We analyze an iterative method for minimizing general nonlinear function with equality and inequality constraints by interior point method. Applying the barrier penalty technique, we solve a sequence of unconstrained minimization problems with primal-dual merit function. We develop an iterative method to compute the Newton steps. Descent direction and direction of negative curvature are found by using alternatively Conjugate Gradient and Lanczos methods. Various computational enhancements to improve these directions are provided. Our strategy is well adapted to deal with indefiniteness and can be applied to large scale nonlinear optimization problems. Promising preliminary numerical results are presented.

We consider strategies for selecting the barrier parameter at every iteration of an interior-point method for nonlinear programming. Numerical experiments suggest that adaptive choices outperform static strategies that hold the barrier parameter fixed until a barrier optimality test is satisfied. A new adaptive strategy is proposed based on the minimization of a quality function which measures the KKT error violation. We also propose a globalization framework that ensures the convergence of adaptive interior methods. Numerical results will be presented comparing various strategies.

Quadratic programs (QPs) play a fundamental role in optimization. They arise as subproblems in algorithms for nonlinear optimization, and in their own right represent a rich class of applications. I discuss an active-set approach for solving large-scale QPs based on gradient projection and augmented Lagrangian methods. Inexpensive gradient-projection iterations identify an active constraint set; they are followed by an approximate solution of a saddle-point system in a subspace. I present numerical results and comment on the suitability of the approach for use in sequential quadratic programming algorithms for nonlinearly constrained optimization.

In this paper we discuss the polynomiality of Mehrotra-type predictor-corrector algorithms. We consider a variant of the original prototype of the algorithm that has been widely used in several IPM based optimization packages, for which no complexity result is known to date. By an example we show that in this variant the usual Mehrotra-type adaptive choice of the parameter $mu$ might force the algorithm to take many small steps to keep the iterates in a certain neighborhood of the central path, which is essential to prove the polynomiality of the algorithm. This example has motivated us to incorporate a safeguard in the algorithm that keeps the iterates in the prescribed neighborhood, while it allows us to warrantee a minimal step size. This safeguard strategy is used also when the affine scaling step performs poorly that forces the algorithm to take pure centering steps. To ensure the asymptotic convergence of the algorithm, we changed Mehrotra's heuristic slightly. The new algorithms have the same iteration complexity as the previous one, but enjoy superlinear convergence rate as well.