Financial Optimization and Asset Management Lecture 8

Financial Optimization and Asset Management

• Lecture 8 covered Linear Programming (LP) Standard Form
• Equivalent transformations of LP models were discussed
• Reference forms for LP (Lect. 3) were presented
• The max form (1) aims to maximize output with limited input (resources)
• The min form (2) aims to minimize input to achieve a pre-defined level of output (performance)
• Any LP problem can be transformed into one of these two forms

Equivalent Transformations of Linear Constraints

• Examples show how to transform constraints into the standard LP form (Ax ≤ b, x ≥ 0)
  • Inequalities in the correct form are left as is
  • Equations are rewritten as two inequalities in opposite directions (e.g. x + y = 6 becomes x + y ≤ 6 and x + y ≥ 6)
  • Inequalities with the incorrect direction have both sides multiplied by -1

LP in Arbitrary Form (Lect. 3)

• Converting arbitrary LP forms to the standard form (Ax ≤ b , x ≥ 0 or Ax ≥ b, x ≥ 0)
• Propositions in the lecture demonstrate how arbitrary LP forms can be converted to standard form.

LP Standard Form (SF)

• Definition of a linear system in Standard Form (SF)
  • All variables are non-negative
  • All other constraints are equations
• A LP is in Standard Form if its system of constraints is in SF.
• The objective function is not impacted by converting to SF

Example of Standard Form Conversion

• Show cases converting an example LP to Standard Form
• Show examples of adding slack variables to inequalities to make them equalities, to allow conversion to the Standard Form

Active Constraints in the Standard Form

• Definition of an active constraint (a constraint where the LHS equals the RHS)
• If the LP is in SF, then active constraints can be detected visually in the sign of the corresponding slack variables within the constraints

Advantages of Standard Form

• In SF, inequalities don't disappear; instead, they are transformed to equalities, preserving information about activity in each constraint.
• Slack variables are now non-negative, which simplifies the problem
• Information from original constraints is transferred to non-negative slack variables

Matrix Standard Form

• Matrix representation for standard form of minimization & maximization problems
• Introduces slack variables as part of the transformation
• Ensures every inequality has a corresponding slack variable in the matrix

Sign-Free Variables and Standard Form

• Explain sign-free variables and how to deal with them in converting an LP form to SF
• These variables are replaced by a pair of non-negative variables (y, z) as part of the reformulation
• The variables are converted to the positive or negative components to allow them only to take positive values. This requires an additional variable in the model.

Exercises (Example Problems)

• Provided exercises in max, min formats for conversion to SF to clarify the conversion process

This quiz covers Lecture 8 on Linear Programming (LP) Standard Form, focusing on its equivalent transformations and reference forms. You'll learn how to maximize and minimize output while adhering to resource constraints, as well as transforming various constraints into the standard LP form for problem-solving. Test your understanding of essential LP concepts and applications in asset management.