Applying Infeasible Interior Point Method To SQP For Constrained Nonlinear Programming

The field of constrained nonlinear programming (NLP) has been principally challenging to various gradient based optimization techniques. The Sequential quadratic programming algorithm (SQP) that uses active set strategy in solving quadratic programming (QP) subproblems proves to be efficient in locating the points of local optima. However, its efficient determination of the optimal active set heavily relies on the initial guess of the starting point. This remains a serious drawback to both primal and dual active set approaches especially for NLPs with several inequality constraints.Active set strategy is a discrete based technique that originates from the simplex algorithm for linear programming (LP). We have investigated both the primal and dual active set methods (ASM). The primal ASM proves to be more intuitive and is widely applicable since it only requires the Hessian of the QP problem to be at least positive semidefinite. However, the possibility of cycling poses a great set-back to the success of the primal ASM. This is contrary to the dual ASM that requires a strictly convex (i.e. positive definite) QP. Besides its immunity form cycling, dual active set method also assures finite convergence for any given feasible problem. However, the main drawback of the dual ASM is its limited applicability to only convex QPs. This is crucial as majority of engineering optimization problems tends to be non-convex in nature. Consequently, active set methods in general suffer deteriorating performance and premature convergence especially when faced with an NLP consisting having several inequality constraints.Thus, we propose an SQP/IIPM algorithm that uses infeasible interior point method (IIPM) for solving QP subproblems. In this approach inequality constraints can be solved directly, alleviating the burden for choosing a feasible starting point necessary for efficient convergence to optimal active set. At every iteration k, we evaluate step length adaptively via a simple line search and/or a quadratic search algorithm depending on the reason for the termination of our infeasible interior point method QP solver. Our SQP/IIPM algorithm falls to quadratic search only if the termination of the IIPM QP solver satisfies the complementary slackness condition (i.e. duality measure). This means that neither the Primal nor the Dual feasibility conditions are satisfied during the termination of the IIPM QP solver. In addition, the algorithm can also switch to quadratic search whenever the line search algorithm exceeds its maximum iteration limit before getting an acceptable value for the step length parameter.Moreover, in order to facilitate efficient and robust determination of a step length parameter that can ensure significant decrease in the search direction and at the same time avoid any constraint violation, we employ different merit functions; one is suitable for the quadratic search and another will suite the line search algorithm. Although the standard SQP algorithm uses only line search algorithm to evaluate its step length parameter, our use of both line search and quadratic search algorithms follows the fact that, line search algorithm suffers irregular failures when it is faced with NLP having several nonlinear inequality constraints. Hence, although quadratic search algorithm tends to be more expensive, it can offer guarantee for the successful determination of a qualitative step length parameter. Therefore, our switching technique is designed to adaptively make a trade-off between either of the two minimization algorithms.Furthermore, SQP/IIPM algorithm is designed to handle QP problems having both equality and inequality constraints concurrently. Our utilization of the damped BFGS update procedure has given us the freedom to adequately approximate and update the Hessian matrix at every major iteration of the SQP/IIPM algorithm. Thus, the capability of the SQP/IIPM algorithm of delivering reliable search directions for both convex and non convex QP problems is guaranteed.To eliminate the difficulties associated with the process of solving NLPs, we have designed a graphical user interface (GUI) for our SQP/IIPM algorithm based on Matlab's GUI development environment. Titled "SQP/IIPM Optimization System", our system is implemented with robust error handling procedures that will ensure safe operation. Our implementation also features simplicity and clarity for enhancing user friendliness.Benchmark numerical problems (i.e. constrained NLPs) are used for performance assessment. The numerical performance of the proposed SQP/IIPM algorithm is compared against that of the standard SQP algorithm. The test results show that, SQP/IIPM algorithm has a satisfactory numerical performance. It requires less number of iterations, function evaluations and remarkably less cpu-time for convergence to optimum solution. We finally conducted a control system engineering test where the proposed algorithm is applied in the optimization of Proportional-Integral-Derivative (PID) controllers. The quality of the obtained controller parameters is compared with that of the classical Gain and Phase Margin (GPM) method and Ziegler-Nichols (ZN) method. Our test results reveal that, SQP/IIPM algorithm has overcome the overshoot encountered in the GPM method and the quite lengthy settling-time associated with the ZN method. Thus, our system is robust, efficient and promising in dealing with parameter optimization of PI/PID controlled systems.