The purpose of this exercise is to provide practice using the
LINGO or Excel solvers. Find the values of X and Y that minimize
the function

Do not assume nonnegativity of the X and Y variables. Recall
that by default LINGO assumes nonnegative variables. In order to
allow the variables to take on negative values you can add

@ FREE (X); @ FREE (Y);

Alternatively, if you want LINGO to allow for negative values by
default, in the LINGO menu select Options and then
click General Solver, and then uncheck the
Variables assumed nonnegative tab. To allow for
negative values in Excel Solver, make sure that the Make
Unconstrained Variables Non-Negative box is not checked in
the Solver Parameters dialog box. Round your
answers to the nearest whole number. If negative answer is
required, enter the minus sign before the number.

Optimal solution is

X = ? , Y = ? , for an optimal solution value
of 0.