Final exam practice problems by rua13781


									                         CO 367/CM 442: Nonlinear Optimization
                                     Winter 2007
                            Final exam practice problems
                                      S. Vavasis

Handed out: 2007-Apr-3 on the web.

  1. Suppose one is given n distinct points in the plane (x1 , y1 ), . . . , (xn , yn ) with n ≥ 3.
     Consider the problem of finding the point (x, y) that minimizes the maximum distance
     to the other points, i.e., find (x, y) such that maxi=1,...,n (x, y)−(xi , yi ) 2 is minimized.
     (a) As was done in lecture, introduce an auxiliary variable t to get rid of the “max”
     in the objective function and change the problem to a differentiable, constrained opti-
     mization problem. [Hint: the constraints have the form t ≥ · · ·]
     (b) Write the KKT conditions for the resulting problem.
     (c) Argue based on the KKT conditions that the optimum solution is either the mid-
     point of two of the input points (xi , yi ) and (xj , yj ) or else is equidistant from three
     (or more) input points.

  2. Name the main tradeoffs between:
     (a) the steepest descent method versus Newton’s method
     (b) the l1 penalty term k     i=1   max(gi (x), 0) versus the l2 (Courant-Beltrami) penalty
     term k n max(gi (x), 0)2 .

  3. Consider the l1 penalty function method applied to the univariate example from lecture,
     namely minimize x2 subject to x ≥ 1. For which nonnegative values of k is 0 a
     subgradient of Fk (x) at x∗ = 1?

  4. The following theorem is well known: if f (x) is a C 1 function of R, then the minimum
     of f (x) over the interval [a, b] is attained either at a critical point of f between a and b
     or at an endpoint (either a or b). Write the KKT conditions for this problem and use
     them to prove the theorem.

  5. Let f (x) be a C 2 function with a negative definite Hessian at all points. Show that
     the 2nd Wolfe condition can never be satisfied for any starting point and any search

  6. Same setting as the previous question: show that there is never a way to choose t(k) to
     satisfy the requirements of the BFGS method.

  7. Consider minimizing a C 1 function f : R2 → R subject to the constraints that x2 +y 2 ≤
     1, y = a, where a is a given real number in (−1, 1)
     (a) Write down the KKT conditions for this problem.
     (b) Show that every feasible point is a regular point.

   (c) Suppose f has the property that df /dx > 0 for all (x, y). Show that the KKT
   conditions can be satisfied only at a point where x < 0.

8. Consider the quadratic programming problem of minimizing x2 + 2x2 subject to (1)
                                                                    1       2
   x1 ≥ 4, (2) x1 ≤ 5, (3) x2 ≤ x1 , and (4) x2 ≥ −x1 . Suppose an active set method is
   used to solve this problem, and suppose the current working set is is {2} (i.e., constraint
   (2) only). Determine what will be the next working set.


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