Consider the following optimization problem:

subject to: , ,

where , , and are given. Assume that for all .

This is not a linear program since the objective function involves absolute values.

Is it possible to formulate it equivalently as a linear program?

Let

be a real matrix, where is a real number.

Suppose is a quadratic function where .

Determine the range of such that has a global minimum value.

Assume . We would like to use the gradient descent method with fixed step size to minimize . Determine the range of the step size so that the algorithm will converge no matter which initial point we choose, and give the reason.

Suppose . Compute the condition number of .

If goes to , what happens to the condition number? Does the result given by the gradient descent with fixed step size converge if ?