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Say, I have some convex objective function w.r.t an optimization value vector x(e.x linear w.r.t x)

I'd like to minimize the objective plus, make outer product of x with itself to be similar to given positive semidefinite matrix X (i.e $xx^T \sim X$)

To make it, I want a suitable convex loss(penalty) function

Is there any suitable convex loss function?

it's okay if you can make a convex problem by adding some constraint though the loss itself is not always convex

(I checked frobenius norm but the hessian of $ ||X-xx^T||$ gives $-4X+4(x^Tx)I+8xx^T$. Thus it is not convex. If I addition the constraint $X \le (x^Tx)I+2xx^T$, the loss function is convex in the feasible set. However, the constraint is not suit to a convex problem because right hand side is convex function. Constraint for a convex problem should be (convex function <= 0) form )

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    How about [Schatten Norm](https://en.wikipedia.org/wiki/Schatten_norm)? And specifically Nuclear Norm? – 2017-01-02
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    hi @Royi anything is ok if it is convex. But then is it a convex? Nuclear_Norm(X-xx^T) is convex?? how can I prove it? – 2017-01-02
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    I'm not sure as you plug in non linear function of $ x $. What about trying to linearize one of the Norms? – 2017-01-02
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    yes nonlinear term makes me hard to solve it. What does "linearizing one of the norms" mean? – 2017-01-02
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    Try using approximation of the norm with respect to your term. Just like a Taylor Series of your object function. – 2017-01-02
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    It might be helpful because in my problem setting x is above -1 and below 1 but not sure whether it works. Thanks anyway! – 2017-01-02
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    You're out of luck here. There is no convex loss function in this case. You'll have to go with a relaxation method. – 2017-01-02

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