Question

Given three numbers n, m, r and a constant matrix Z E R"Xm, consider the optimization problem minimize Z- XY subject toX20, Y20 (note that the sign"2" means that all elements of the corresponding matrix are nonnegative, and thatIF denotes the Frobenius norm). (10 points) Write the first-order optimality conditions for (1). (10 points) Describe how to solve (1) using the gradient projection method with the step size along the feasible direction chosen to be and the step size along the projection arc to be designed using the Armijo rule. Derive the iterative equations and simplify them as much as possible. (5 points) Describe how to solve (1) using the block coordinate descent method where we solve for X and Y alternatively. Discuss (without having to write the equations in details) how to find a global solution (up to a given precision) of each of the optimization sub-problems that need to solved in the iterations of the block coordinate descent.

Answer 1

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SOLUTION :

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