Methods for Constrained Optimization - MIT Certificate Error

Lagrange also increased the run-time, as shown in Figure 4. Quadratic programming stillperforms very fast because it simply treats the active set of inequality constraints asequality constraints, and then runs a classical Lagrange multiplier procedure, which forquadratic optimizer and linear constraints, is quite a simple and fast task. The gradientprojection method first checked if the global minimum satisfied the constraints, and sinceit did not, the algorithm chose some point on the active constraint set—in this casex ′ = [ 1.75, 0]—and made a projection of the steepest path, which overshot anotherconstraint. It found the point at which the two constraints intersected and made anotherprojection from there. After detecting that the projection went in an infeasible direction,the algorithm concluded that the corner point must be the solution.Figure 4. Comparison of run-time for methods handling inequality constraints4.3 Using Both Equality and Inequality ConstraintsI implemented two methods that can handle both equality and inequality constraintssimultaneously, which are the penalty function method and the QP method. Figure 5shows the results of these two methods on the full example problem.