Hi I am trying to minimize the following objective function over F. Any suggestion is appreciated.

min_F ( Trace( (I-F)^-1 V 1^T) ),

subject to

0 <= F(i,j) <= 1,

F*1 = 1 - (V + K), K is constant vector given.

here, I is identity matrix, F is n by n matrix, V is n by 1 given vector, 1 is a column vector of all ones.

I believe this is non-convex. Do you have a proof that I am wrong?

Thanks Mark. Yes, this is a nonlinear optimization problem. Actually original objective was 1^T (I-F)^-1 V. I rewrote it in this way, But still it seems like it is nonlinear. Not sure how to proceed with this.

I believe you will have to use a nonlinear optimization solver which can deal with non-convex nonlinear objective subject to linear constraints. You should be able to apply matrix-level automatic differentiation to the objective function, such as by using http://www.sc.informatik.tu-darmstadt.de/res/sw/adimat/index.en.jsp .