Kuhn Tucker Optimality Conditions with inequality constraints.

the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied.

Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints. Similar to the Lagrange approach, the constrained maximization (minimization) problem is rewritten as a Lagrange function whose optimal point is a saddle point, i.e. a global maximum (minimum) over the domain of the choice variables and a global minimum (maximum) over the multipliers, which is why the Karush–Kuhn–Tucker theorem is sometimes referred to as the saddle-point theorem.
Consider the case of a two-good world where both goods, x and y. are rationed. Let the consumer’s utility function be U = U(x,y). The consumer has a fixed money budget of B and faces the money prices Px and Py. Further, the consumer has an allotment of coupons, denoted C, which can be used to purchase both x or y at a coupon price of cx and cy. Therefore the consumer’s maximization problem is Maximize U =U(x,y) Subject to and B ≥Pxx+Pyy C ≥cxx+cyy in addition, the non-negativity constraint x ≥ 0 and y ≥ 0. The Lagrangian for the problem is Z =U(x,y)+λ(B−Pxx−Pyy)+λ2(C −cxx+cyy) where λ,λ2 are the Lagrange multiplier on the budget and coupon constraints respectively. The Kuhn-Tucker conditions are Zx =Ux−λ1Px−λ2cx =0 Zy =Uy −λ1Py −λ2cy =0 Zλ1 = B−Pxx−Pyy ≥0 λ1≥0 Zλ2 = C −cxx−cyy ≥0 λ2≥0 Numerical Example Let’s suppose the utility function is of the form U = x · y2. Further, let B =100,Px = Py =1while C =120and cx =2,cy =1. The Lagrangian becomes Z =xy2+λ1(100−x−y)+λ2(120−2x−y) The Kuhn-Tucker conditions are now Zx =y2 −λ1−2λ2 ≤0 x≥0 x·Zx=0 Zy =2xy−λ1−λ2 ≤0 y ≥0 y·Zy =0 Zλ1 =100−x−y ≥0 λ1 ≥0 λ1·Zλ1 =0 Zλ2 =120−2x−y ≥0 λ2 ≥0 λ2·Zλ2 =0