Part 3. Linear Programming 3.2 Algorithm

General Formulation Convex function Convex region

Graphical Solution

Degenerate Problems Non-unique solutions Unbounded minimum

Degenerate Problems – No Feasible Region

Remarks The solution obtained from a cannonical system by setting the non-basic variables to zero is called a basic solution. A basic feasible solution is a basic solution in which the values of the basi variables are nonnegative. Every corner point of the feasible region corresponds to a basic feasible solution of the constraint equations. Thus, the optimum solution is a basic feasible solution.

Full Rank Assumption

Fundamental Theorem of Linear Programming Given a linear program in standard form where A is an mxn matrix of rank m. If there is a feasible solution, there is a basic feasible solution; If there is an optimal solution, there is an optimal basic feasible solution.

Implication of Fundamental Theorem

Extreme Point

Theorem (Equivalence of extreme points and basic solutions)

Corollary If there is a finite optimal solution to a linear programming problem, there is a finite optimal solution which is an extreme point of the constraint set.

Step 2 x1 and x2 are selected as non-basic variables

Step 3: select new basic and non-basic variables new basic variable

Which one of x3, x4, x5 should be selected as the new non-basic variables?