2. ESTRUCTURA ECONÓMICA DEL SECTOR
2.2. Tendencias de la industria
Theorem 4.3.1 (Primal Theorem of alternative [11, 18]) SupposeA:Sk+ → Rmis a linear transformation, b ∈ Rm, P ∈ Sk and Z ∈ Sk. Then exactly one of the following alternative
systems is consistent:
(I) 0≺ P∈F :={P∈ Sk :A(P)=b,P 0} (Slater) (4.9a)
(II) 0,Z ∈D:={Z ∈ Sk :Z =A∗y 0,bTy=0}. (Auxiliary) (4.9b)
Proof Note that if (II) is consistent, thenZexposes a face ofSn
+that contains the minimal face
(F,Sn
+). That is, for P∈Fwe have
traceZP=trace(A∗y)P= yTb= 0. The remainder of the proof can be found in [11, 18] or Appendix A.
Equation (4.9a) is called theprimal problemand equation (4.9b) is called theauxiliary problem. The theorem of alternative for the dual form follows.
Theorem 4.3.2 ( Theorem of alternative for dual form [8, 9]) Suppose A : Sk
+ → Rm is a
linear transformation, P∈ Sk,Z ∈ Sk. Then exactly one of the following alternative systems is
consistent:
(I) Z =C− A∗y0 (4.10a)
(II) A(X)=0,hC,Xi=0,X 0 =⇒ X= 0. (4.10b) See [11] for a detailed proof.
4.3.3
Facial reduction
Recall Theorem 4.3.1, that when (4.9a) is true, theSlater conditionholds. The Slater condition is an important concept in optimization. The failure of the Slater condition usually results in poor performance of algorithms such as interior point methods and the Douglas-Rachford method. Facial reduction aims to regularize an SDP problem so that the Slater condition holds on a minimal face.
4.3. SDPand facial reduction 99
Lemma 4.3.3 (Facial reduction on the primal form) Suppose FPis non-empty. Then
A(P)=b,P∈ Sk + 0, Z= A∗ y∈ Sk +,bTy=0 ⇒ P∈ {Z}⊥∩ Sk+ (4.11)
Such a Z is called anexposing vector of Sk
+. By solving the second problem in the bracket,
we can get an exposing vector which reduces the primal problem (4.9a) to a smaller face, i.e.,
{Z}⊥∩ Sk
+ which is reformulated as a primal feasibility problem on a smaller cone Sk+¯,k¯ < k
described in Theorem 4.3.4. The process is repeated until we get face (Fp,Sk+), the minimal
face ofSk
+ containingFp, and the Slater condition (4.9a) holds.
Based on the two statements of the theorems of alternative, we can always apply facial reduction to the dual or primal form to reduce the dimension of the problem. In this chapter, we express our moment matrix problem in the primal form yielding greater accuracy in our examples when solved using facial reductions and the Douglas-Rachford (DR) method. Details of DR are given later in Section 4.4.
Suppose an exposing vector is found. The following theorems shows how to use the ex- posing vector to get an equivalent problem on a smaller positive semidefinite cone so that an additional facial reduction can be done.
Theorem 4.3.4 SupposeA:Sk →
Rmis a linear transformation as in Problem 4.2.1, P ∈ Sk,
Z ∈ Sk+is an exposing vector, Z = h U V i Dl 0 0 0 h U V iT
is the spectral decomposition. Suppose A¯t := VTAtV and A¯ : Sd → Rm is the linear transformation induced by A¯t where
d+l=k. Then
∃P∈ Sk,A(P)=b,ZTP=0,P0 (4.12a)
⇐⇒
∃P¯ ∈ Sd,A¯( ¯P)=b¯,P¯ 0. (4.12b)
Proof First, we assume ¯b=b.
To show (4.12a) implies (4.12b). Suppose there exists P 0 satisfying (4.12a). Apply the spectral decomposition to P. Then we have P = U1P1U1T whereU
T
1U1 = I,ZTU1 = 0,
transformation such thatV Q= U1. Then trace(AtV QP1QTVT) = trace(AtU1P1U1T)= A(P) =
b. Hence we conclude∃P¯ = QP1QT,A¯( ¯P)= b,P¯ 0.
