SQPmenetelmiä

SQPmetodi, which stands for Sequential Quadratic Programming, is a class of algorithms used for solving nonlinear optimization problems. These problems involve finding the minimum or maximum of an objective function subject to a set of constraints, which may also be nonlinear. SQP methods are iterative, meaning they generate a sequence of solutions that ideally converge to the optimal solution.

At each iteration, an SQP method approximates the original nonlinear problem with a simpler subproblem, typically a quadratic programming (QP) problem. This QP subproblem is then solved to determine a search direction. The objective function is approximated using a quadratic model, and the constraints are approximated using linear models. The QP subproblem incorporates information about the gradients and Hessians of the objective function and constraints of the original problem.

The solution to the QP subproblem provides an estimate of the next point in the sequence and a descent direction. A line search or a trust-region method is then used to determine a step size along this direction, ensuring that progress is made towards a solution. The algorithm then updates its approximations of the objective function and constraints and repeats the process until a convergence criterion is met. SQP methods are known for their efficiency and robustness, particularly for problems with a moderate number of variables and constraints. They are widely used in various fields, including engineering design, economics, and machine learning.