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Short version

Is there a duality gap between a QCQP problem and the SDP problem obtained through Lagrangian relaxation?

A paper I'm studying is using this fact, but I cannot achieve the authors' results.

Longer version

I've been trying to reproduce the results from a paper entitled "Supervised Rank Aggregation" [pdf] that devises a formalism to aggregate ranks as a non-convex optimization problem, then transforms it to a non-convex QCQP problem and finally to a convex SDP problem through Lagrangian relaxation.

Since I could not achieve the results of the authors, I dug deeper in the proof and I am not 100% sure about a part of it.

The idea is to give a weight to each ranking so that the resulting rank is as close as possible to a target ranking, minimizing the error.

For example, given only one ranking input: [2, 1] the target ranking [2,1], the algorithm should return the list of weights [1] with error 0. This is far away from what I get: error is close to 0 (actually 3.17e-5) but the weight is [-0.448].

I implemented it in python, using picos and cvxopt to solve the SDP problem. This gist is the source code

Usage is simple:

>>> mc = MarkovChain(columns=[[2,1]], target=[2,1])
>>> alpha, x, t = mc.solve()

I know the QCQP formulation (8.2.2 in the paper) is correct. The relaxation seems fullproof too. Courses available on the Internet[1] explain the maths behind it. The SDP formulation (8.2.5) seems correct.

Then the authors claim the QCQP problem is strictly feasible (there is a solution in the interior of the domain) and therefore the Slater condition for the SDP problem holds, so there is no duality gap, and the solution of the dual is the solution of the primal.

Here is my question: is the last affirmation true? I am not sure and it seems like the most plausible reason why I cannot reproduce any result.

Thank you very much.

1: google "Notes on relaxation and randomized methods for nonconvex QCQP", §2.2

Rodrigo de Azevedo
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user73053
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  • I am having a hard time fully understanding if the matrices $H_0$ and $D$ are not PSD. Either way, the authors of the paper are correct, but for the wrong reasons (taking what your said above at face value): in fact Slater's condition is not (immediately, at least) why strong duality holds in this case. The case given is a very special one where a nonconvex QCQP with exactly one nonconvex inequality constraint still satisfies strong duality. The result is somewhat technical, but a basic and complete exposition is given in appendix B of Boyd & Vandenberghe. – Guillermo Angeris Feb 05 '20 at 19:40

1 Answers1

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Alright, I believe that the result can be found in the book "Handbook of Semidefinite Programming: Theory, Algorithms, and Applications" edited by Wolkowicz, Saigal, and Vandenberghe. It's really expensive, so sorry for that. Anyway, chapter 4 is entitled "Duality and Optimality Conditions" and is written by Shapiro and Scheinberg. There are two results you need.

Theorem 4.1.3 Suppose that the primal problem (P) is convex and that the regularity condition (4.1.28) holds. Then there is no duality gap between the primal and dual problems and, moreover, if their common optimal value is finite, then the set Sol(D), of optimal solutions of the dual problem, is nonempty and bounded.

Proposition 4.1.4 Suppose that the mapping G(x) is convex with respect to the cone Q:=-K and that K has a nonempty interior. Then (4.1.28) is equivalent to the Slater condition.

This is on page 76 and page 77 of the aforementioned book.

Looking for some more easily accessible references, there's a research paper entitled "Complementarity and Nondegeneracy in Semidefinite Programming" by Alizadeh, Haeberly, and Overton that references this result on Assumption 2, but they refer to Nesterov and Nemirovskii's book, "Interior Point Polynomial Algorithms in Convex Programming". I find their stuff hard to read, but it's probably somewhere around Theorem 4.2.1 in their book on page 108 and 109.

Anyway, it's a moderately technical result and there's a bunch of assumptions involved, so I'm not vouching for the paper that you cited. Mostly, these are the references that I could think of off the top of my head that deal with a zero duality gap and Slater's constraint qualification.

wyer33
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