Consider the parametric SCQP problem of min xT Hx/2 + (g0 + sg1)T x subject to eTx = 1 and x 0. Here, H Rnn is a positive semidefinite matrix, g0,g1 are vectors, g1 = 0, and s is a parameter.

Show that there is an s such that for all s s , the optimal solution is of the form ei, where i is the position of the minimum entry of g1. To simplify the question, assume that i is unique, i.e., there is not a tie for the smallest entry. [Hint: Determine values of the multipliers and z to satisfy the KKT conditions for the specified vector x. The ith row of the gradient equations determines since complementarity forces zi = 0. Once is specified, the remaining entries of z are obtained from the gradient equation. Argue that for s sufficiently large, these remaining entries are all nonnegative.]