Nonlinear Optimization Exam Sample

  1. Write the KT problem corresponding to the NLP problem:

  2. Given the NLP problem

    Check if the constraint qualification holds at

  3. Take the NLP problem

    Use the appropriate KT theorem to classify the point State clearly the requirements of the theorem and test if they apply on this problem.

  4. The minimizer of the single-variable function is being sought. It is known that the function has its minimizer in the interval [-1, 0]. Reduce the interval of uncertainty by applying the Fibonacci method with n = 4 and report the limits of the final interval.
  5. Consider the unconstrained function of two variables

    Which of the following two directions is a descent direction at ?

  6. We seek to find the minimizer of the function

    If , find using the steepest descent method.

  7. We want to use the conjugate gradient method to solve the unconstrained NLP problem

    Starting from , it has been calculated , . Starting from this information, find .  Is an optimal solution?

  8. Take the NLP problem

    1. Write the composite function consisting of the exterior penalty function.
    2. Without carrying the actual calculations, indicate how the starting solution should be selected if the exterior penalty function will be used to solve the problem?
    3. What would be a good starting value for the penalty parameter?
  9. DFP quasi-Newton method will be used to solve the unconstrained problem

    Given:

    calculate the direction Maintain 3 decimal places.

  10. Use linear approximation to approximate the NLP problem

    by a linear model at .

  11. We want to solve the NLP problem

    Let . If the starting basic variables are chosen to be x1, x3, and x5, use the reduced gradient method to find x1.

    Note: use the method of augmented matrix to find the inverse of a matrix:

Dr Muhammad Al-Salamah
Industrial Engineering
Systems Engineering Department
King Fahd University
Saudi Arabia
2008