9.2 Geometric interpretation of linear programs

Linear programming uses geometry to solve optimization problems. It visualizes decision variables as axes, constraints as lines or planes, and the objective function as parallel lines or surfaces. This approach helps identify feasible regions and optimal solutions graphically.

The geometric interpretation highlights key concepts like extreme points, which are crucial for solving linear programs efficiently. Understanding this visual representation aids in grasping the simplex method and other algorithms used in linear programming.

Geometric Interpretation of Linear Programs

Geometric interpretation of linear programming

Top images from around the web for Geometric interpretation of linear programming

• 

  Derivatives and the Shape of a Graph · Calculus View original

  Is this image relevant?

• 

  3.2c. Examples: Solving Linear Programming Graphically | Finite Math View original

  Is this image relevant?

• 

  3.2a. Solving Linear Programming Problems Graphically | Finite Math View original

  Is this image relevant?

• 

  Derivatives and the Shape of a Graph · Calculus View original

  Is this image relevant?

• 

  3.2c. Examples: Solving Linear Programming Graphically | Finite Math View original

  Is this image relevant?

1 of 3

Top images from around the web for Geometric interpretation of linear programming

• 

  Derivatives and the Shape of a Graph · Calculus View original

  Is this image relevant?

• 

  3.2c. Examples: Solving Linear Programming Graphically | Finite Math View original

  Is this image relevant?

• 

  3.2a. Solving Linear Programming Problems Graphically | Finite Math View original

  Is this image relevant?

• 

  Derivatives and the Shape of a Graph · Calculus View original

  Is this image relevant?

• 

  3.2c. Examples: Solving Linear Programming Graphically | Finite Math View original

  Is this image relevant?

1 of 3

• Decision variables visualized as axes in coordinate system x and y for 2D, x, y, and z for 3D
• Constraints represented as lines (2D) or planes (3D) equality constraints form exact lines/planes, inequalities create half-planes/spaces
• Objective function depicted as family of parallel lines/planes showing constant values
• Level curves/surfaces concept illustrates points with equal objective function values

Graphical determination of feasible regions

• Feasible region encompasses all points satisfying constraints plotted on coordinate system
• Constraint lines/planes drawn and area satisfying inequalities shaded
• Intersection of all constraint regions forms feasible region
• Special cases include unbounded regions extending infinitely, empty regions (infeasible problems), single point regions

Optimal solutions through graphical methods

• Direction of improvement for objective function determined
• Level lines/planes for objective function utilized
• Level line/plane moved in improvement direction
• Furthest feasible point in improvement direction identified
• Optimal solution found where level line/plane last intersects feasible region
• Special cases: multiple optima (level line aligns with feasible edge), unbounded solutions (feasible region extends infinitely in improvement direction)

Extreme points in linear programming

• Extreme points defined as corners/vertices of feasible region formed by constraint intersections
• Optimal solution always occurs at extreme point (Fundamental Theorem of Linear Programming)
• Finite number of extreme points in bounded feasible region
• Extreme points correspond to basic feasible solutions in algebraic representation
• Crucial for simplex method algorithms efficiency