# Complexity Analysis of Interior Point Algorithms for Non-Lipschitz and Nonconvex Minimization

We propose a first order interior point algorithm for a class of non-Lipschitz and nonconvex minimization problems with box constraints, which arise from applications in variable selection and regularized optimization. The objective functions of these problems are continuously differentiable typically at interior points of the feasible set. Our algorithm is easy to implement and the objective function value is reduced monotonically along the iteration points. We show that the worst-case complexity for finding an $\epsilon$ scaled first order stationary point is $O(\epsilon^{-2})$. Moreover, we develop a second order interior point algorithm using the Hessian matrix, and solve a quadratic program with ball constraint at each iteration. Although the second order interior point algorithm costs more computational time than that of the first order algorithm in each iteration, its worst-case complexity for finding an $\epsilon$ scaled second order stationary point is reduced to $O(\epsilon^{-3/2})$. An $\epsilon$ scaled second order stationary point is an $\epsilon$ scaled first order stationary point.

## Citation

Department of Applied Mathematics, The Hong Kong Polytechnic University, July, 2012