To show (4.12b) implies (4.12a), note that the existence of ¯Psatisfying (4.12b) implies that
P= VPV¯ T satisfies (4.12a).
We assumeAis linearly independent, however, ¯Ais not necessarily linearly independent. We can remove the redundant linear constraints in ¯Aand the corresponding elements inbto obtain ¯b. So without loss of generality, we have ¯A( ¯P)=b¯,P¯ 0, ¯Ais linearly independent.
4.3.4
Facial reduction maximum rank algorithm
Recall from the Primal Form Feasibility Problem 4.2.1, we can just setZ = BBT as the exposing vector to do the first facial reduction as described in Theorem 4.3.4.
To do more facial reductions, after the first facial reduction, the problem is considered in the form of (4.9a). Then according to Theorem 4.3.1, we need to determine if (4.9a) is strictly feasible, i.e. to determine if there exists a P 0. We need to solve the following auxiliary problem:1 p∗(A,b) := min y 1 2(b T y)2 s.t. Z =A∗ y 0 traceA∗y=1. (4.13)
We set traceA∗y = 1 because we need to rule outybeing the zero solution. If we solve this problem successfully with |bTy| = 0 with a non-zero y, we have Z = A∗y , 0. By Theo- rem 4.3.1, (4.9a) only admits a positive semidefinite but no positive definite solution which indicates Slater condition fails and a second facial reduction is needed. We then use this Z
as the exposing vector to do the second facial reduction as described in Theorem 4.3.4. We repeat this process until p∗(A,b) is strictly positive which means there exists a positive definite solution of (4.12b) and that the slater condition holds.
The algorithm to use facial reduction to find maximum rank solutions is summarized as
1This can be implemented in e.g., CVX using thenormfunction or absolute value function for the objective,
4.3. SDPand facial reduction 101
follows:
Algorithm 4.3.1:Facial reduction on the primal.
1 Input(A:Sn→ Rm,b∈Rm,B∈Rk×las in Problem 4.2.1, set p∗(A,b)=0, W = I); 2 repeat
3 Find the exposing vectorZby settingZ = BBT(first facial reduction) or solving the
auxiliary problem (4.13) for p∗(A,b).
4 if p∗(A,b)> 0then
5 STOP, facial reduction finished, Slater condition holds
6 else
7 Apply eigenvalue decomposition toZto obtainV such thatV is the nullspace of
Z andVTV = I.
8 UpdateAsuch thatAi ←VTAiV,∀i∈ E, then updateA,bby removing
redundant relations.
9 UpdateW byW ←W·V.
10 end if
11 until p∗(A,b)> 0;
12 SolveA(P)= b,P0. Recover the moment matrix M= W PWT. 13 Output(M which is maximum rank)
Theorem 4.3.5 (Maximum rank) No further facial reductions can be done if and only if p∗(A,b)>0. Algorithm 4.3.1 returns a maximum rank solution of Problem 4.2.1.
Proof After the first facial reduction, Problem 4.2.1 is transformed into (4.12a). By the proof
of Theorem 4.3.1, each time when we find a solutionZ 0 from (4.9b), we can find the feasible solutions of (4.9a) lie in the nullspace ofZ. By Theorem 4.3.4, we can reduce the problem to an equivalent smaller problem without loss of information. By Theorem 4.3.1 when we have
p∗(A,b) > 0, we have reduced the problem to a minimal face where (4.9a) admits a positive semidefinite solution, which is equivalent to saying no further facial reductions can be done. So if p∗(A,b) > 0, there exists P 0 such that A(P) = b. Since the minimal face contains all the feasible solutions of Problem 4.2.1 andPis the maximum rank solution on the minimal face, Mis also the maximum rank solution of Problem 4.2.1.
Singularity degree The minimal number of facial reduction steps is calledsingularity degree. The examples in Section 4.10 show that some examples with singularity more than 1 can be accurately solved by Facial reduction heuristics. For more details, see [45, 17].