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Exploring 2D Linear Programming: Optimal Solutions and Graphical Methods

This resource provides an in-depth exploration of two-dimensional linear programming (LP), focusing on graphing techniques to identify optimal solutions. We delve into various examples, including problems of maximizing and minimizing objectives subject to constraints, and examine the feasible regions defined by these constraints. Furthermore, we discuss the significance of corner points in determining optimal solutions and the role of algorithms such as the Simplex Method. This study aims to demystify LP solutions through visual representations and fundamental geometric principles.

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Exploring 2D Linear Programming: Optimal Solutions and Graphical Methods

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  1. Linear Programming Piyush Kumar

  2. Graphing 2-Dimensional LPs Optimal Solution y Example 1: 4 Maximize x + y 3 x + 2 y ³ 2 Subject to: Feasible Region x £ 3 2 y £ 4 1 x ³0y ³0 0 x 0 1 2 3 These LP animations were created by Keely Crowston.

  3. Graphing 2-Dimensional LPs Multiple Optimal Solutions! y Example 2: 4 Minimize ** x - y 3 1/3 x + y £ 4 Subject to: -2 x + 2 y £ 4 2 Feasible Region x £ 3 1 x ³0y ³0 0 x 0 1 2 3

  4. Graphing 2-Dimensional LPs y Example 3: 40 Minimize x + 1/3 y 30 x + y ³ 20 Subject to: Feasible Region -2 x + 5 y £ 150 20 x ³ 5 10 x ³ 0y ³ 0 x 0 Optimal Solution 40 0 10 20 30

  5. Do We Notice Anything From These 3 Examples? Extreme point y y y 4 40 4 3 3 30 2 2 20 1 1 10 0 0 0 x x x 0 1 0 1 2 0 10 20 2 3 3 30 40

  6. A Fundamental Point y y y 4 40 4 3 3 30 2 2 20 1 1 10 0 0 0 x x x 0 1 0 1 2 0 10 20 2 3 3 30 40 If an optimal solution exists, there is always a corner point optimal solution!

  7. Graphing 2-Dimensional LPs Optimal Solution Second Corner pt. y Example 1: 4 Maximize x + y 3 x + 2 y ³ 2 Subject to: Feasible Region x £ 3 2 y £ 4 1 x ³ 0y ³ 0 Initial Corner pt. 0 x 0 1 2 3

  8. And We Can Extend this to Higher Dimensions

  9. Then How Might We Solve an LP? • The constraints of an LP give rise to a geometrical shape - we call it a polyhedron. • If we can determine all the corner points of the polyhedron, then we can calculate the objective value at these points and take the best one as our optimal solution. • The Simplex Method intelligently moves from corner to corner until it can prove that it has found the optimal solution.

  10. Linear Programs in higher dimensions maximize z = -4x1 + x2 - x3 subject to -7x1 + 5x2 + x3 <= 8 -2x1 + 4x2 + 2x3 <= 10 x1, x2, x3 0

  11. In Matrix terms

  12. LP Geometry • Forms a d dimensional polyhedron • Is convex : If z1 and z2 are two feasible solutions then λz1+ (1- λ)z2 is also feasible. • Extreme points can not be written as a convex combination of two feasible points.

  13. LP Geometry • Extreme point theorem: If there exists an optimal solution to an LP Problem, then there exists one extreme point where the optimum is achieved. • Local optimum = Global Optimum

  14. LP: Algorithms • Simplex. (Dantzig 1947) • Developed shortly after WWII in response to logistical problems:used for 1948 Berlin airlift. • Practical solution method that moves from one extreme point to a neighboring extreme point. • Finite (exponential) complexity, but no polynomial implementation known. Courtesy Kevin Wayne

  15. LP: Polynomial Algorithms • Ellipsoid. (Khachian 1979, 1980) • Solvable in polynomial time: O(n4 L) bit operations. • n = # variables • L = # bits in input • Theoretical tour de force. • Not remotely practical. • Karmarkar's algorithm. (Karmarkar 1984) • O(n3.5 L). • Polynomial and reasonably efficientimplementations possible. • Interior point algorithms. • O(n3 L). • Competitive with simplex! • Dominates on simplex for large problems. • Extends to even more general problems.

  16. LP: The 2D case Wlog, we can assume that c=(0,-1). So now we want to find the Extreme point with the smallest y coordinate. Lets also assume, no degeneracies, the solution is given by two Halfplanes intersecting at a point.

  17. Incremental Algorithm? • How would it work to solve a 2D LP Problem? • How much time would it take in the worst case? • Can we do better?

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