48 If X is a matrix, then X is the direct sum of matrices A and B, written X = A® B, if X has the block diagonal form • a 0 B Theorem 4.1. (Maschke’s Theorem) Let G be a finite group and let V be a nonzero G-module. Then V = VK(1) © W (2) © • • • © W (fc), where each is an irreducible G-submodule of V. Corollary 4.2. Let G be a finite group and let X be a matrix representation of G of dimension d > 0. Then there is a fixed m,atrix T such that every matrix X (g), g 6 G, has the form x (1)Go o 0 0 X&(g) 0 TX{g)T -l o ... where each X ^ is an irreducible matrix representation of G. This means that every representation of a finite group having positive dimension 49 is completely reducible. Example 4.7. Consider the defining representation from Example 4.3. The S3- module corresponding to this representation is 1 1 s 1 \ :1 0 1 O ..... V = span < 0 1 1 ) 0 0 1 , 0 1 1 \ 1 > / \ 1 We saw in Example 4.5 that the space W ^ = span < 1 > is a submodule of V r-H 1 < 1 > with the corresponding representation A (1) : S 3 ->• G L i A (1)( According to Theorem 4.1, we can find a submodule U such that V = © U. As found in [9], this space / 1 1 \ rH 1 - 1 U = span < 1 5 0 > O 1 < 1 1 / 50 So if X is the defining representation of S3 and we let 1 - 1 - 1 T = 1 1 0 1 0 1 then X where X ^ : S3 —»• GL2 is the representation corresponding to U. The matrices for each element of S3 are shown below. 1 0 0 1 0 0 X(e) = 0 1 0 , *((1,2)) = 0 -1 -1 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 * ((2 ,3 )) 0 0 1 0 -1 -1 0 1 0 1 0 0 1 0 0 *((1,2,3)) = 0 - 1 - 1 , *((3,2,1)) = 0 0 1 0 1 0 0 - 1 - 1 We see that X ^ is just the trivial representation from Example 4.1 which is clearly 51 irreducible as it is of degree one. X (2) is also an irreducible representation of S3. Unfortunately, two representations could look different from one another when they are essentially the same representation. To remedy this, we have the following definitions: Definition 4.6. Let V and W be G-modules. Then a G-homomorphism (or simply a homomorphism) is a linear transformation 9 : V —> W such that = 90{v) for all g G G and v G V. We also say that 9 preserves or respects the action of G. We can think of this definition in terms of the matrix representations X and Y corresponding to the given G-modules. Let T be the matrix of 9 in terms of the two bases of these representations. Then the homomorphism property becomes simply TX(g) = Y(g)T VgeG. Definition 4.7. Let V and W be modules for a group G. A G-isomorphism is a G-homomorphism 6 : V —> W that is bijective. In this case we say that V and W are G-isomorphic, or G-equivalent, written V = W. Otherwise we say that V and W are G-inequivalant. Again we can think of this definition in matrix terms. If 9 is a bijection then our 52 matrix T invertible. So, the representations X and Y are G-equivalent if and only if there exists a matrix T such that Y{g) = TX(g)T~1 Vg G (7. In other words we will consider two representations to be equivalent if they are the same under a change of basis. Now that we can differentiate between representations, we introduce the follow ing tools from [9] that we can use to organize and classify our representations. Definition 4.8. Let X(g), g G G, be a matrix representation. Then the character of X is X(g) = trace(X(g)), Example 4.8. If we look again at the defining representation from Example 4.3, we see it has the following character e (1,2) (1,3) (2,3) (1,2,3) (3,2,1) 3 1 1 1 0 0 E xam ple 4.9. Similarly, the characters for the two subrepresentations that we found in Example 4.7 are shown here e (1,2) (1,3) (2,3) (1,2,3) (3,2,1) x(1) 1 1 1 1 1 1 x(2) 2 0 0 0 -1 -1 53 This definition gives several special properties about characters that we can use to our