Final exam practice problems by rua13781

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									                         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
                                   n
     (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 .
               i=1

  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
     direction.

  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.

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   (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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