Linear Optimization

Lecture 1: IntroductionInstructor: Tsvetan Asamov

• Elementary Linear Programming with Applications, 2nd edition by Bernard Kolman and Robert E. Beck

Example

Need: Energy (2000 kcal), Protein (55 g), Calcium (800 mg)

Example

Diet Problem Formulation

Linear Programming Examples

Linear Programming

• Objective function

• Linear equations (equalities)

• Linear inequalities

• Linear equations and linear inequalities are referred to as linear constraints

Standard Form

Diet Problem

• Objective function

• Feasible solution

• Optimal solution

• Optimal value: 92.5

Linear programming problems

• Unique optimal solution• Many optimal solutions• No optimal solutions– Infeasible problems

Linear programming problems

• Unique optimal solution• Many optimal solutions• No optimal solutions– Infeasible problems– Unbounded problems

Problems

Problems

Reading Assignment

• Review linear algebra– Vectors– Matrices– Matrix multiplication– Matrix inversion– Gauss-Jordan reduction– Linear independence and basis– Subspaces