disposal. Proposition 4.3. Let X be a matrix representation of a group G of degree d with character x- Then 1- X(e) =d 2. If K is a conjugacy class of G, then g ,h e K ==> x(g) = x(h)- 3. IfY is a representation of G with character^, then x^Y x(g)=^(g) VgcG Clearly properties 1 and 2 hold for our characters in Example 4.9. These prop erties allow us to efficiently organize the representations of a given group Definition 4.9. Let G be a group. The character table of G is an array with rows indexed by the inequivalant irreducible characters of G and columns indexed by the conjugacy classes. The table entry in row x and column K is Xk Example 4.10. Here is the character table for S3. Note that these are all repre sentations we have already seen, is the trivial representation, xstgn is the sign 54 representation from Example 4.2, and x® is the irreducible representation that we found in Example 4.7. x ^ is als° called the standard representation. Ke K(a,fc) K(a,6,c) x(1) 1 1 1 ^Sign 1 -1 1 x(2) 2 0 -1 here Ke is the conjugacy class of the identity element, is the conjugacy class of two cycles (reflections), and K ^ c ) is the conjugacy class of three cycles (rotations). As shown in [9], we can define an inner product on characters of representations of the symmetric group to determine which characters are irreducible. Definition 4.10. Let x and (f> be characters of a permutation group Sn. We define the inner product of x and (f> to be (x> 0) = T7T7 X ] \K \XK(t>K, I n I K where the sum is over all conjugacy classes of Sn Example 4.11. (xm,xm ) = = g(l(l(l)) +3(1(1)) + 2(1(1))) = 1 (xm ,x"sn) = i(l(l(l)) + 3(l(-l)) + 2(1(1))) = 0 (X"” ',X,2)) = i(l(l(2)) + 3(-l(0)) + 2(l(-l))) = 0 55 This definition along with the following theorem allow us to differentiate among different characters. The following relations are from [9]: Theorem 4.4. (Character Relations of the First Kind) Let x and 0 for x ^ 4> 1 for x = Corollary 4.5. Let X be a matrix representation of G with character x ■ Suppose X “ © m2X (2) © • • • © mkX {k\ where the X ^ are pairwise inequivalant irreducibles with characters x ■ Then 1. x = + m2x(2) H------h r n .k X ^ . 2- (x,x{j)) = mj f or all j. 3- (x, x) = m\ + m\ + • • • + m\. 4. X is irreducible if and only if (x?x) — 1- 5. Let Y be another matrix representation of G with character 4>. Then X * Y « x(g) = 4>{g) for all g £ G. 56 Example 4.12. Let X be the defining representation from Example 4.3. We saw that in Example 4.7, X = X ^ © X (2) under a change of basis. We also have the corresponding characters x, X ^\ X('2> from Examples 4.8 and 4.9. Now we can make the following computations: 1- X = i(x(1) + l(x(2) 111111 + 1 2 0 0 0 -1 2- 3. (x, X) = §(1(3(3)) + 3(1(1)) + 2(0(0))) = 2 = l 2 + l 2 4- Example 4.13. For a more complicated example, let us decompose the representa tion X we defined in Example 4.4. After computing all six representation matrices we see that X has character x shown here: K e K(a,b) K(a,fc,c) x (1) 10 2 1 We also have the characters for each irreducible representation of S3: K e K(a,&) K(a,i>,c) x(1) 1 1 1 ■y/sign 1 -1 1 x(2) 2 0 -1 First, we can see that our representation is reducible by checking its inner product with itself, (x, x) = 19 7^ 1. Since x is reducible, we will use corollary 4.5 to see how it can be decomposed into irreducibles. 57 (x,xm) = i(l(10(l)) + 3(2(1)) + 2(1(1))) = 3 (x,x"n = i(l(10(l)) + 3(2(-l)) + 2(1(1))) = 1 (x,xm) = j(l(10(2))+3(2(0)) + 2(l(-l)))=3 So now we can see that X = 3x(1) + lx sign + 3x(2) Using this brief introduction to representation theory, and in particular Example 4.13, we will simplify our semidefinite programs into much smaller problems. 