A Primal-Dual Augmented Lagrangian Penalty-Interior-Point Algorithm for Nonlinear Programming
|Other Titles:||Ein primal-dualer erweiterter Lagrange Bestrafungs-Innere-Punkte Algorithmus für die nichtlineare Optimierung||Authors:||Kuhlmann, Renke||Supervisor:||Büskens, Christof||1. Expert:||Büskens, Christof||Experts:||Gill, Philip E.||Abstract:||
This thesis treats a new numerical solution method for large-scale nonlinear optimization problems. Nonlinear programs occur in a wide range of engineering and academic applications like discretized optimal control processes and parameter identification of physical systems. The most efficient and robust solution approaches for this problem class have been shown to be sequential quadratic programming and primal-dual interior-point methods. The proposed algorithm combines a variant of the latter with a special penalty function to increase its robustness due to an automatic regularization of the nonlinear constraints caused by the penalty term. In detail, a modified barrier function and a primal-dual augmented Lagrangian approach with an exact l2-penalty is used. Both share the property that for certain Lagrangian multiplier estimates the barrier and penalty parameter do not have to converge to zero or diverge, respectively. This improves the conditioning of the internal linear equation systems near the optimal solution, handles rank-deficiency of the constraint derivatives for all non-feasible iterates and helps with identifying infeasible problem formulations. Although the resulting merit function is non-smooth, a certain step direction is a guaranteed descent. The algorithm includes an adaptive update strategy for the barrier and penalty parameters as well as the Lagrangian multiplier estimates based on a sensitivity analysis. Global convergence is proven to yield a first-order optimal solution, a certificate of infeasibility or a Fritz-John point and is maintained by combining the merit function with a filter or piecewise linear penalty function. Unlike the majority of filter methods, no separate feasibility restoration phase is required. For a fixed barrier parameter the method has a quadratic order of convergence. Furthermore, a sensitivity based iterative refinement strategy is developed to approximate the optimal solution of a parameter dependent nonlinear program under parameter changes. It exploits special sensitivity derivative approximations and converges locally with a linear convergence order to a feasible point that further satisfies the perturbed complementarity condition of the modified barrier method. Thereby, active-set changes from active to inactive can be handled. Due to a certain update of the Lagrangian multiplier estimate, the refinement is suitable in the context of warmstarting the penalty-interior-point approach. A special focus of the thesis is the development of an algorithm with excellent performance in practice. Details on an implementation of the proposed primal-dual penalty-interior-point algorithm in the nonlinear programming solver WORHP and a numerical study based on the CUTEst test collection is provided. The efficiency and robustness of the algorithm is further compared to state-of-the-art nonlinear programming solvers, in particular the interior-point solvers IPOPT and KNITRO as well as the sequential quadratic programming solvers SNOPT and WORHP.
|Keywords:||Nonlinear Programming; Large-Scale Optimization; Primal-Dual Penalty-Interior-Point Algorithm; Augmented Lagrangian Method; Modified Barrier Method; Parametric Sensitivity Analysis; WORHP||Issue Date:||6-Dec-2018||Type:||Dissertation||URN:||urn:nbn:de:gbv:46-00106924-13||Institution:||Universität Bremen||Faculty:||FB3 Mathematik/Informatik|
|Appears in Collections:||Dissertationen|
checked on Jan 28, 2023
checked on Jan 28, 2023
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