A **quadratic programming problem** is a mathematical
programming problem, which consists of an objective function composed
of linear and quadratic terms and a set of linear constraints. A quadratic programming problem can be solved by a modification of
the simplex procedure.

The general *quadratic programming problem* is given by

Maximize f = c_{j}x_{j} + c_{jk}x_{j}x_{k}

subject to

a_{ij}x_{j} - b_{i} ≤ 0 , i = 1, 2, ...., m

x_{j} ≥ 0, j = 1, 2, ....,
n

If the quadratic form is negative semidefinite and symmetric, then it is a concave function of the decision variables. Since the constraints are linear, the feasible region is convex. Thus, Kuhn-Tucker conditions in this case are both necessary and sufficient.

To derive these conditions, we use multipliers λ_{i } ( i = 1, 2, ...., m) corresponding to the constraints

a_{ij}x_{j} - b_{i} ≤ 0;

and μ_{j} ( j = 1, 2, ....,
n) corresponding to the non-negativity constraints.

df/dx_{j} = c_{j} + 2 c_{jk}x_{k} , j = 1, 2, ...., n

d ---- dx _{j} |
a_{ij}x_{j} - b_{i} |
= | a_{ij} ; i = 1, 2, ...., m; j
= 1, 2, ...., n |

Let S_{i} = b_{i} - a_{ij}x_{j}; i = 1, 2, ...., m .....(i)

Where S_{i} is a slack variable.

Hence, Kuhn-Tucker conditions are given by

c_{j} + 2 c_{jk}x_{k} - λ_{i}a_{ij} + μ_{j} = 0; j = 1, 2, ...., n

or - 2 c_{jk}x_{k} + λ_{i}a_{ij} - μ_{j} = c_{j}; j = 1, 2, ...., n .....(ii)

S_{i}λ_{i} = 0, x_{j}μ_{j} = 0

S_{i}, λ_{i}, μ_{j},
x_{j} ≥ 0, i = 1, 2, ...., m;
j = 1, 2, ...., n

If we ensure that the pairs ( λ_{i},
S_{i}) and (μ_{j}, x_{j})
are not into the basis simultaneously, then the conditions S_{i}λ_{i} = 0 and x_{j}μ_{j} =
0 will be automatically satisfied. Therefore, we use simplex procedure
with a restricted entry rule.