4.2 Symmetry Adaptation The invariance under the symmetric group that our semidefinite programs have, allow us to use representation theory to simplify our problems. The requirements and method for how to do so are outlined in [5] as shown in this section. To begin, let us recall the definition of a semidefinite program. 58 An SDP is of the form min trace (C A) s.t. X y 0 trace(AjA) = £>*> i = 1,..., m For the examples in this section, we will use the running example of the semidef inite program used in 3.1. min trace (Q1) s.t. Q' y 0 W W = where [y\ = y\ y\y2 yfy3 yiy2 yiy2y3 yxy\ y\ y%y3 y2y\ y\ The first requirement that we need in order to reduce our problems using rep resentation theory is that a representation a maps the cone of positive semidefinite matrices to itself, i.e., <7fo)(s;)cs; V3 6 G For our problem, we use the representation °(9)(X) '■= p{d)TXp{g) 59 where X G §10, g G S3, and p is the representation defined in Example 4.4. Proposition 4.6. Let a(g)(X) p(g)TXp(g), where X G § 10, g G S3, and p is the representation defined in Example 4-4 satisfies the first requirement cr(gf)(S") C §” VgGG. Proof. Let X G §+, a(g)(X) := p(g)TXp(g), and v G R 10 First note that (a(g)(X))T := (p(g)TXp(g))T = p(g)TX Tp(g) = p(g)TXp(g) = Thus o{g)(X) G S10. Also, vTa{g)(X)v = vT p(g)T X p(g)v = {y')TXv' where v' = p(g)v. Since X >z 0, {y')TXv' > 0 Vi; G M10. Hence cr(g){X) X 0 Mg G S3 and thus o(g)(§ } ° )C § f \/g G S3. 60 □ For our next restriction, we must look at a subspace of the set of all n x n symmetric matrices §n. Definition 4.11. A subset L C §n is called invariant with respect to a if X G L implies a(g)(X) G L Vg G G. Example 4.14. Let L = {Q1 G §+° : [y]Q'[y]T = ^ _ y\y\y\ Vy G R3}. We have already showed that a(g)(Q') G §+. The rest of the requirement is satisfied due to the fact that our representation simply permutes the indicies of the variables. For example, [yMUMQOIyF = i Since our multivariate polynomials are invariant under such permutations, a(g)(Q') G L MQ' G L, g G S3 and the feasible set L to our semidefinite program is invariant. Our next definition refers not to the feasible set, but the objective function of semidefinite programs. Definition 4.12. A linear mapping F : —> R is called invariant with respect to cr if for all X E §N we have F(X) = F(a(g)(X)) for all g G G. For the cost function to be invariant, we require trace(CX) = trace(Ca(g)(X)) \/g G G, V I G SN. 61 This brings us to the definition used to describe the special kind of semidefinite programs that are the focus of our study. Definition 4.13. Given a finite group G, and associated linear representation a : G —> Aut{SN), a semidefinite optimization problem as defined in definition 2.19 is called invariant with respect to cr, if the following conditions are satisfied: • the set of feasible matrices {X : trace(AiX) = bt i = 1 fl §+ is invariant with respect to a • the cost function trace(CX) is invariant with respect to a We have already shown in Example 4.14 that the feasible set LflS^0 is invariant. Now we will show that our cost function trace(CQ') = trace(Q') is invariant with respect to a. Proposition 4.7. The cost function trace(Q') is invariant with respect to a i.e. trace(Q') = trace(a(g)(Q'), where a is the representation defined by a(g)(Q') = pT(g)Q'p(g) where p is the representation defined in Example 4-4- Proof. As shown in [6], trace(A5) = trace (BA) if A is an n x rn matrix and B is an m x n matrix. Since pT(g)Q' and p(g) are both 10 x 10 matrices, trace(o-(5')(<5/)) = trac e(pT (g)Q'p(g)) = tT&ce(p(g)pT(g)Q') = trac e(Q') 62 the last equation coming from the fact that p(g) is an orthogonal matrix (p 1(g) = pT(g)). Therefore, our semidefinite program is invariant with respect to a. □ In addition to invariant sets and problems we define, as in [5], invariant matrices. Definition 