Here, you can find a (non-exhaustive!) list of the contents and take-aways of Chapter 4: Optimization. This list serves as an opportunity to assess both how thoroughly you should read the chapter before opening it for the first time, and how well you have managed to follow along once you have read it.
Chapter 4: Optimization discusses
- the formal representation of and basic concepts related to mathematical optimization problems
- the step-by-step procedure used to solve unconstrained optimization problems, and the justification of this procedure
- the step-by-step procedure used to solve optimization problems with one equality constraint, and the justification of this procedure
- the generalization of the solution approach to multiple equality constraints and inequality constraints
- various approaches to simplifying problems before solving them, and how to avoid an excessive amount of computations in solving them
Someone with profound knowledge of the contents of this chapter should
- first and foremost: be able to solve unconstrained and constrained optimization problems
- be able to correctly write down an optimization problem and use the correct terminology to describe it (objective, choice variables, constraint types)
- know how , the restriction of a function to a set , is formally defined and how the concept is helpful in the optimization context
- be able to decscribe the objects and and how they are related ( is the constraint set of the problem )
- be able to verbally describe how the first and second order necessary conditions for local extremizers come to be in the unconstrained problem
- know how to classify solution candidates (critical points and border solutions) using first and second order conditions
- be familiar with the univariate implicit function theorem and the intuition of how it helps in deriving first and second order conditions analogous to those of the unconstrained problem for problems with one equality constraint
- know where the Lagrangian function comes from, and how to write it down correctly
- be able to graphically illustrate level sets, especially those relevant to economics, e.g. indifference “curves”
- be familiar with the “value-cost interpretation” of the Lagrangian multipliers, and when and how it can be used as a very simplistic sufficient condition for local extremizers
- be aware of the statement of the Karush-Kuhn-Tucker theorem
and be able to answer a number of related questions, including
- Why is the topic of mathematical optimization important for economists?
- What are maximizers and minimizers? How are they different from maxima and minima?
- What is the statement of the Weierstrass Extreme Value Theorem? What is its role in optimization?
- How does a “saddle point” of a bivariate function look like? Where does the label come from?
- How does convexity or concavity of the objective function help in unconstrained optimization problems?
- True or false: equality-constrained optimization can be viewed as optimization on the zero-level-set(s) of the constraint function(s).
- Which additional “irregular” or border solution candidates do we have to take into account in equality-constrained optimization problems compared to unconstrained problems?
- When and how can we re-write inequality-constrained problems as equality-constrained ones? Why would we be interested in doing so?
- How are “explicit function representations” of equality constraints helpful in facilitating the problem?
- What role does solution existence play in concrete applications, i.e. how does the issue relate to justifying our solution for the global extremizer of interest?