I was studying a problem in a paper in which the author tries to use the semidefinite relaxation (SDR) technique in order to solve it. After changing the original problem using the SDR technique, a rank-1 constraint is added to the problem. The author states that this constraint is non-convex, and it should be relaxed. I do not know why the rank-1 constraint is a non-convex constraint. Could someone help me out?
In an optimization problem, why is a rank-1 constraint non-convex?
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optimization
convex-analysis
convex-optimization
relaxations
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0I guess you can find plenty of material out there... – 2017-01-02
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0I could not find, if you know materials which answer my question, please let me know. – 2017-01-02
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0Given the fact that LinAlg's answer was not immediately clear to you, I would suggest that Boyd & Vandenberghe's book "Convex Optimization", specifically through chapter 4, would serve you well. – 2017-01-02
1 Answers
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The matrices $A = (-1)$ and $B=(1)$ are both rank one. However, the matrix $\lambda A + (1-\lambda)B$ with $\lambda=0.5$ does not have rank one. The set of rank one matrices is therefore not convex.
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0how this answers my question ? – 2017-01-02
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0A constraint is convex only if the set of matrices that satisfy the constraint is convex. – 2017-01-02
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0Are A and B matrices or scalars? – 2017-01-02
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0In this example, they are $1 \times 1$ matrices, but you can easily extend the example to larger matrices. For example, take $A=I$ and $B=-I$. – 2017-01-02
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0If we assume $A = I$ and $B=-I$ then the linear combination you mentioned would bring us to a zero matrix which is also rank one. I mean the matrix $\lambda A + (1-\lambda)B$ with $\lambda=0.5$ is a zero matrix which is rank one – 2017-01-02
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0I cannot relate your answer to my question. I would be pleased if you explain your answer more – 2017-01-02
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0A zero matrix is not 'rank one'. What property do you expect from a "convex constraint"? – 2017-01-02
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0Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/51087/discussion-between-ekh-and-linalg). – 2017-01-02