In this paper, the filter technique of Fletcher and Leyffer (1997) is used to globalize the primaldual interior-point algorithm for nonlinear programming, avoiding the use of merit functions and the updating of penalty parameters. The new algorithm decomposes the primal-dual step obtained from the perturbed first-order necessary conditions into a normal and a tangential step, whose sizes are controlled by a trust-region type parameter. Each entry in the filter is a pair of coordinates: one resulting from feasibility and centrality, and associated with the normal step; the other resulting from optimality (complementarity and duality), and related with the tangential step. Global convergence to first-order critical points is proved for the new primal-dual interior-point filter algorithm.

Jorge Nocedal z We study the local convergence of a primal-dual interior point method for nonlinear programming. A linearly convergent version of this algorithm has been shown in [2] to be capable of solving large and di cult non-convex problems. But for the algorithm to reach its full potential, it must converge rapidly to the solution. In this paper we describe how to design the algorithm so that it converges superlinearly on regular problems. Key words: constrained optimization, interior point method, large-scale optimization, nonlinear programming, primal method, primal-dual method, successive quadratic programming.

Abstract. An active set algorithm (ASA) for box constrained optimization is developed. The algorithm consists of a nonmonotone gradient projection step, an unconstrained optimization step, and a set of rules for branching between the two steps. Global convergence to a stationary point is established. For a nondegenerate stationary point, the algorithm eventually reduces to unconstrained optimization without restarts. Similarly, for a degenerate stationary point, where the strong secondorder sufficient optimality condition holds, the algorithm eventually reduces to unconstrained optimization without restarts. A specific implementation of the ASA is given which exploits the recently developed cyclic Barzilai–Borwein (CBB) algorithm for the gradient projection step and the recently developed conjugate gradient algorithm CG DESCENT for unconstrained optimization. Numerical experiments are presented using box constrained problems in the CUTEr and MINPACK-2 test problem libraries. Key words. nonmonotone gradient projection, box constrained optimization, active set algorithm,

A class of affine-scaling interior-point methods for bound constrained optimization problems is introduced which are locally q-superlinear or q-quadratic convergent. It is assumed that the strong...

Recently the authors have proposed a homogeneous and self-dual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interior-point type method, nevertheless it yields either an approximate optimal solution or detects a possible infeasibility of the problem. In this paper we specialize the algorithm to the solution of general smooth convex optimization problems that also possess nonlinear inequality constraints and free variables. We discuss an implementation of the algorithm for large-scale sparse convex optimization. Moreover, we present computational results for solving quadratically constrained quadratic programming and geometric programming problems, where some of the problems contain more than 100,000 constraints and variables. The results indicate that the proposed algorithm is also practically efficient. Department of Management, Odense University, Campusvej 55, DK-5230 Odense M, Denmark. E-mail: eda@busieco.ou.dk y ...

A slack-based feasible interior point method is described which can be derived as a modification of infeasible methods. The modification is minor for most line search methods, but trust region methods require special attention. It is shown how the Cauchy point, which is often computed in trust region methods, must be modified so that the feasible method is effective for problems containing both equality and inequality constraints. The relationship between slack-based methods and traditional feasible methods is discussed. Numerical results showing the relative performance of feasible versus infeasible interior point methods are presented.

We propose a BFGS primal-dual interior point method for minimizing a convex function on a convex set defined by equality and inequality constraints. The algorithm generates feasible iterates and consists in computing approximate solutions of the optimality conditions perturbed by a sequence of positive parameters µ converging to zero. We prove that it converges q-superlinearly for each fixed µ. We also show that it is globally convergent to the analytic center of the primal-dual optimalset when µ tends to 0 and strict complementarity holds.

The local convergence properties of a class of primal-dual interior point methods are analyzed. These methods are designed to minimize a nonlinear, nonconvex, objective function subject to linear equality constraints and general inequalities. They involve an inner iteration in which the log-barrier merit function is approximately minimized subject to satisfying the linear equality constraints, and an outer iteration that species both the decrease in the barrier parameter and the level of accuracy for the inner minimization. It is shown that, asymptotically, for each value of the barrier parameter, solving a single primal-dual linear system is enough to produce an iterate that already matches the barrier subproblem accuracy requirements. The asymptotic rate of convergence of the resulting algorithm is Q-superlinear and may be chosen arbitrarily close to quadratic. Furthermore, this rate applies componentwise. These results hold in particular for the method described by Conn, Gould, Orb...

In recent work, the local convergence behavior of path-following interior-point methods and sequential quadratic programming methods for nonlinear programming has been investigated for the case in which the active constraint gradients may fail to be linearly independent at the solution, but the Mangasarian-Fromovitz constraint qualication is satisfied. In this paper, we describe a stabilization of the primal-dual interior-point approach that ensures rapid local convergence under these conditions without enforcing the usual centrality condition associated with path-following methods. The stabilization takes the form of perturbations to the coefficient matrix in the step equations that vanish as the iterates converge to the solution.

We compare and contrast a number of recent sequential quadratic programming (SQP) methods that have been proposed for the solution of large-scale nonlinear programming problems. Both line-search and trust-region approaches are considered, as are the implications of interior-point and quadratic programming methods.