Lecture08-2003 - Lecture VIII Linear Inequality and...

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Lecture VIII Linear Inequality and Nonlinear Equality Constraints I. A Review of Linear Equality Constraints A. As previously developed, the first order necessary conditions for a linearly constrained equality can be written as Zf x x '( ) = 0 An alternative specification for this constraint is = x fx A () ' λ where λ are a vector of lagrange multipliers. Another way to look at the problem is that the lagrange multipliers are that set of constants which equate the gradient with the linear constraints. B. To offer an intuitive proof Lfx b a x x Lf x aa xx = + + ∇= = ( ) ( ) λλ 11 1 2 2 2 2 0 . Thus, = + = x fx a a A ' . The general concept of the lagrange multiplier and the null space matrix then work together. C. Another concept that I want to discuss is the fact that a equality constrained problem can have lagrange multipliers of either side. Intuitively, the lagrange multiplier is the change in the objective function associated with a one unit increase in the right hand side of the constraint. Therefore, in a maximization problem: 1. If the constraint cuts the frontier below the global maximum (assuming that the objective function resembles a quadratic function), the lagrange multiplier will take on a positive value implying that an increase in the right hand side of the constraint will increase the objective function value. 2. Similarly, if the constraint cuts the frontier above the global maximum (again assuming that the objective function resembles a quadratic function), the
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Lecture08-2003 - Lecture VIII Linear Inequality and...

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