How can the frank-wolfe algorithm be efficiently combined

Question

Suppose that a new objective function is to be considered that is a nonlinear function of the holdings entering the final period. More precisely, assume that we wish to maximize the expected utility of these holdings, with the utility function given by:

u(x) = ax^1-e.

where a > 0 and 0

_{e_N∈E_N}∑p(e_N) u {_k=1∑^K[_m=0∑^N-1 (y_m^k (e_m) + v_m,N^k (e_N)) h_m,N^k(e_N) + (y_N^k(e_N) + V_N,N^k(e_N))b_N^k(e_N)]}.

a) Show that this problem cannot be handled directly by decomposition. [Hint. Is this objective function separable in the appropriate way?]

b) If the nonlinear-programming problem were solved by the Frank-Wolfe algorithm, a sequence of linear programs would be solved. How can the decomposition approach presented in this chapter be used to solve one of these linear programs?

c) How can the Frank-Wolfe algorithm be efficiently combined with the decomposition approach presented in this chapter, to find the optimal solution to the nonlinear program defined by maximizing the expected utility given above?

d) Does your proposed method generalize to other nonlinear problems?