What Is Optimization Toolbox?

3 Standard Algorithmsis called the projected Hessian. Assuming the Hessian matrix H ispositive definite (which is the case in this implementation of SQP), then theminimum of the function q(p) in the subspace defined by occurs whengradient of,which is the solution of the system of linear equationsAstepisthentaken of the form(3-33)(3-34)At each iteration, because of the quadratic nature of the objective function,there are only twochoicesofsteplength . A step of unity along is theexact step to the minimum of the function restricted to the null space of .If such a step can be taken, without violation of the constraints, then thisisthesolutiontoQP(Equation3-30). Otherwise, the step along to thenearest constraint is less than unity and a new constraint is included in theactive set at the next iteration. The distance to the constraint boundaries inany direction is given by(3-35)which is defined for constraints not in the active set, and where the directionis towards the constraint boundary, i.e., .When n independent constraints are included in the active set, withoutlocation of the minimum, Lagrange multipliers, , are calculated that satisfythe nonsingular set of linear equations(3-36)If all elementsof are positive, is the optimal solution of QP (Equation3-30). However, if any component of is negative, and the component doesnot correspond to an equality constraint, then the corresponding element isdeleted from the active set and a new iterate is sought.3-36