Problem 1
Find a Karush-Kuhn-Tucker point of the following quadratic programming problem:
Problem 2
Consider the design of a storage vessel that has the form. of a cylinder. The required volume is 25 m3. The cost of the side of the cylinder is $150/m2, while the top and bottom cost $190/m2 and 260/m2 respectively. Formulate a nonlinear programming problem to determine the optimal dimensions of this vessel, and solve with GAMS/Pyomo using a number of NLP solvers that you can identify (e.g. MINOS, CONOPT, KNITRO and IPOPT if you use GAMS). Do you get the same solution by changing the starting points and using different solvers? Please submit your source code file as well.
Problem 3
Consider the nonlinear programming problem:
Assuming that f(x) and h(x) are continuous and differentiable in the n-dimensional vector x, derive the Karush-Kuhn-Tucker conditions for the above problem.
Problem 4
Suppose you are given the steady-state model of a process that is represented by the system of nonlinear equations, h(x) = 0, where dim(h) = dim(x). If you have an NLP algorithm available for solving this problem, how would you formulate the NLP to find the solution for the system of equations?
Problem 6
Consider the following nonlinear optimization problem.
a) Please plot the feasible region and objective function, and then determine the optimal solution graphically.
b) Write down the Karush-Kuhn-Tucker (KKT) conditions of the above problem, and check if the optimal solution satisfies the KKT conditions.
c) Solve this problem using at least two NLP solvers (e.g. CONOPT, MINOS, SNOPT, IPOPT, etc.) and compare the resulting solutions.
d) What conclusion can you draw from this example on solving NLP problems using standard NLP algorithms?
Problem 7
Consider the following nonlinear programming problem.
a) What are the optimality conditions that any optimal solution of this problem must satisfy?
b) Does the solution point at x1 = 2, x2 = 1 satisfy the optimality condition? Why or why not?
c) Does the solution point at x1 = 1, x2 = 2 satisfy the optimality condition? Why or why not?