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Notes 4 NonNegLinSys 2010 correction

Notes 4 NonNegLinSys 2010 correction - File...

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File: NonNegLinSys.Doc Non-negative Solutions to Linear Systems Consider a typical constraint set for a linear program. These constraints generally take the form ,0 Cy b y . We can convert this system of inequalities into a system of linear equations with non- negativity restrictions on the variables by adding to each constraint a non-negative variable s i , where each s i 0. (The s i variables have a unique name: slack variables ). The system then becomes , 0 Is b y s + =≥ . Combining the variables into one notation, ( ) , TT T x ys = , and letting () , AC I = , we have a generalization of our previous system which now includes non-negative variables: Ax b x = . Again, A is an mn × matrix which we assume has full rank m . Our goal is to extend the concepts of general solutions of linear systems and specifically the special particular solution called a basic solution to incorporate non-negativity restriction on the solution variables. A basic solution which is also non-negative is called a basic feasible solution . The use of the term feasibility refers to the validity of the constraints ( ) 0 x . Note that the resulting system now has m + n restrictions, m for b = and n more for 0 x . To solve for n variables, we need n equality constraints. We obtain these n constraints, m from b = and ( n - m ) from among the non-negative restrictions by setting n - m of these variables to zero (such as x j = 0). Denoting the indexes of these restrictions to zero as N , then x j N j = 0 for all . Letting this last set of variables in matrix form be N x , we have then the n equality constraints , N b x = = or , BN N Bx Nx b x + == reducing finally to . b x = =
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Thus, if we have properly selected the variables in x B so that their coefficients matrix B has an inverse, the particular or basic solution becomes 1 B x Bb = and 0 N x = . If we are lucky and the basic variables are nonnegative, 0 B x , then this is called a basic feasible solution . We cannot go about solving this system of equations merely hoping that the particular basic solution we construct is also feasible. What is needed is a test that, a priori the solution, would insure that the resulting basic solution would also be feasible. This is very difficult if not impossible task for a general system of linear equalities. However, there is a simple procedure for creating a new basic feasible solution from an existing basic feasible solution. An initial basic feasible solution frequently is easy to find, merely setting B x sb == and 0 N xy = = for the initial inequality system above is a basic feasible solution if 0 b . At this point, we have not really established a reason for finding a sequence of basic feasible solutions. This is, however, the solution structure underlying the simplex method for linear programming problems .
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Notes 4 NonNegLinSys 2010 correction - File...

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