Optimization Methods: Optimization using Calculus Kuhn-Tucker Conditions 1. Module - 2 Lecture Notes 5. Kuhn-Tucker Conditions

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1 Optimization Methods: Optimization using Calculus Kuhn-Tucker Conditions Module - Lecture Notes 5 Kuhn-Tucker Conditions Introduction In the previous lecture the optimization of functions of multiple variables subected to equality constraints using the method of constrained variation and the method of Lagrange multipliers was dealt. In this lecture the Kuhn-Tucker conditions will be discussed with examples for a point to be a local optimum in case of a function subect to inequality constraints. Kuhn-Tucker Conditions It was previously established that for both an unconstrained optimization problem and an optimization problem with an equality constraint the first-order conditions are sufficient for a global optimum when the obective and constraint functions satisfy appropriate concavity/convexity conditions. The same is true for an optimization problem with inequality constraints. The Kuhn-Tucker conditions are both necessary and sufficient if the obective function is concave and each constraint is linear or each constraint function is concave, the problems belong to a class called the convex programming problems. Consider the following optimization problem: Minimize f(x) subect to g (X) 0 for =,,,p ; where X = [x x... x n ] Then the Kuhn-Tucker conditions for X * = [x * x *... x n * ] to be a local minimum are f x i m g + λ = 0 i =,,..., n x = i λ g g = 0 =,,..., m 0 =,,..., m λ 0 =,,..., m () ML5

2 Optimization Methods: Optimization using Calculus Kuhn-Tucker Conditions In case of minimization problems, if the constraints are of the form g (X) 0, then λ have to be nonpositive in (). On the other hand, if the problem is one of maximization with the constraints in the form g (X) 0, then λ have to be nonnegative. It may be noted that sign convention has to be strictly followed for the Kuhn-Tucker conditions to be applicable. Example Minimize f = x + x + 3x 3 subect to the constraints g = x x x 3 g = x + x 3x 8 3 using Kuhn-Tucker conditions. Solution: The Kuhn-Tucker conditions are given by f g g + + = a) λ λ 0 xi xi xi x + λ + λ = 0 () 4x λ + λ = 0 (3) 6x λ 3λ = 0 (4) 3 b) λ g = 0 λ ( x x x ) = 0 (5) 3 λ ( x + x 3x 8) = 0 (6) 3 c) g 0 ML5

3 Optimization Methods: Optimization using Calculus Kuhn-Tucker Conditions 3, x x x 0 (7) 3 x + x 3x 8 0 (8) 3 d) λ 0, λ 0 (9) λ 0 (0) From (5) either λ = 0 or, x x x3 =0 Case : λ = 0 From (), (3) and (4) we have x = x = λ /and x 3 = λ /. Using these in (6) we get λ + 8λ = 0, λ = 0or 8 From (0), λ 0, therefore, λ =0, X * = [ 0, 0, 0 ], this solution set satisfies all of (6) to (9) Case : x x x3 = 0 λ λ λ λ λ+ 3λ Using (), (3) and (4), we have = 0 or, 4 3 7λ + λ = 44. But conditions (9) and (0) give us λ 0 and λ 0 simultaneously, which cannot be possible with 7λ+ λ = 44. Hence the solution set for this optimization problem is X * = [ ] ML5

4 Optimization Methods: Optimization using Calculus Kuhn-Tucker Conditions 4 Example Minimize f = x + x + x subect to the constraints 60 g = x 80 0 g = x + x 0 0 using Kuhn-Tucker conditions. Solution The Kuhn-Tucker conditions are given by f g g g3 a) + λ + λ + λ = x x x x 3 0 i i i i x λ + λ = 0 () x + λ = 0 () b) λ g = 0 λ ( x 80) = 0 (3) λ ( x + x 0) = 0 (4) c) g 0, x 80 0 (5) x+ x (6) d) λ 0, ML5

5 Optimization Methods: Optimization using Calculus Kuhn-Tucker Conditions 5 λ 0 (7) λ 0 (8) From (3) either λ = 0 or, ( x 80) = 0 Case : λ = 0 λ From () and () we have λ x = 30 and x = ( ) Using these in (4) we get λ λ 50 = 0; λ = 0 or 50 Considering λ = 0, X * = [ 30, 0]. But this solution set violates (5) and (6) For λ = 50, X * = [ 45, 75]. But this solution set violates (5). Case : ( x 80) = 0 Using x = 80 in () and (), we have λ = x λ = x 0 (9) Substitute (9) in (4), we have ( x ) x 40 = 0. For this to be true, either x = 0 or x 40=0 For x = 0, λ = 0. This solution set violates (5) and (6) ML5

6 Optimization Methods: Optimization using Calculus Kuhn-Tucker Conditions 6 For x 40 = 0, λ = 40 and λ = 80. This solution set is satisfying all equations from (5) to (9) and hence the desired. Therefore, the solution set for this optimization problem is X * = [ ]. References / Further Reading:. Rao S.S., Engineering Optimization Theory and Practice, Third Edition, New Age International Limited, New Delhi, Ravindran A., D.T. Phillips and J.J. Solberg, Operations Research Principles and Practice, John Wiley & Sons, New York, Taha H.A., Operations Research An Introduction, Prentice-Hall of India Pvt. Ltd., New Delhi, Vedula S., and P.P. Muumdar, Water Resources Systems: Modelling Techniques and Analysis, Tata McGraw Hill, New Delhi, 005. ML5