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Design and Analysis of Inexact Nonmonotone Sequential Quadratic Programming Algorithms for Large-Scale Nonlinear Optimization

Ahmadzadeh Salot, Hani | 2024

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 57405 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahdavi Amiri, Nezameddin
  7. Abstract:
  8. Two algorithms for solving constrained nonlinear programming problems (NLPs) are proposed. The first algorithm is an inexact nonmonotone filter sequential quadratic programming iNFSQP method. In each iteration of the algorithm, the search direction is computed using an inexact solution of a strictly convex quadratic program (QP). The search direction is a descent direction for the objective function (in feasible iterations) or a descent direction for the constraints violation function (in infeasible iterations). In this algorithm, the penalty parameter is updated in such a way that the search direction is a descent direction for the penalty function, which is a combination of the objective function and the penalty function. In order to achieve a superlinear rate of local convergence, an accelerator direction is calculated near the stationary point of the problem. Using a nonmonotone filter strategy, the global convergence of the algorithm is guaranteed. The main advantage of this algorithm is due to the establishment of its global convergence, despite the use of inexact solutions of QP subproblems. In addition, using inexact solutions instead of exact solutions of subproblems enhances the robustness and efficiency of the algorithm. However, the \textit{i}NFSQP algorithm requires the calculation and storage of the full Hessian matrices. In addition, the inexact solution of the subproblems requires the use of QP solvers as a black box. Therefore, the efficiency of this algorithm strongly depends on the performance of the QP solver. The second algorithm is a nonmonotone trust region-line search projected exact penalty method (NTRLSEP). This method leverages the correspondence between NLP solutions and minimizers of the associated \(\ell_1\)-exact penalty function. Algorithm iterations are categorized as ``infeasible", ``almost feasible", and ``local". The main purpose of infeasible iterations is to approach the feasible region. For this purpose, the penalty parameter is updated adaptively in the infeasible iterations. In almost feasible iterations, the goal is to reach a neighborhood of a stationary point of the exact penalty function. In local iterations, when we are near a stationary point of the penalty function, the Lagrange multipliers are calculated by solving a linear least squares problem. If the Lagrange multipliers satisfy the first-order optimality conditions, a Newton step is computed to obtain a superlinear convergence rate; otherwise, a dropping step is calculated to reduce the value of the penalty function. A novel nonmonotone strategy for evaluating the step directions and updating the iterates is proposed. Both algorithms have been implemented in the MATLAB software environment and have been tested on a wide range of test problems. Comparative numerical results on a large number of test problems from the CUTEst library confirm the efficiency of the two proposed algorithms as compared to some similar algorithms. In addition, the numerical results show that NTRLSEP is more efficient and more stable than iNFSQP
  9. Keywords:
  10. Exact Penalty Function ; Constrained Nonlinear Programming ; Constrained Optimization ; Sequential Quadratic Programming (SQP) ; Filters ; Quasi-Newton Methods ; Projected Structured Hessian Update ; Nonmonotone Quasi-Newton Method ; Inexact Newton Method

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