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I've formulated some problem as follow and I want to check whether it is a convex optimization problem.

the optimization value is a vector, x and V is a fixed positive semidefinite matrix

$$ minimize \quad b^Tx $$ $$ subject \; to $$ $$ V \; \le \; (x^Tx)I+2xx^T $$

The inequality in the constraint means generalized inequality (not elementwise ineq)

Of course, the objective is linear thus, convex but I'm not sure about the constraint

Please tell me whether it is a convex problem or not.

Additionally, if it is, how can I optimize the vector x? the interior-point method with log determinant barrier?

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    I think I proved that the rhs (including -V) of the constraint is convex. thus it does not satisfy a convex problem setting. To whom might get interested I will show how I proved it. $(x^Tx)I$ is convex w.r.t S+. I showed $2xx^T-V$ is convex w.r.t S+. First substitute x+tv to x. Then I showed it is convex w.r.t one-dimensional variable t for any vector x and v. I used the definition of the convex. In short, I proved it is convex for any line thus it is convex. – 2017-01-02
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    Just take V=0 and a single scalar variable x: x^2+2x>=0, which is not convex. – 2017-01-02

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