A Number of Jacobian entries

Question

I'm trying to understand an optimization problem from IPOPT package.

 * Problem hs071 looks like this
 *
 *     min   x1*x4*(x1 + x2 + x3)  +  x3
 *     s.t.  x1*x2*x3*x4                   >=  25
 *           x1**2 + x2**2 + x3**2 + x4**2  =  40
 *           1 <=  x1,x2,x3,x4  <= 5
 *
 *     Starting point:
 *        x = (1, 5, 5, 1)
 *
 *     Optimal solution:
 *        x = (1.00000000, 4.74299963, 3.82114998, 1.37940829)

However, in the other file it is stated

   // The problem described in HS071_NLP.hpp has 4 variables, x[0] through x[3]
   n = 4;

   // one equality constraint and one inequality constraint
   m = 2;

   // in this example the jacobian is dense and contains 8 nonzeros
   nnz_jac_g = 8;

   // the Hessian is also dense and has 16 total nonzeros, but we
   // only need the lower left corner (since it is symmetric)
   nnz_h_lag = 10;

So we have $4$ variables and $2$ constraints. But then somehow, they put $8$ non-zeroes in Jacobian. Why? Should not it be only $4$, as cost function is $f: \mathbb{R}^4 \to \mathbb{R}$?

Moreover, Hessian is clearly stated as $16$, i.e. a $4 \times 4$ matrix, as expected.

Answer 1

Apparently, they are not talking about the $\nabla f$, but rather about the Jacobian of the constraints. Which then are treated as $G: \mathbb{R}^4 \to \mathbb{R}^2$, hence $J_G$ has $8$ entries.