4.14. A matrix is called invariant with respect to a if it is fixed under the group action, i.e., X = cr(g)(X) V The fixed point subspace of SN is the subspace of all invariant matrices: ¥ := {X E SN : X = a(g)(x) g e G} Example 4.15. Recall that our cost matrix is simply the 10 x 10 identity matrix. So under our representation, a(g)C = pT(g)Cp{g) = PT(9)IP(9) = pt (9)p(9) = I = C Mg € S3 with the final equality coming from the fact that p(g) is an orthogonal permutation matrix for all g e S3. We can also see that the fixed point subspace of our running 63 example is F = {Q' € L n §+ ; p(g)TQ'p(g) = Q' Vg G S3}. Since p(g) is orthogonal for all g, we can rewrite this set as: F={Q'eLn§“ : p(g)Q' - Q'p(g) = 0 Vg G S3}. In order to explicitly define the fixed point subspace of our optimization problem, we simply need to show that p(g)Q' — Q'p(g) = 0 (or p(g)TQ'p(g) = Q') for the generators of S3. Proposition 4.8. Given a representation p : Sn —» GL(R") and an n x n matrix Q, if pT(a)Qp(a) = Q for all generators a G Sn then pT{g)Qp{g)= Q V Proof. Let p be a representation of a symmetric group Sn, Q be a n x n matrix, p(a)TQp(a) = Q for all generators a of Sn, and let g G Sn. Our goal is to show that with only this knowledge, it is clear that p(g)TQp(g) = Q. Since g G Sn, we can write it as a product of the generators of Sn. Let g = ai • • • afe, where «i,..., a*, are all generators of Sn. Recall that by definition 4.1, this means p(e) = I and p(gh) = p(g)p(h) Wg, h G Sn. This means P(d) = P(a 1 • • • Ofc) = p(ai) ■ ■ ■ p(ak) 64 So applying this to our desired equation we see that p(9)tQ'p(9) = p-\g)Qp{g) = (p(ai) ■ ■ ■ /o K ))-1<2p(ai) • • • p(ak) = p~l(ak) ■ ■ ■ p~l (a{)Qp{ai) ■ ■ ■ p(ak) = pT(ak) ■ ■ ■ pT{ax)Qp{al) ■ ■ ■ p{ak) = pT{ak) • • • pT{a2)Qp{a2) • • • p(ak) with the final equality holding due to our assumption that pT{a\)Q'p{a\) — Q as a\ is a generator of Sn. After repeating the final step k times, we conclude that pT(g)Qp(g) = Q'- □ Example 4.16. In our running example we are looking at the matrix Q' as notated here: #3,0,0;3,0,0 Since we are looking at the group S3 with the generators (1, 2) and (1,3), we need only restrict that p((l, 2))Q' — Q'p((l, 2)) = 0 and p((l, 3))Q' — Q'p((l, 3)) = 0. This gives us the following, explicit, definition of our fixed point subspace: F = {Q' e L fl §+} such that 93,0,0;3,0,0 -- 90,3,0;0,3,0 = 90,0,3;0,0,3 = a *73,0,0:2,1,0 = 93,0,0;2,0,1 — 9l,2,0;0,3,0 = 9l,0,2;0,0,3 — 90,3,0;0,2,1 — 90,1,2;0,0,3 — b 90,2,1;0,0,3 = 90,3,0;0,1,2 = 92,0,1;0,0,3 = 92,1,0;0,3,0 = 93,0,0;i,0,2 = 93,0,0;1,2,0 = c 9l,l,l;0,0,3 = 9l,l,l;0,3,0 = 93,0,0;1,1,1 = d 90,3,0;0,0,3 = 93,0,0;0,0,3 = 93,0,0;0,3,0 = 6 9l,0,2;0,3,0 = 9l,2,0;0,0,3 = 92,0,1;0,3,0 = 92,1,0;0,0,3 = 93,0,0;0,i,2 = 93,0,0;0,2,i = / 90,1,2;0,1,2 = 90,2,1;0,2,1 = 9l,0,2;l,0,2 = 9l,2,0;l,2,0 = 92,o,i;2,o,i = 92,i,0;2,i,0 — 9 9l,0,2;0,l,2 = 9l,2,0;0,2,l = 92,1,0;2,0,1 = h 90,2,1;0,1,2 = 92,0,1;1,0,2 = <72,1,0;1,2,0 = * 9l,l,l;0,l,2 — 9l,l,l;0,2,l = 9l,l,l;l,0,2 = 9l,2,0;l,l,l — 92,o,i;i,i,i = 92,i,0;i,i,i = j 9l,0,2;0,2,l = 9l,2,0;0,l,2 = 92,0,1:0,1,2 = 92,0,1;1,2,0 = 92,1,0:0,2,1 = 92,i,0;i,0,2 = k 9l,2,0;l,0,2 = 92,0,i;0,2,i = 92,i,0;0,i,2 = I 9l,l,l;l,l,l = m, 66 Notice that these constraints restrict the 55 variable Q' matrix (which is already a reduction of the 210 variable matrix Q to just 13 variables. We renamed them as o, b, c, d, e, /, g, h, i, j , k, Z, m for the sake of simplicity. As shown in [5], the optimal value to our semidefinite program is achievable when restricting our feasible set to the fixed point subspace F. This greatly reduces our problem size. Definition 4.15. Given an SDP min(trace(CX)) s.t.X £ Ln§+, the fixed point restricted semidefinite program is min(trace(C'X)) si.X € F