Wednesday, November 6, 2013

Complexity of solving conic quadratic problems

First a clarification conic quadratic optimization and second order cone optimization is the same thing. I prefer the name conic quadratic optimization though.

Frequently it is asked on the internet what is the computational complexity of solving conic quadratic problems. Or the related questions what is the complexity of the algorithms implemented in MOSEK, SeDuMi or SDPT3.

Here are a some typical questions
To the best of my knowledge almost all open source and commercial software employ a primal-dual interior-point algorithm using for instance the so-called Nesterov-Todd scaling.

A conic quadratic problem can be stated on the form
\[
\begin{array}{lccl}
\mbox{min} & \sum_{j=1}^d (c^j)^T x^j & \\
\mbox{st} & \sum_{j=1}^d A^j x^j & = & b \\
& x^j \in K^j & \\
\end{array}
\]
where \(K_j\) is a \(n^j\) dimensional quadratic cone. Moreover, I will use \(A = [A^1,\ldots, A^d ]\) and \(n=\sum_j n^j\). Note that \(d \leq n\). First observe the problem cannot be solved exactly on a computer using floating numbers since the solution might be irrational. This is in contrast to linear problems that always have rational solution if the data is rational.

Using for instance the primal-dual interior point algorithm the problem can be solved to \(\varepsilon\) accuracy in \(O(\sqrt{d} \ln(\varepsilon^{-1}))\) interior-point iterations, where \(\varepsilon\) is the accepted duality gap. The most famous variant having that iteration complexity is based on Nesterov and Todds beautiful work on symmetric cones.

Each iteration requires solution of a linear system with the coefficient matrix
\[ \label{neweq} \left [ \begin{array}{cc} H & A^T \\ A & 0 \\ \end{array} \right ] \mbox{ (*)} \]
This is the most expensive operation and that can be done in \(O(n^3)\) complexity using Gaussian elimination so we end at the complexity \(O(n^{3.5}\ln(\varepsilon^{-1}))\).
That is the theoretical result. In practice the algorithms usually works much better because they normally finish in something like 10 to 100 iterations and rarely employs more than 200 iterations. In fact if the algorithm requires more than 200 iterations then typically numerical issues prevent the software from solving the problem.

Finally, typically conic quadratic problem is sparse and that implies the linear system mentioned above can be solved must faster when the sparsity is exploited. Figuring our to solve the linear equation system (*) in the lowest complexity when exploiting sparsity is NP hard and therefore optimization only employs various heuristics such minimum degree order that helps cutting the iteration complexity. If you want to know more then read my Mathematical Programming publication mentioned below. One important fact is that it is impossible to predict the iteration complexity without knowing the problem structure and then doing a complicated analysis of that. I.e. the iteration complexity is not a simple function of the number constraints and variables unless A is completely dense.

To summarize primal-dual interior-point algorithms solve a conic quadratic problem in less 200 times the cost of solving the linear equation system (*) in practice.

So can the best proven polynomial complexity bound be proven for software like MOSEK. In general the answer is no because the software employ an bunch of tricks that speed up the practical performance but unfortunately they destroy the theoretical complexity proof. In fact, it is commonly accepted that if the algorithm is implemented strictly as theory suggest then it will be hopelessly slow.

I have spend of lot time on implementing interior-point methods as documented by the Mathematical Programming publication and my view on the practical implementations are that are they very close to theory. 

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