Methods for Constrained Optimization - MIT Certificate Error

QP methods only take one step to reach the solution. The gradient projection methodstarts from x [ 0.5, 0]on the constraint and makes a projection to the solution.0=Figure 2 shows that the Lagrange multiplier and QP methods derive the solution slightlyfaster than the gradient projection method and much faster than the penalty functionmethod. The Lagrange multiplier and QP methods are so fast because in my case ofquadratic optimizer function and linear constraints, the partial derivatives are trivial tofind. More complicated optimizer and constraint functions may cause these two methodsto take more time in determining the partial derivatives. The scaling on Figure 2 mayseem large, but that is because it is set for easy comparison with the time results forproblems with inequality constraints in sections 4.2 and 4.3.Figure 2. Comparison of run-time for methods handling equality constraints4.2 Using Only Inequality ConstraintsThe algorithms that I implemented for optimization with inequality constraints includethe penalty function method, augmented Lagrange method, QP method, and gradientprojection method. Figure 3 shows the results of these methods on the example problemwithout equality constraints.