fl L fl §+. Theorem 4.9. [[5]] Given an orthogonal linear representation a : —» Aut(SN) of a finite group G consider a semidefinite program which is invariant with respect to a. Then the optimal values of the original semidefinite program and the fixed point restricted semidefinite program, are equal. Recall: In Example 4.13, we discovered that the representation a with character X has the following decomposition: X = 3 x (1) + Xsign + 3 x (2)- This of course means that we can decompose o into a direct sum of its pairwise 67 inequivalant irreducible sub representations corresponding to Xs*9”, and a = Saw ® a sl9n ® 3a{2). Using corollary 4.2, we know that there exists a 10 x 10 matrix T such that 0-(1)($) 0 0 0 0 7 (1 )($) 0 0 0 0 7(1)G?) 0 0 0 0 ysi9n{g) Tcr(g)T~l = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2\g) 0 0 0 0 0 0 0 0 for all g G S3. An algorithm for finding this change of basis matrix is outlined in [4] yielding the following T matrix: 68 1 2 73 0 0 0 V6 0 0 0 0 0 1 1 i 1 0 76 0 76 0 0 73 0 0 73 1 i 1 1 0 Ve 0 v/6 0 0 71 0 0 "75 i 1 __i 1 1 i 0 v/6 0 v/6 0 0 2y/3 2 2 2y/3 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 76 0 76 0 0 2y/3 2 2 2^3 i 1 i 7! 0 0 0 "75 75 0 0 0 0 i i 1 1 1 1 0 76 0 75 0 0 2^3 2 2 273 1 i 1 1 1 1 0 75 0 V6 0 0 2^3 2 2 2y/3 i 1 i 73 0 0 0 "75 “ 72 0 0 0 0 Amazingly, all the invariant matrices with respect to a are block diagonolized under this change of basis and thus matrices of this form are all we need to solve our semidefinite program as stated in Theorem 4.9. This results in the associated theorem from [5]. Theorem 4.10. Let G be a finite group with h real irreducible representations. Let a be a linear representation of G on Sn induced as in Example 4-6 by an orthogonal linear representation p on Mn . Given a semidefinite program which is invariant with respect to a then there exists an explicitly computable coordinate transforma tion under which all invariant matrices are block diagonal. As a consequence, the 69 semidefinite program can be simplified to h min m,jtrace(CiXi), over X £ L, X* £ § Z=1 where ,rnl are the multiplicities in p and ni the dimensions of the real irreducible representations. Ci denote the blocks of C as well as X t denote the blocks of X in the new coordinates, respectively. This new problem is precisely the fixed-point restricted version of the original semidefinite program, under a simpler semidefinite program. Notice that in our case, this reduces the number of variables from 55 to 13 creating a much smaller problem. Example 4.17. We will denote the block diagonalized, version of Q' as: Xi 0 0 TtQ'T = 0 X 2 0 0 0 X; where a 2c by/2 -f- cy/2 -I- f \/2 dy/3 X, by/2 “H cy/2 f \/2 Q h i *2k, -\-I j y/S dy/3 jy/6 m 70 -h + g — i + 2k — I cy/6 fs/ 6 a - e 0 bV2 - b& - 0 2 2 U - n o 1 1 1 0 a — e 0 c- -B, “Is, 1 by/2 -£ & -M 0 g + h — | — k — | 0 o **& “ S. SH ■9" ■a- 1 o i 1 0 l 0 g+h-k-± is/3 IV3 g+h-^-k-^ 0 2 2 0 9 — h ^ — k + ^ 0 is/3 ls/3 0 g + h - \- k - { 0 2 2 0 g — h ^ — k ^ It is important to notice that the linear restraints that we used to define our feasible space L in 4.14 can be simplified due to the reduction of variables. So now we have L' = {£' G §V° : U Q 'W = - y\y\y\ V[y] G R3}. Which is simply / CL 27 5 ^ ^5 2c + g = d + h = 0, c -\- i — 0, j + fc + / = 0, 6/ + m = — | With this final restriction we see that with a fixed a = ^ and 6 = 0, we have simplified our original 210 variable problem to just 11 variables. Our fully reduced semidefinite program is min trace(*i)+trace(*2)+trace(X 3), over Q' G L'. Xi G §3 ,X 2 G , X 3 G §+ . Yielding the following SOS decomposition: Ei,i,i(ylylyl) ~ EM ,ylyl) = ^ — .8166 ^ („ J + y |+y|)j \/6 / o 2 2 2 2 2\'\ ^ + .5773 ( — (^1^2 + viv3 + y \V2 + 2/12/3 + 