• You want to find the minimum or maximum of a function.
• Possibly subject to some constraints.

\[ \min_x f(x)\] \[ \text{s.t. } g(x) \ge 0\] - Maximizing a function \(f\) is equivalent to minimizing \(-f\).

• Agent’s choice problem:
  • Maximize utility subject to some constraints.
• Estimation:
  • Ordinary Least Squares (OLS)
  • Simulated Method of Moments (SMM): Choose parameters to minimize distance between data moments and model moments.
  • Maximum Likelihood Estimation (MLE): Choose parameters to maximize likelihood of observing data.

• Pencil and paper: Take derivates, use Lagragian, …
  • This is what you did in first year.
  • Not always possible.
• Grid search: Guess and check.
  • This is what we did last week in Lecture 2.
  • Accurate solution requires a very fine grid.
  • This can be very slow, especially in many dimensions.
• Optimization algorithms: “Smart” guess and check.
  • Use limited information about function (such as derivatives) and previous guesses to decide next guess.
  • Lots of choices here.
  • Sometimes doesn’t work; no perfect algorithm.

• We know solutions exists in \([a,b]\).
  • Search intensity normalized between 0 and 1
  • Hours of work between 0 and 80.
  • Savings is betwen 0 and net worth.
• Common algorithm: Brent’s method.
• Does not require derivatives.
• Uses bisections, secants, and inverse quadratic interpolation…

• Optimization in multiple dimension can be challenging..
  • Curse of dimensionality.
• If function is twice differentiable and has analytic gradient/Hessian, you can use Newton’s method.
• In Economics, you will rarely be dealing with such a nice function in your optimization.

• Numerically approximates gradient and Hessian and applies similar updating rule to Newton’s method.
  • Finite differences: For a small enough \(\Delta\) \[ f'(x) \approx \frac{f(x+\Delta) - f(x)}{(\Delta)} \]
  • Automatic differentiation: Computer attempts to apply chain run to components of the function.
• Most common algorithm of this type is BFGS or (L-BFGS).
• Problems:
  • Approximation of derivative might not be well-behaved.
  • Can be computationally expensive.

• Most common method: Nelder-Mead
  • Easy to implement. Does not necessarily work well.

https://en.wikipedia.org/wiki/Nelder–Mead_method

• Operationalizes the intuition of “rolling down hill” by updating a simplex until converges.
• Often referred to as “downhill simplex method”.
  
• Can easily be confused by local minimum.
• Can also get stuck in flat areas or go around in circles.

• Randomization can improve the performance of our optimizers.
  • Many local minimum
  • Function is not smooth.
• Basin hopping
  1. Guess initial point and run algorithm (such as NM).
  2. From candidate solution, randomly “hop” to new initial point and run algorithm again.
• Laplace-type estimator
  • Randomly jump around parameter space.
  • Accept (with some randomization) better guesses.
  • Size of jump updates based on fraction of accepted guesses.

• Many (but not all) optimization alogrithms end after reaching some preset tolerance level.
• Lower tolerance improves quality of answer but increase time algorithm will run.
• Important to keep nested optimization in mind.
• Common procedure in economics:
  • Inner loop solves model given a parameter guess: make choices to maximize utility.
  • Outer loop chooses parameters to minimize difference between model and data (maximum likehood/SMM).
• Low tolerance in outer loop may lead to bad parameter guesses.
• Low tolerance in inner loop may lead to slower convergence.

• Try to reduce dimensions, especially when getting started.
• You can plot in 2D and 3D, but 4D becomes quite challenging.
• Understand the shape of the function you are trying to optimize:
  • Is it smooth? Does it have kink points?
• There is no perfect algorithm.
• Transformation of variables and scale may be helpful. Log(), exp(), …

https://jblevins.org/notes/bijections