Just select a subset by using indexing (in the following example: classic python-based slicing; but more complex indexing / numpy-style is possible):

Example:

from cvxpy import *

x = Variable(5)
constraints = []
constraints.append(x >= 0)   # all vars
constraints.append(x <= 10)  # all vars
constraints.append(sum_entries(x[:3]) <= 3)  # only part of vector; sum(first-three) <=3
objective = Maximize(sum_entries(x))
problem = Problem(objective, constraints)
problem.solve()
print(problem.status)
print(x.value.T)

Output:

optimal
[[  1.   1.   1.  10.  10.]]

Note: as of cvxpy 1.0, sum_entries has been changed to sum

I also suspect you are misunderstanding the problem here, but that formula-image is of course incomplete to be implemented.

You can construct the cost function in a loop. It also seems that g should be positive semidefinite, so I added PSD=True. Here's a complete example:

from cvxpy import *
n = 10
g = Variable((n, n), PSD=True)
eta = Variable((n, n))
reg = norm(eta, 1)
d = np.ones((n, n))
lamb = 1.

cost = 0
for i in range(n):
    for j in range(n):
        cost += (g[i, i] + g[j, j] - 2 * g[i, j] + eta[i, j] - d[i, j]**2) ** 2

cost = 0.5 * cost + lamb * reg
prob = Problem(Minimize(cost))
prob.solve()