Linear Programming Learning Notes (1) Introduction

Linear Programming Learning Notes (1) Introduction

All the resources come from Linear Programming: Foundations and Extensions by Professor Robert J. Vanderbei.
Explore the link below for further information:
LP Book Resources

Part 1

Basic Theory: The Simplex Method and Duality

Chapter 1 Introduction

Examples

• Resource Allocation
• Blending Problem(Diet Problem)
• Shape Optimization(Telescope Design) (Not quite understood)
• FIR Filter Design(Not quite understood either)

• Portfolio Optimization
  -A Markowitz Type Problem
  If we need to optimize two objectives, we could make a compromise to set a bound of one and maximize(minimize) another.
  -Efficient Frontier
  Varying risk bound μ produces the so-called ecient frontier.
  Portfolios on the ecient frontier are reasonable.
  Portfolios not on the ecient frontier can be strictly improved.

  Definitions

• Optimist and Pessimist
  -Production manager as Optimist, to maximize the product quantity.
  -Comptroller as Pessimist, to minimize the cost.
• The Linear Programming Problem
  -Decision Variables:
  xj,j=1,2,3,...,n
  -Objective Function:
  ζ=c1x1+c2x2+…+cnxn
  In real-world problems are most naturally formulated as minimizations(since real-world planners always seem to be pessimists), but when discussint mathematics it is usually nicer to work with maximization problems.
  -Constraints:
  a1x1+a2x2+...+anxn=⎧⎩⎨⎪⎪≤=≥⎫⎭⎬⎪⎪b.
  -Mathematically preferred presentation:
  maximize CTX
  subject to AX≤B,X≥0
  where C∈IRn,X∈IRn,A∈IRm×n,B∈IRm
  m constraints, n variables
  -Infeasible problems
  -Unbounded: Feasible solutions with arbitrarily large objective values.