Ask AI: the SQP method should start from a feasible design point

Sequential Quadratic Programming (SQP) is a popular and effective method for solving constrained optimization problems. One of the key aspects of the SQP method is that it involves solving a sequence of quadratic programming subproblems, each of which approximates the original nonlinear problem. Starting from a feasible design point is generally recommended for the following reasons: 1. **Feasibility Maintenance**: SQP methods are designed to handle constraints efficiently, but starting from a feasible point simplifies the process of maintaining feasibility throughout the iterations. This ensures that the iterates remain within the feasible region, avoiding violations of constraints that could complicate the solution process. 2. **Improved Convergence**: While SQP methods can, in theory, handle infeasible starting points, starting from a feasible point often leads to faster convergence to the optimal solution. When the initial point is feasible, the algorithm can focus on improving the objective function value without having to first bring the solution into the feasible region. 3. **Numerical Stability**: Starting from a feasible point can enhance the numerical stability of the method. Infeasible points may lead to issues such as ill-conditioned QP subproblems or large infeasibility measures, which can degrade the performance and robustness of the algorithm. 4. **Reduction of Computational Effort**: An initial feasible point can reduce the computational effort required in the first few iterations of the SQP method. The algorithm can take more meaningful steps toward optimality rather than having to deal with constraint violations. However, there are also strategies in SQP methods to deal with infeasible starting points if no feasible point is readily available. These strategies include: 1. **Feasibility Restoration Phase**: Some SQP implementations include a preliminary phase where the initial goal is to find a feasible point. Once feasibility is restored, the algorithm proceeds with the standard SQP iterations. 2. **Penalty Functions**: SQP methods can incorporate penalty functions to handle infeasible iterates by penalizing constraint violations in the objective function. This encourages the iterates to move toward feasibility while optimizing the objective. 3. **Filter Methods**: These methods use a filter approach to manage trade-offs between objective improvement and constraint satisfaction. This way, the algorithm can accept iterates that do not strictly improve both criteria at every step but contribute to overall progress. In summary, while starting from a feasible design point is advantageous and can lead to more efficient and stable convergence in the SQP method, there are techniques available to handle infeasible starting points when necessary.