Optimization

• Minimum of 1/a + 4/b + 9/c When a + b + c = 6
  Apply Cauchy–Schwarz to show the minimum is 6, attained at (a, b, c) = (1, 2, 3), which satisfies the positive-variable constraint.
• How to Derive Demand from Utility and an Endowment
  Derive a consumer's demand by valuing the endowment, maximizing utility subject to the budget, and distinguishing chosen consumption from net trade.
• How to Find the Distance from an Ellipsoid to a Plane
  Use Lagrange multipliers to find the ellipsoid’s extreme projection onto the plane normal, then divide the remaining gap by the normal’s length.
• How to Convert a Constrained Optimization Problem into an Unconstrained One
  Convert equality-constrained quadratic programs by parameterizing the feasible set; use penalties or barriers carefully, since they are not generally exact.
• How to Solve Calculus Word Problems
  Learn a reliable method for calculus word problems: identify the goal, define variables, model the situation, differentiate or integrate, and check units.
• How to Convert a Primal Linear Program to Its Dual
  Convert a linear program to its dual by transposing the constraint matrix, reversing the objective direction, and matching row and variable signs.
• Maximize xy Subject to x² + xy + y² = 1
  For real x and y satisfying x² + xy + y² = 1, the maximum of xy is 1/3, attained when x and y are both 1/√3 or both −1/√3.
• How to Minimize a Sum-of-Squares Error Function
  Minimize a sum-of-squares error by setting its gradient to zero; for linear models, solve the least-squares equations and check uniqueness.
• What Is the Steepest Descent Method?
  The steepest descent method minimizes a differentiable function by moving opposite its gradient; learn how to choose a step size and avoid slow convergence.