Lecture12-2003 - Lecture XII Linear Equality and Inequality...

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Lecture XII Linear Equality and Inequality Constraints I. Linear Equality Constraints A. The general optimization problem for the linear equality constraints can be stated as: LEP f x st Ax b x min ( ) = B. Like the unconstrained problem, we are again looking for the point where the projected gradient vanishes. However, this time instead of searching over dimension n , we only have to search over dimension n - t where t is the number of nonredundant equations in A . 1. In the vernacular of the problem, we want to decompose the vector x into a range-specific portion which is required to solve the constraints and a null-space portion which can be varied. a. Specifically, xY x Z x YZ = + where Y x Y denotes the range-specific portion of x and Z x Z denotes the null-space portion of x . b. By the definition of the null-space, we know that AY x b Y * = if x * is feasible, since AZ =0. 2. The numerical example given in the text uses a single constraint xxx 123 3 + + = There are several ways to derive the null-space of a matrix. The null- space matrix is 33 66 3333 Z  −−  +− =− −+  A numerical equivalent to this matrix can be computed using Gauss. Specifically, (gauss) a={1 1 1}; (gauss) null(a); -0.57735027 -0.57735027 0.78867513 -0.21132487 -0.21132487 0.78867513
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AEB 6533 – Static and Dynamic Optimization Lecture 12 Professor Charles B. Moss 2 Thus, the solution of the constraint can be expressed as: ** 33 1 1 66 1 3333 Z x x  −− 
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Lecture12-2003 - Lecture XII Linear Equality and Inequality...

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