A Globally Convergent Conjugate Gradient Method for Minimizing Self-Concordant Functions with Application to Constrained Optimisation Problems

Self-concordant functions are a special class of convex functions introduced by Nesterov and Nemirovskii and used in interior point methods. This paper proposes a damped conjugate gradient method for optimization of self-concordant functions. This method is an ordinary conjugate gradient method but with a novel step-size selection rule which is proved to ensure the algorithm converges to the global minimum. As an example, the algorithm is applied to a quadratically constrained quadratic optimization problem.