Lecture 8

Learning objectives

After this class, you should be able to:

1. Define a linear program and its standard form.
2. Define the following terms: feasible solution, feasible region, objective function, objective value, integer linear program, and constraint.
3. Given a linear program, write an equivalent one in standard form.
4. Given a problem, formulate it as a linear program or as an integer linear program.
5. Define the terms: (i) primal program, and (ii) dual program.
6. Given a linear program, construct its dual.
7. Explain the implications of the duality theorem.
8. Prove the weak duality theorem.

Reading assignment

1. Chapter 12, sections 12.1.
2. None.

Exercises and review questions

• Questions on current lecture's material
  1. Give a feasible solution for the following linear program. Maximize 2x - 3y + 3z subject to: (i) x + y - z <= 7, (ii) -x -y + z <= -7, (iii) x - 2y + 2z <= 4, (iv) x, y, z >= 0.
  2. Write the following linear program in standard (maximization) form. Minimize 2x + 7 y subject to: (i) x = 7, (ii) 3x + y >= 24, (iii) y >= 0, (iv) z <= 0.
  3. Give the dual of the above (standard form) linear program.
  4. Exercise 12.2. (Show how you will deal with variables that don't have non-negativity constraints, how you will deal with equality constraints, and how you will deal with consraints that are <=. rather than >=.)
  5. Give an example of a linear program where the best integer solution has objective value greater than that for the optimal solution.
  6. Formulate Vertex Cover as an integer linear program.
  7. Exercise 12.1.
  8. (Post your answer on the discussion board) Give an application for linear programming.

• Questions on next lecture's material
  1. None.