Trust-region interior-point SQP algorithms for a class of nonlinear programming problems A family of trust-region interior-point sequential quadratic programming (SQP) algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise, e.g., from the discretization of optimal control problems. The algorithms treat states and controls as independent variables. They are designed to take advantage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems arising from optimal control problems governed by partial differential equations.par The algorithms keep strict feasibility with respect to the bound constraints by using an affine scaling method proposed, for a different class of problems, by {it T. F. Coleman} and {it Y. Li} [SIAM J. Optim. 6, 418-445 (1996; Zbl 0855.65063)] and they exploit trust-region techniques for equality-constrained optimization. Thus, they allow the computation of the steps using a variety of methods, including many iterative techniques.par Global convergence of these algorithms to a first-order Karush-Kuhn-Tucker (KKT) limit point is proved under very mild conditions on the trial steps. Under reasonable, but more stringent, conditions on the quadratic model and on the trial steps, the sequence of iterates generated by the algorithms is shown to have a limit point satisfying the second-order necessary KKT conditions. The local rate of convergence to a nondegenerate strict local minimizer is $q$-quadratic. The results given here include, as special cases, current results for only equality constraints and for only simple bounds.par Numerical results for the solution of an optimal control problem governed by a nonlinear heat equation are reported. (Source:

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  1. van Leeuwen, T.; Herrmann, F.J.: A penalty method for PDE-constrained optimization in inverse problems (2016)
  2. Ma, Yan-Bo: Newton method for estimation of the Robin coefficient (2015)
  3. Lantoine, Gregory; Russell, Ryan P.: A hybrid differential dynamic programming algorithm for constrained optimal control problems. I: Theory (2012)
  4. Andreani, R.; Birgin, E.G.; Martínez, J.M.; Schuverdt, M.L.: Second-order negative-curvature methods for box-constrained and general constrained optimization (2010)
  5. Liu, Xinwei; Yuan, Yaxiang: A null-space primal-dual interior-point algorithm for nonlinear optimization with nice convergence properties (2010)
  6. Lee, Hyung-Chun: Optimal control problems for the two dimensional Rayleigh-Bénard type convection by a gradient method (2009)
  7. Lu, Wenting; Yao, Yirong; Zhang, Liansheng: A penalized interior point approach for constrained nonlinear programming (2009)
  8. Antil, H.; Hoppe, R.H.W.; Linsenmann, Chr.: Path-following primal-dual interior-point methods for shape optimization of stationary flow problems (2007)
  9. Audet, Charles; Orban, Dominique: Finding optimal algorithmic parameters using derivative-free optimization (2006)
  10. Chen, L.; Goldfarb, D.: Interior-point $\ell_2$-penalty methods for nonlinear programming with strong global convergence properties (2006)
  11. Chen, Z.W.: A penalty-free-type nonmonotone trust-region method for nonlinear constrained optimization (2006)
  12. Hager, William W.; Zhang, Hongchao: A new active set algorithm for box constrained optimization (2006)
  13. Hoppe, Ronald H.W.; Linsenmann, Christopher; Petrova, Svetozara I.: Primal-dual Newton methods in structural optimization (2006)
  14. Leibfritz, F.; Volkwein, S.: Reduced order output feedback control design for PDE systems using proper orthogonal decomposition and nonlinear semidefinite programming (2006)
  15. Di Pillo, Gianni; Lucidi, Stefano; Palagi, Laura: Convergence to second-order stationary points of a primal-dual algorithm model for nonlinear programming (2005)
  16. Gilbert, J.Charles; Gonzaga, Clovis C.; Karas, Elizabeth: Examples of ill-behaved central paths in convex optimization (2005)
  17. Mostafa, El-Sayed M.E.: A trust region method for solving the decentralized static output feedback design problem (2005)
  18. Yamashita, Hiroshi; Yabe, Hiroshi; Tanabe, Takahito: A globally and superlinearly convergent primal-dual interior point trust region method for large scale constrained optimization (2005)
  19. Zhu, Zhibin: A feasible interior-point algorithm for nonconvex nonlinear programming (2005)
  20. El-Alem, M.M.; El-Sayed, S.; El-Sobky, B.: Local convergence of the interior-point Newton method for general nonlinear programming (2004)

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