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I am looking for practical applications of semidefinite- programming. So far, I found that the low-rank matrix completion problem (recomendendattion matrices) can be expressed as a semidefinite program. The same goes for the combinatorial problem MAX-CUT.

  1. What is a practical applications of the MAX-CUT problem?
  2. What would be a third practical applications of semidefinite programming?

Would be nice if anyone could recommend any references. Thanks.

Rodrigo de Azevedo
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P.Müller
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    Do they use semidefinite programming in industry? – Rodrigo de Azevedo Apr 02 '19 at 06:49
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    One application is determining whether a polynomial can be expressed as a sum of squares. Take a look at this. – Rodrigo de Azevedo Apr 02 '19 at 06:51
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    If you haven't already read Boyd and Vandenberghe, it discusses a bunch of applications of SDPs (including in the exercises and the additional exercises). – littleO Apr 02 '19 at 09:19
  • I've looked into Boyd and Vandenberghe. Problem is, these problems are convex or quasiconvex and since semidef. problems are just a subfield, things dont really fit. Or am I wrong here? – P.Müller Apr 03 '19 at 04:12
  • @Rodrigo the sum of squares is quite interesting. I read https://math.stackexchange.com/questions/2410994/write-x2-y2-z22-3-x3-y-y3-z-z3-x-as-a-sum-of-three-squ/2414847#2414847 .
    How would I have to choose A_i if I want to transform min tr(Q) s.t. A(Q)=b,Q⪰O to the standard sdp form min tr(C,Q) s.t. Q⪰O ,tr(A_i,Q)=b_i .     – P.Müller Apr 03 '19 at 04:49
  • @P.Müller It's already in standard form! Look at the code. – Rodrigo de Azevedo Apr 07 '19 at 03:07
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    you are right :) thanks. Sadly your matrix Q is only positiv semidefinite so I cant do the cholesky decomposition. How do you get the matrix 6x2 square root =
    Matrix([[ 1, 0], [-1/2, sqrt(3)/2], [-1/2, -sqrt(3)/2], [-3/2, -sqrt(3)/2], [ 0, sqrt(3)], [ 3/2, -sqrt(3)/2]])?     – P.Müller Apr 08 '19 at 06:42

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