Reading Assignment:

• Optimality Conditions for Constrained Problems: Due Friday, February 6.
• Quadratic Programming Notes: Due Wednesday, February 25.
• Probability Notes: Due Monday, March 1
• Investment Science: Sec. 6.1-6.3, Due Wednesday, March 3.

Homework Assignment:

• Quadratic Programming
  • Do all of the exercises in the notes on quadratic probramming.
  • Be able to state both necessary and sufficient conditions for optimality for a QP in standard form as specified in the notes.
  • Be able to transform any QP to one in standard form.
  • Be able to transform a quadratic function into matrix form.
  • Be able to determine if a 2X2 symmetric matrix is positive definite, positive semi-definite, or neither.
  • Be able to write down the KKT conditions for a concretely specified QP.
  • Be able to derive the KKT conditions for a general QP.
• Basic Probability
  • Do all of the exercises in the notes on probability.
  • Be able to determine the pdf for a concretely specified random variable.
  • Be able to compute means, variances, and covariances for concretely specified random variables.
  • Be able to sketch the mean-standard deviation curve for two concretely specified random variables.
  • Be able to derive the Cartesian formula for the hyperbola defining the mean-standard deviation curve.

Vocabulary List:

• Constrained Optimization
  • feasible directions
  • tangent cone
  • regularity
  • The Lagrangian
  • first-order conditions for optimality
  • KKT conditions (KKT pairs)
  • second-order conditions for optimality
  • first-order necessary and sufficient conditions under convexity
  • convexity implies the local solutions are global solutions
• Quadratic Programming
  • quadratic function
  • symmetric matrix
  • positive definite, positive semi-definite
  • quadratic program
  • QP standard form
  • descent direction
  • feasible direction
  • first-order necessary conditions for optimality in QP
  • first-order sufficient conditions for optimality in QP
  • KKT conditions for QP
• Basic Probability
  • Probability Space
  • Random Variable
  • Expectation operator
  • mean, variance, covariance, standard deviation, correlation coefficient
  • stochastic independence
  • probability density function
  • joint probability density function
  • random vector
  • covariance matrix
  • mean-standard deviation curve

Key Concepts:

• Constrained Optimization
  • KKT conditions
  • Second-order Optimality conditions.
  • Convexity and it implications for optimality
• Quadratic Programming
  • first- and second-order optimality conditions for QP
  • KKT conditions
• Basic Probability
  • probability space
  • random variable
  • expectation operator
  • probability density function
  • mean-standard deviation curve
  • covariance matrix

Skills to Master:

• Checking optimality conditions
• checking regularity
• checking convexity
• Writing a quadratic function in matrix form.
• Determine if a 2X2 symmetric matrix is positive definite, positive semi-definite, or neither.
• computing the KKT conditions and finding KKT points.
• computing means, variances, standard deviations, covariances, and correlation coefficients of random variable
• computing the pdf of a random variable
• computing the covariance matrix of a random vector
• graphing the mean-variance curve for two rvs

Quiz: