SELF-CONCORDANT EXPONENTIAL KERNEL FUNCTION BASED INTERIOR-POINT ALGORITHM FOR SEMIDEFINITE OPTIMIZATION

Semidefinite optimization (SDO) problem is an extension of linear optimization (LO) problem by replacing the vector of variables with a symmetric matrix and the nonnegativity constraints with a positive semidefinite constraint. Most of interior-point methods (IPMs) with polynomial-time complexity can be generalized to the case of semidefinite optimization. In this paper, we present a primal-dual interior-point algorithm for semidefinite optimization problems based on a new self-concordant (SC) exponential kernel function. Combining both properties of self-concordance and kernel function properties for this function, we design and analyze the algorithm and derive the complexity bound for large-update methods. The obtained complexity bounds are analogous to the result in [6] for LO. Finally, we implement the method with different kernel functions and use it to solve the numerical examples. We also provide a comparison of the performances of these different versions of the algorithm when applied to the numerical examples.