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Part 3. Linear Programming

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Presentation on theme: "Part 3. Linear Programming"— Presentation transcript:

1 Part 3. Linear Programming
3.2 Algorithm

2 General Formulation Convex function Convex region

3 Graphical Solution

4 Degenerate Problems Non-unique solutions Unbounded minimum

5 Degenerate Problems – No Feasible Region

6 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.

7 Full Rank Assumption

8

9 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.

10 Implication of Fundamental Theorem

11 Extreme Point

12 Theorem (Equivalence of extreme points and basic solutions)

13 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.

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

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

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


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