## On Formulating Semidefinite Programming Problems as Smooth Convex
Nonlinear Optimization Problems

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R.J. Vanderbei and H. Yurttan Benson

Consider the diagonal entries $d_j$, $j=1,2,\ldots,n$, of the matrix
$D$ in an $LDL^T$ factorization of an $n \times n$ matrix $X$. As a
function of $X$, each $d_j$ is well-defined on the closed domain of
positive semidefinite matrices. We show that these functions are
twice continuously differentiable and concave throughout the interior
of this domain. Using these facts, we show how to formulate
semidefinite programming problems as standard convex optimization
problems that can be solved using an interior-point method for
nonlinear programming.
ORFE 99-01, Dept. of Operations Research and Financial Engineering, Princeton
University, Princeton NJ

Contact: rvdb@princeton.edu