2/22/3 + 2/22/3 J I ) \/6 / 2 2 2 , 2 .2 2 + .2778 — (yiV2 - 2/12/3 - 2/12/2 + 2/12/3 + 2^22/3 - 2/22/3 ( 6 v /%/g / 3 1 3 1 3\\ +.0556 ( t I"1 - 5y* - 2V3)) ( v/3 / 2 1 2 1 2 2 1 2 1 2 \ ^ + .2886 ( — ^2/12/2 - -yiy2 - -yiy3 + VlV3 - - V 2V3 ~ ^22/3J J -.4999 (yly3 - yiy% + y2y3 - yiy|)) ^ +.0556 (-8166 ( t _ y3^) +.2886 Q (yiy2 - yiyf - V2vl + vlvs)'} ( V^3 / 2 2 1 2 1 2 1 2 1 2\ A \ -.4 9 9 9 I — ( yxy2 - ViV3 + -yiy2 “ ~VlV3 ~ ~V2V3 + -V 2V3J J J ( ( (2 1 2 ^ 2 2 1 2 ^ 2 'N ^ +.2778 I - .8660 I — ^yxy2 - -yiy2 ~ -yiy3 + viv* ~ ~V2V3 ~ -2/22/3J I -.5000 (-y\yz + yiy2 - y2y3 4- yiy^ )^ ^ + .2778 ^.8660 [y\yl - yiyf - y2y3 + y ^ )) ( / 9 2 1 2 1 2 1 2 1 2^ A + .5000 I — [y 1y2 ~ y iV3 + ~V 1V2 ~ “ V ^3 “ ~ 2/2 2/3 + ~V2 y3 J J J The rest of the results for n — r = 3 are shown below. 72 Ei,i,i(ybyhvi) £ 2,1 (yl,ylvi) .0568 ('8 y \ y i + y \y z + y \y \ + y i y \ + y\y?> + y z y \ ( ? 0 »)’ + .0568 ( f < y\v2 - y\yz - y\y\ + 2/11/3 + ~ 2/22/3»' + .0542 ^.4286 + .3868 (t (•!-'!)) -.3 0 3 2 ( f3 tyvty2 - -yiy% ~ - v iv l + 2/12/3 - -V2yl - —2/22/3^ / l / 2 2 2 2\'\ -.2 7 3 6 ( - (^yiy2 ~ 2/12/3 ~ 2/2y3 + y2y3J J -.5 2 4 9 { - y l v 3 + 2/12/2 ~ V2V3 + 2/12/!)) ( V 3 / 2 2 ,1 2 1 2 12 ,1 2N -.4 7 3 7 I — yyiV2 - yiV3 + -yiy2 ~ “yi2/3 “ ~^V2y* 2V2V3y + .0542 3868 (- + .4286 *'»)) ( V^3 / 2 1 2 ^ 2 2 1 2 1 2 \ + .2736 I — ^2/12/2 - -yiy2 - -yiy3 + yiy3 - -y2y3 - -V2V3J -.3 0 3 2 ■ yiy3 - y^y\ + yfys)) (5( t (yiy2 + .4737 ( _y2y3 + 2/1 yf - y2y3 + yiyi)) / \/3 / 2 2,121212,12\ -.5 2 4 9 I — (^yiy2- yiy3 + - y i y 2 - - yiJ/3 - -V 2 V3 + -y2 y3J r +.0555 (-* / V3 / 2 1 2 1 2 2 ++.8164 .8 I — (^yiy2 - -yiy2 - -yiy3 + yiy -y2y3 - r + .0555 (, +.8164 (yiy2 - vivl - y*y\ + y^)) ^ 73 E^ylvlvi) ~ Ez{yl,ym) = / 2 2 2 2 2 2\'\ .2222 — (^2/i2/2 - V\V3 - VW2 + 2/l2/3 + 2/22/3 - V2V3) I / ( \ / 3 / o ^ 2 ^ 2 2 ^ 2^2^^ + .2222 I .8660 I — f y xy2 - - y 12/2 - —2/12/3 + ViV3 - - y 2V3 ~ “ 2/22/3J I +.5000 (-y2?/3 + 2/12/2 - v 2 vl + 2/12/3) ) ^ +.2222 ^ . 8 6 6 0 (yivl - yiyl - 2/2 y | + 2/2 2 /3 ) ) ( v/3 / 2 2 1 2 1 2 ^ 2 ^ 2^ \ \ + .5000 I — (^2/^2 - ViV3 + -y iy 2 “ ~V lV 3 ~ “ 2/2 2/3 + - 2/22/3 J J j 74 2„2\ _ P2M , y l y D - 1,1,1 (yl vlvi) .1136 (-8166 (t (vi + yf + vt (V6 -.5 7 7 2 --- V 6 \/6 / 2 2 2 2 2 2\'\ + .1136 ( — [yiy2 - viv3 - ywi + yiy3 + 2/2V3 - y^y^) 1 + .1086 + .5772 ( t (*■-*■)) -.°102 ((f(» — (yiy2 - -1/12/2 - - -.4 0 8 1 / 1 / 2 2 2 2\'\ (^yiy2 “ 2/12/3 - V22/3 + y2V3 J J - .0 1 7 7 (2 ( _ y 2^3 + 1/1V2 - 2/22/3 + 2/iy|)) 1 2 1 -.7068 -yiv2- 5i (*(• +.1086 ( t <*•-*!>) \/3 / 2 ^2^2 2 - .4 0 8 1 ( — ^yiy2 ~ “ 2/12/2 “ -V12/3 + ViV3 +.0102 (yiya - 2/12/3 - y2y| + y iy s )) -.7 0 6 8 Q ( - y lv s + yiyi - y2y3 + yiy^)) f y/3 / 2 2 . 1 21 2 + .0177 ( — (^iy2 - y iy 3 + - y iy 2 - - y iy 3 + .1111 ( f (V|-V33)) +.8165 Q (yiy| - yiy 3 - y2vl + ylyz)') ^ + .1111 (.5774 ( v ! - \ * l - ± * t) ) a/3 / 2 ^2^2 2 + .8165 ( — (2/12/2 - -yiy2 _ ~ yiy3 + VlV3 ' 75 Pa(yl> vh vi) - p 2,M , vlvl) = .3386 .8165 (v? + y\ + 2/3)^ \/6 / 2 2 2 2 2 2\'\ \ -.5 7 7 3 ( — {ViV2 + ViV3 + V12/2 + 2/12/3 + 2/2 2/3 + 2/22/3) J J + .1164 ( ^ (2 2 2. 2 ,2 2\^2 I — [y12/2 - yiV3 - yiV2 + viv3 + v2V3 - 2/22/3J J +.1084 .0097 / 3 1 3 1 3\\ C271 - 2^ - 2^ ; J ( Y^3 / 2 1 2 1 2 2 1 2 ^ 2 I — ^2/i 2/2 ~ ~ 2/12/2 ~ -2/12/3 + VlV3 ~ “ 2/2y3 - —2/22/3 .5059 ( ” ( —^22/3 + 2/12/2 “ V2V3 + 2/12/f)^ +.1084 ( - .0097 ( T (V* _y»)) +.8625 ( “ (2/12/2 ~ 2/12/3 - V2V3 + 3/2 2/3)) / / 2 2 ^ 2 ^ 2 ^ 2 ^ 2^^ ^ + .5059 [ ~ \yiV2 ~ 2/12/3 + 2V1V2 ~ 2 2/12/3 ” 2 2/2 2/3 + i y2y3 J J J + .3307 8165 (- / \/3 / 2 1 2 1 2 , 2 1 2 1 2 I — ^2/i 2/2 “ “ VI2/2 “ 2271273 + y l J/3 " 2 2/22/3 “ 2 2/22/3 + .4940 (2 ( “ V2V3 + 2/12/2 _ V22/3 + vivi)) ^ +.3307 f - .8165 (^(v !-v l)) -.2 9 8 9 doyi2/2 - V12/3 - v22/3 + 2/2V3 )) +.4940 (?