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I have an optimization problem of the following form:

$$\min_{X\succeq0} \mathrm{trace\;} MX$$

under the linear constraint $\mbox{diag} (X) = \mathrm{Id}$ and the non-convex constraint $\mbox{rank} (X) = 1$. The matrix $M$ is square and rank-deficient.

The convex relaxation of this problem corresponds to dropping the rank-1 constraint, and merely keeping $X$ positive semidefinite.

I tried running a standard SDP solver (Mosek) on this problem but it yields a matrix $X$ which, despite satisfying the linear constraint and being positive semidefinite, is not of rank one. Instead, it is typically of rank $(n - \mathrm{rank\;} M)$ where $n$ is the number of rows of $X$.

Can you explain why I am getting this result?

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You removed the rank-one constraint. The optimization will no longer find a rank-one matrix. However, for your problem, you may not need a solver. Let $M$ be a $n\times n$ matrix. Observe that $X\geq 0$ and $rank(X)=1$ implies that $X=xx^T$ for some vector $x\in\mathbb{R}^n$. Then $trace\{MX\}=x^TMx $. Also, $diag(X)=Id$ will imply $x_i^2=1$ where $x=[x_1,\dots,x_n]$. Thus, your optimization problem becomes $$\min_{x\in\mathbb{R}^n}~x^TMx \\s.t.~x_i\in\{-1,1\} \forall i .$$ Let $M$ be symmetric (try convincing yourself that this doesn't lose generality). Then $M$ has a decomposition of form $$M =\sum_{i=1}^{n}\lambda_iv_iv_i^T$$ where $\lambda_1<\dots<\lambda_n$ are eigenvalues and $v_i$ are corresponding eigenvectors. Thus, we have $$x^TMx=\sum_{i=1}^{n}\lambda_i(x^Tv_i)^2$$ Also, note that $\sum_{i}x_i<=n$. Given this, can you try and show that $x = sign(v_1)$. Here $sign(.)$ is +1 or -1 corresponding to positive and negative entries of the argument vector.

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Let $\mathrm Q : = \frac 12 \left( \mathrm M + \mathrm M^{\top}\right)$ be the symmetric part of $\mathrm M$. Consider the following binary quadratic program

$$\begin{array}{ll} \text{minimize} & \mathrm x^{\top} \mathrm Q \,\mathrm x\\ \text{subject to} & \mathrm x \in \{\pm 1\}^n\end{array}$$

or, equivalently, the following quadratically constrained quadratic program (QCQP)

$$\begin{array}{ll} \text{minimize} & \mathrm x^{\top} \mathrm Q \,\mathrm x\\ \text{subject to} & x_i^2 = 1 \quad \forall i \in [n]\\ & \mathrm x \in \mathbb R^n\end{array}\tag{QCQP}$$

where $[n] := \{1,2,\dots,n\}$. Both optimization problems above are hard (even when $\mathrm Q \succeq \mathrm O_n$).


Primal relaxation

$$\mathrm x^{\top} \mathrm Q \,\mathrm x = \mbox{tr} (\mathrm x^{\top} \mathrm Q \,\mathrm x) = \mbox{tr} (\mathrm Q \,\mathrm x \mathrm x^{\top})$$

where $\mathrm x \mathrm x^{\top}$ is symmetric, positive semidefinite, rank-$1$ and has $n$ ones on its main diagonal. Lifting, we obtain the following (non-convex) optimization problem

$$\begin{array}{ll} \text{minimize} & \mbox{tr} (\mathrm Q \mathrm X)\\ \text{subject to} & \mathrm X_{ii} = 1 \quad \forall i \in [n]\\ & \mathrm X \succeq \mathrm O_n\\ & \mbox{rank} (\mathrm X) = 1\end{array}$$

Dropping the non-convex rank constraint, we obtain the following semidefinite program (SDP)

$$\begin{array}{ll} \text{minimize} & \mbox{tr} (\mathrm Q \mathrm X)\\ \text{subject to} & \mathrm X_{ii} = 1 \quad \forall i \in [n]\\ & \mathrm X \succeq \mathrm O_n\end{array}$$

which is easy to solve. If the solution of this SDP, which we denote by $\mathrm X^*$, is

  • rank-$1$, then we have solved the original QCQP.
  • not rank-$1$, then we do need a rounding scheme.

If matrix $\mathrm Q$ is nonnegative with zeros on the main diagonal, then it is the adjacency matrix of an undirected, weighted graph and we can use Goemans & Williamson's famous randomized rounding scheme for MAX CUT [MG&DW'95, BG&JM'12].


Dual

The QCQP yields the following Lagrangian

$$\mathcal{L} (\mathrm x, \lambda) := \mathrm x^{\top} \mathrm Q \,\mathrm x - \sum_{i=1}^n \lambda_i (x_i^2 - 1) = \mathrm x^{\top} \left( \mathrm Q - \mbox{diag} (\lambda) \right) \mathrm x + 1_n^{\top} \lambda$$

The dual of the QCQP [PP&SL'03, PP&SL'06] is, thus,

$$\begin{array}{ll} \text{maximize} & 1_n^{\top} \lambda\\ \text{subject to} & \mbox{diag} (\lambda) \preceq \mathrm Q\end{array}$$

Hence, we have the following SDP in $\Lambda$

$$\begin{array}{ll} \text{maximize} & \mbox{tr} (\Lambda)\\ \text{subject to} & \Lambda_{ij} = 0 \quad \forall i \neq j\\ & \Lambda \preceq \mathrm Q\end{array}$$

which is convex and, thus, easy (unlike the QCQP). This SDP does provide a lower bound on the minimum of the QCQP. If $\mathrm x \in \{\pm 1\}^n$ is in the feasible region of the QCQP, then

$$\mathrm x^{\top} \mathrm Q \,\mathrm x \geq \mathrm x^{\top} \Lambda \,\mathrm x = \sum_{i=1}^n \Lambda_{ii} x_i^2 = \sum_{i=1}^n \Lambda_{ii} = \mbox{tr} (\Lambda)$$


References

[MG&DW'95] Michel X. Goemans, David P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, Journal of the ACM, Vol. 42, No. 6, November 1995.

[BG&JM'12] Bernd Gärtner, Jiří Matoušek, Approximation Algorithms and Semidefinite Programming, Springer 2012.

[PP&SL'03] Pablo Parrilo, Sanjay Lall, Quadratically Constrained Quadratic Programming, 2003.

[PP&SL'06] Pablo Parrilo, Sanjay Lall, Quadratically Constrained Quadratic Programming, 2006.

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When there are multiple optimal solutions to an SDP (and this also occurs in primal-dual methods for LP), the primal-dual interior method (used by Mosek and most other SDP solvers) will typically not converge to an extreme point of the set of optimal solutions but rather to a point on the interior of the set of optimal solutions. In SDP the resulting solutions generically have the maximum possible rank among all optimal solutions.

There's really nothing that can be done about this in the context of using a primal-dual method to solve the SDP relaxation.

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