A primal-dual trust-region algorithm for non-convex nonlinear programming (Q1591480)

The authors propose a primal-dual algorithm for the problem \(\min f(x)\) subject to \(Ax=b, c(x) \geq 0\), where \(f: \mathbb{R}^n \to \mathbb{R}\) and \(c: \mathbb{R}^n \to \mathbb{R}^m\) are twice continuously differentiable and an \(m\times n\)- matrix \(A\) has full rank. The algorithm is basically a sequential minimization of a logarithmic barrier function \(\phi(x,\mu_k)=f(x)-\mu_k \langle e, \log(c(x))\rangle, e=(1, \dots, 1)\), for a sequence \(\mu_k \to 0\). Convergence is proved to second-order critical points from arbitrary starting points. Numerical results are presented for general quadratic problems.