## The Single Server QueueWhen Mary and her son come to stay with Katy and her divorced father, Katy has a hard time learning to share her father's love. Suggested level: primary. |

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### Contents

1 The watchdog technique for forcing convergence in algorithms | 1 |

2 Reduced quasiNewton methods with feasibility improvement | 18 |

3 A superlinearly convergent algorithm for constrained optimization | 60 |

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### Common terms and phrases

active constraints active set active set strategy Ak+1 applied approximation augmented Lagrangian bounded codes computational conjugate gradient methods constrained problems defined descent direction equality constraints equality-constrained equations exact penalty function exterior penalty feasible point formula global convergence Hence Hessian I(zk inequality constraints infeasible Jacobian Kuhn-Tucker conditions Lagrange multiplier estimates Lagrange multipliers Lagrangian function Lemma line search linear constraints linear programming linearised major iteration manifold Mathematical Programming matrix minimization MINOR ITERATIONS Murtagh and Saunders Newton's method nonlinear constraints nonlinear programming objective function obtained Optimization Theory original problem penalty function penalty parameter positive definite procedure Proof Proposition QP sub-problem QP(x quadratic programming quasi-Newton method rate of convergence reduced gradient reduced problem relaxed criterion S(zk satisfied search direction Section solution solving step length strict complementary slackness subproblem superbasic superbasic variable superlinear convergence test problems Theorem unconstrained update variable metric vector W(xk watchdog technique xk+i zero