Chapter V: Conclusions
5.2 Future Directions
Identifying Additional Useful Convex Graph Invariants
In Chapters 2 and 3 we employ convex graph invariants based on graph spec- trum, stability number and maximum-cut value for producing effective convex relaxations to two difficult problems arising from graphs. Evidently many other convex graph invariants may be useful in this context. Extending the list of suitable convex graph invariants would increase the applicability and approximation quality of relaxations arising from our framework.
Theoretical analysis of relaxations based on stability number and maximum-cut
In this thesis we extensively study optimality conditions pertaining to opti- mizing over the Schur-Horn orbitope, which is an invariant convex set based on the graph spectrum. Our analysis is based on Schur-Horn orbitope’s ge- ometric aspects, as we closely investigate the structure of its normal cones.
However, our analysis for the relaxations based on the stability number and the maximum-cut invariants are mainly expository and limited in depth, since we do not have an as deep understanding of these invariants’ geometric/facial structures. Obtaining a better understanding of the geometric properties of these constraint sets may facilitate producing theoretical results regarding the optimality conditions of the corresponding convex relaxations.
Extending our results to nonsymmetric matrices
In Chapters 2 and 3 we investigate problems arising from undirected graphs which can be represented by symmetric adjacency matrices. Similarly, in Chapter 4, our focus is solely limited on inverse eigenvalue problems aris- ing from symmetric matrices. However, in various applications nonsymmetric matrices are of significant importance, and questions similar to the ones we have investigated in this thesis can be raised for their nonsymmetric counter- parts. It may be possible to adapt some of the methods described in this thesis to answer such questions. For instance, in order to solve a particular “inverse singular value problem” on nonsymmetric n×n matrices, one might consider utilizing majorization inequalities on the eigenvalues of the 2n ×2n matrix whose eigenvalues correspond to plus and minus of the desired singular val- ues. Such approaches may enable extending the applicability of our framework from undirected graphs to directed graphs.
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A p p e n d i x A
APPENDIX FOR CHAPTER 2
In the appendix, we further investigate the connection between our method and Ames and Vavasis’ nuclear norm minimization approach [5], for the special case of the plantedk-clique problem. In order to identify ak-clique, Ames and Vavasis propose solving the following optimization problem:
A∈minSn
kAk∗ s.t. X
i,j∈{1,...,n}
Ai,j ≥k2,
Ai,j = 0 if (AG)i,j = 0, i6=j.
(AV)
We establish the claim below:
Proposition 40. Consider an instance of the planted k-clique problem. The Schur-Horn relaxation (P) succeeds in uniquely identifying the planted k-clique whenever the optimization program (AV) succeeds.
Proof. We establish the result by presenting a sequence of optimization prob- lems which relate the optimization problem (AV) to the Schur-Horn relaxation (P). Assume that the hiddenk-clique (with 1’s on the diagonal) is the unique optimal solution of (AV). Consider the first intermediate optimization problem below:
max X
i,j∈{1,...,n}
Ai,j s.t. X
i,j∈{1,...,n}
Ai,j ≥k2,
Ai,j = 0 if (AG)i,j = 0, i6=j, kAk∗ =k.
(I1)
If the k-clique is the unique optimal solution of (AV), then (I1) has a single feasible point. That is because of the additional constraint on the nuclear norm of the variable, and the fact that the nuclear norm of the k-clique (with 1’s on the diagonal) is equal to k. Further, the only feasible point of (I1) is the adjacency matrix of the planted clique.
Given that the optimization problem (I1) is feasible, its first constraint is redundant, as that constraint and the objective function overlap. More- over, one can replace the objective function with Tr(A · AG), since by the planted model, AG is equal to 1 on every index where A is equal to 1, i.e.,
P
i,j∈{1,...,n}
Ai,j = Tr(A·1k1Tk) = Tr(A·AG). These modifications lead to the second intermediate optimization problem given below:
max Tr(A·AG)
s.t. Ai,j = 0 if (AG)i,j = 0, i6=j, kAk∗ =k.
(I2)
The optimization problem (I2) is feasible in a potentially bigger set than the optimization problem (I1), but its unique optimal value is still attained by the same matrix – the adjacency matrix of the planted clique.
Now consider adding the constraint A 0 to the constraint set of the opti- mization problem (I2). Since the adjacency matrix of the k-clique (including the 1’s on the diagonal) satisfies this constraint, it is still the unique optimal solution of the resulting problem. Furthermore, under the positive semidef- initeness of A, one can replace the nuclear norm constraint kAk∗ = k with the trace constraint Tr(A) =k. With this final change, we obtain exactly the Schur-Horn relaxation (P), where the Schur-Horn orbitope is as described in equation (2.3).
A p p e n d i x B