The following code does symbolic minimization using Reduce to solve the Karush-Kuhn-Tucker conditions. No sign conditions are placed on the Lagrange multipliers, so it gives all maxima and minima. The subscripts of the multipliers are the constraint they are multiplying.
consconvrule = {
x_ >= y_ -> y - x,
x_ == y_ -> x - y ,
x_ <= y_ -> x - y,
lb_ <= x_ <= ub_ -> (x - lb) (x - ub),
ub_ >= x_ >= lb_ -> (x - lb) (x - ub)
};
KKT[obj_, cons_List, vars_List] :=
Module[
{convcons, gradobj, gradcons, \[CapitalLambda]},
convcons = (cons /. consconvrule);
{gradobj, gradcons} = D[{obj, convcons}, {vars}];
\[CapitalLambda] = Subscript[\[Lambda], #] & /@ cons;
LogicalExpand @ Reduce[
Flatten[{
Thread[gradobj == \[CapitalLambda].gradcons],
Thread[\[CapitalLambda]*convcons == 0] /.
Subscript[\[Lambda], Equal[a_, b_]] -> 0,
cons,
{objval == obj}
}],
Join[{objval}, vars, \[CapitalLambda]],
Reals,
Backsubstitution -> True
]
]
Here’s an example application
KKT[x + 2 y + 3 z, {x^2 + y^2 + z^2 == 1, 3 x + 2 y + z <= 1}, {x, y,
z}]
(objval == -Sqrt[14] && x == -(1/Sqrt[14]) && y == -Sqrt[(2/7)] &&
z == -(3/Sqrt[14]) &&
Subscript[\[Lambda], x^2 + y^2 + z^2 == 1] == -Sqrt[(7/2)] &&
Subscript[\[Lambda], 3 x + 2 y + z <= 1] == 0) ||
(objval ==
1/7 (5 - 2 Sqrt[78]) && x == 1/42 (9 + 2 Sqrt[78]) &&
y == 1/42 (6 - Sqrt[78]) && z == 1/42 (3 - 4 Sqrt[78]) &&
Subscript[\[Lambda], x^2 + y^2 + z^2 == 1] == -2 Sqrt[6/13] &&
Subscript[\[Lambda], 3 x + 2 y + z <= 1] ==
1/91 (65 + 2 Sqrt[78])) ||
(objval == 1/7 (5 + 2 Sqrt[78]) &&
x == 1/42 (9 - 2 Sqrt[78]) && y == 1/42 (6 + Sqrt[78]) &&
z == 1/42 (3 + 4 Sqrt[78]) &&
Subscript[\[Lambda], x^2 + y^2 + z^2 == 1] == 2 Sqrt[6/13] &&
Subscript[\[Lambda], 3 x + 2 y + z <= 1] == 1/91 (65 - 2 Sqrt[78]))
