Select 10 of the characteristics below that best describe the Sequential Quadratic Programming algorithm for finding the optimal solution to nonlinear problems.

This method converges quickly and is the fastest of the NLP solvers.

If the method is stopped before it converges, the solution that is available may not be feasible, as the method does not require all iterations to be feasible.

Because the algorithm requires iterative solutions to be feasible the algorithm works will with linked simulation models that require feasible state variable values.

The algorithm does not require that all iterations are feasible. In fact, this can be a weakness of the algorithm.

This algorithm allows the constraints to be violated, but requires that the violation be lessened in each iteration. The magnitude of the violation is measured by a penalty function. If the current delta x will cause the new point to have a larger penalty function, then delta x is made smaller until the new point will have a smaller penalty function.

The first step is to make an approximation of the problem so that the problem can be represented as a quadratic programming problem with a quadratic objective function and linear constraints. The initial guess for the Lagrangian Hessian is the identity matrix.

The optimization problem becomes a quadratic programming problem, (an optimization problem with a quadratic objective and linear constraints) which is straight forward to solve as the KKT conditions create a set of linear equations.

It can be a more robust solver than some others, as it uses a second order (quadratic) approximation to the objective function.

The algorithm stays feasible through all iterations. If it starts at an infeasible point, then it will work to find a constraint to get to the feasible region and then work along a constraint.

The second step of the algorithm is to find the Lagrange multipliers by solving the Quadratic Programming problem.

The main idea of the method is that it travels in the steepest direction that stays tangent to the constraints. If inequality constraints are included, then the algorithm incorporates slack variables The main idea of the method is that it travels in the steepest direction that stays tangent to the constraints. If inequality constraints are included, then the algorithm incorporates slack variables

The fourth step is to compute the reduced gradient by taking the derivative of the objective function and the constraints with respect to the dependent and independent variables. The reduced gradient is created by including the gradient of the constraints into the gradient of the objective function

The method works by identifying independent and dependent decision variables. The independent decision variables are used to improve the objective function and the dependent variables are used to make sure constraints are satisfied. The reduced gradient is formed by incorporating the first derivative of the constraints into the derivative of the objective function.

The method implicitly reduces variables - similar to the substitution method, but implicitly. It is implicit, by including the derivative of the constraints into the derivative of the objective function.

If the method is stopped before it converges, the solution that is available will be feasible, and thus, possibly useful.

The method works by identifying independent and dependent decision variables. The independent decision variables are used to improve the objective function and the dependent variables are used to make sure constraints are satisfied. The reduced gradient is formed by incorporating the first derivative of the constraints into the derivative of the objective function.

Because the algorithm does not require iterative solutions to be feasible (except for the final solution) the algorithm may not work will with linked simulation models that require feasible state variable values for all iterations.

The sixth step is to take a step with all the independent variables in a line search. This has to be done so that all the constraints are met. So, once the new values of the independent variables is known, then the dependent variables are calculated. A search algorithm such as Newton-Rapson can be used to find a feasible value for the independent and dependent variables. If an independent variable is at its boundary, then set it equal to its upper or lower bound.

The algorithm makes a quadratic approximation of the objective function and linear approximation of the constraints.

The method estimates the next iteration by using the gradient to improve the objective function and by requiring that the derivative of the constraint not be changed - and thus be met.