(■Vl2/2 - 2/1 V3 + - V12/2 -2/12/3 ~ “ V2V3 + 76 HzAvlylyl) ~ Hi,i,i(ylym) = .0268 (-■8167 + y/6 / o 2 2 2 2 2\'\ ^ ( — [yIV2 + ViV3 + y 1 V2 + V1V3 + 2/2^3 + V2V^J J J y/6 ( 2 2 2 2 2 2\\ + .0268 ( — [yiV2 - yiV3 - viy2 + 2/12/3 + 2/22/3 - 2/22/3) J +.0278 (,5775 ( ^ (t,3_ly3_ly3^ ( V3 / 2 1 21 2 2 1 2 1 2 + -8164 ( — ^ 1 2 /2 - “ 2/12/2 - —2/12/3 + 2/12/3 - -2/2V 3 - — 2/2 2/3J I I + .0278 (.577S ^ (v® - y|j) + .8164 ( y i »2 - J/11/3 - y2yl + S/2o t ) ^ ^ + .0287 (-•5723 +.0763 ( f (vi-vl)) ( (2 ^ 2 ^ 2 2 ^ 2 ^ 2 \ ^ +.4048 ( — (^2/^2 - “V12/2 - -2/1V3 + VlV3 - “ 2/2y3 - -^2^3J J -.0540 (yiy| - y\y\ - y2yl + 2/22/3)) +.7008 (~y%y3 + yivl - yivl + vivT)') -.0935 I( —V3 \^y1y2/ 2 ~ yiV32 + -yi2/2 1 2 “ “^1^3 1 2 _ *“ 2/22/32 + 1~y2y3J 2^ j JV + .0287 (°763 + .5723 (t O*-*s)) / \/3 (2 1 2 ^ 2 2 1 2 ^ 2 \ \ I — (2/1 y2 - -^1^2 ~ — 2/1 y3 + y iy 3 - -y2y3 - - y2V3J J (- (yiy2 - yiyf ~~ y^y\ + ylvs)) (- {-ylv3 + yiy2 - yiy\ + yiyf)) / \/3 (2 2 ^ 2 ^ 2 ^ 2 ^ 2\ \ \ ( — yyiy2 - Viy3 + - y i y 2 - -yiy3 - ~y2ys + -y^y3J J J 77 H3(ylyly!)-H2,1(ylyiyl) = .0688 ^.8166 (»? +»|+!(!)j - .5 7 7 2 ( v ?»2 + ViV3 + y i V 2 + VIV3 + y%V3 + 1/21/3 ) ^ ^ + 0466 i yl y2 ~ y*y3 ~ yiy% ■*" y i!/3 ■*" y^V3 ~ V2V3^j + .0433 ^.0403 ( t (y* “ v*)) -.8 5 0 7 Q (yivl - 3/13/3 - V2y\ + 2/23/3)) / \/3 / 2 2 1 2 1 2 1 2 1 2\\ V -.5 2 4 1 1 — ^ 3 / i 3/2 - 3 /i3/3 + “ 3/13/2 “ “ 3/13/3 “ “ 3/23/3 + -V 2 3 /3 J I I m 3/i V2 - “ 3/13/2 - -3 /1 3 /3 + V l3 / 3 - “ 3/23/3 “ “ 2/2 2/3 5241 ( 2 (“y23/3 + yi2/2 “ 3/2V3 + yiyV)^j ^ + .0656 3156 ( ( 2 1 2 ^ 2 2 1 2^2^^ + .3310 I — 2/2 - -y 13/2 “ “ 3/1 ^3 + 3/i3/3 “ “ 3/23/3 “ “ ^2^3J J -.4745 (-V2V3 + yi^2 _ y2yf + yiy^)) ^ + .0656 ^.8156 (yl - yf) j +.3310 ( y iy i - yiy3 - + y ^ ) ) ( / 2 2 1 2 1 2 1 2 1 2\^\ V -•4 7 4 5 ( — ( yiy2 - yiy3 + -yiy2 - ~yiV3 ~ ~y2ys + -y2y3j J J 78 S2,i(ylyly%) - Sltiti(ylylyj) = .2500 ($<■[ViV 2 ■ViV3 ~ 2/12/2 + yivl + y\yz - 2/22/3 + .2500 1 2 .2 1 2 1 2 m ( 2/? 2/2 • - 2/ 12/2 — 2/12/3 +2/12/3 ~ -2/22/3 ~ —2/22/3 + .5000 ( “ 2/2 2/3 + 2/12/§ “ 2/2 2/3 + 2/12/i + .2500 ^.8660 ^ - (yi2/i - 2/12/3 ~ yiy\ + 2/22/3)^ +.5000 [ 2/? 2/2 - 2/12/3 + —2/12/2 “ -2/iy3 ~ 2/2 2/3 "t" ^2/22/3 (¥(■ )))' 79 Szivlvlvl) - S^ylvM) 8166 (-y(»l +»!+»! "\/6 / 2 2 2 2 2 2'\'\ \ (— \yiV2 + ViV3 + y iV2 + yiy3 + 2/22/3 + 2/2 y3 J 1 I + .0871 (—\/6 (yiy2 / 2 - yiy3 2 - yiy2 2 + yiy3 2 + V2V3 2 - yiy^) 2\^\ I +.1014 (.0233 ( ^ ( y» - i y»- i „ l ) ) / ^3 / 2 ^2^2 2 ^2^"r +-8739 I — ^yiy2 - -yiy2 - -yiy3 + yiys - -y2y3 - -yj +.4855 (-y^ys + yiy2 - v^y\ + yiy^)) ^ +.1014 ^ — .0233 — .8739 (yiyf - yiyf - y2y^ + y^ ) ) -.4 8 5 5 ( / 2 2 ^2^2^2 ^ 2^ \ ^ ( — ^iy2 - yiy3 + -yiy2 - -yiys ~ 5^2V3 + -y2y3HI + .1514 (#1 -»!)) +.2638 Q (yiy2 - yiy3 - y2y2 + y2^)) /V3 / 2 2 1 2 1 2 1 2 1 2\ ^ V — .514° I — ^yxy2 - yiy 3 + -V1V2 ~ “ yiy3 “ “y2y3 + 2V22/3J J J +.1514 (.8163 (^(v?-iv|-|vl)) .2638 + ( ^^3 / 2 1 2 ^ 2 2 ^ 2 ^ 2 \ ^ I — \ y1y2 - -yiy2 - -yiy3 +yiy3 - -y2y3 - -y2y3J I ■514° ( i ( “ y ^ 3 + yiy2 - y2y2 + y iy f) ) j Unfortunately these symmetry adapted solutions have complicated SOS decom positions and we see some of the square terms are complicated due to the lack of zeros in the eigenvectors. However, we do recognize some patterns in these results. None of our SOS solutions have more than six square terms. In other words our 80 optimal solutions have no more than six eigenvalues out of a total possible ten. We also see that the first two sub problems corresponding to the trivial representation and the first sub problem corresponding to the regular representation were never needed, i.e., these were our four unused eigenvalues. Regardless of the complex ity, the solutions were easier to compute and we have sufficiently shown that the inequalities from 1.1 hold for n — r = 3. Proposition 4.11. The inequalities from Theorem 1.1 F(x) > G(x)\/x > 0 hold true for n = r = 3 as shown through the fact that F(y2) — G(y2) is a sum of squares. 