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

### 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