81 Chapter 5 Conclusion We have shown that it is possible to verify inequalities between term normalized invariant functions, similar to the ones shown in [3], using semidefinite program ming. After re-organizing these inequalities, we found it sufficient to determine the existence of a sum of squares. This determination can be done using semidefinite programming as shown in Section 2.2. After computing some small examples, we noticed an underlying symmetry about the sum of squares decompositions. This inspired us to use representation theory to separate our programs into independent small parts. Using the methods from [5] shown in Section 4.2, we were able to ex plicitly apply this idea to small examples. The results we found under this added step were symmetric as we had hoped, however they were not the same symmetries that we had seen before. It is our hope that this research is continued in future theses. There is still discovery to be made about the cause of the original symmetries we were finding in 82 our solutions. We also wonder if those symmetries and the symmetries coming from representation theory can be used to prove Conjecture 1.1. Conjecturel.l Let F{xi,..., xn) and G{xi,..., xn) be symmetric functions from the families used in Theorem 1.1. Then F(xi,..., xn) > G(xi,..., xn), for x > 0 for the listed cases if and only if F(yf,..., y2) — G(yf,... , y2) is an SOS. This of course would lead to proof of the majorizations conjectured in [3], that the last two inequalities from Theorem 1.1 can be completed as if and only if statements as shown here. We believe this conjecture could be true based on the many examples uie computed for n e {2,3} and r G {2,3}. We also believe that the symmetries in these poly nomials as well as the resulting symmetries in our semidefinite programs promote the existence of an SOS decomposition. We hope that further research will make a stronger connection in this regard and lead to the proof of our conjecture. Chapter 6 Appendix Our code in Mat Lab: '/ja. = 3, r = 3 cvx_begin sdp va ria b les abcdefghi2j2knm; A = [2*e+a b*sqrt(2)+c*sqrt(2)+f*sqrt(2) d*sqrt(3); b*sqrt(2)+c*sqrt(2)+f*sqrt(2) g+h+i2+2*k+n j2*sqrt(6); d*sqrt(3) j2*sqrt(6) m]; A >= 0; B = [-h+g-i2+2*k-n] B >= 0; C ■ [a-e 0 b*sqrt(2)-c*sqrt(2)/2-f*sqrt(2)/2 0 c*sqrt(6)/2-f*sqrt(6)/2 0; 0 a-e 0 b*sqrt(2)-c*sqrt(2)/2-f*sqrt(2)/2 0 c*sqrt(6)/2-f*sqrt(6)/2; b*sqrt(2)-c*sqrt(2)/2-f*sqrt(2)/2 0 g+h-i2/2-k-n/2 0 i2*sqrt(3)/2-n*sqrt(3)/2 0; 0 b*sqrt(2)-c*sqrt(2)/2-f*sqrt(2)/2 0 g+h-i2/2-k-n/2 0 i2*sqrt(3)/2-n*sqrt(3)/2; c*sqrt(6)/2-f*sqrt(6)/2 0 i2*sqrt(3)/2-n*sqrt(3)/2 0 g-h+i2/2-k+n/2 0; 0 c*sqrt(6)/2-f*sqrt(6)/2 0 i2*sqrt(3)/2-n*sqrt(3)/2 0 g-h+i2/2-k+n/2]; C >= 0; ’/. Constraints '/.Shape 6 ,0 ,0 ( i . e . y l '6 , y2‘ 6, or y3“6 monomial c o e e fic ie n ts) 84 a == 1/10; ‘/.Shape 5 ,1 ,0 2*b == 0; ’/.Shape 4 ,2 ,0 2*c + g ~ -1/40; '/.Shape 4 ,1 ,1 2*d + 2*h == 0; ’/.Shape 3 ,3 ,0 2*e + 2*i2 == 0; ’/.Shape 3 ,2 ,1 2*f + 2*j2 + 2*k == 0; ’/.Shape 2 ,2 ,2 6*n + m == -3/2 0 ; ’/. SDP minimize(trace(A) + trace(C) -h+g-i2+2*k-n) cvx.end ’/.Solutions ’/.Solutions corresponding to trivial representations [UA, DA] = eigs(A ) ’/,Solution corresponding to sign representation [UB, DB] = eigs(B ) ’/.Solution corresponding to regular representations [UC, DC] = eigs(C ) 85 Bibliography [1] Grigoriy Blekherman, Pablo A. Parrilo, and Rekha R. Thomas (eds.), Semidef inite optimization and convex algebraic geometry, MOS-SIAM Series on Op timization, vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2013. [2] Peter S. 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