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Applied Mathematics · Quick Reference
Optimization Cheat Sheet
In one line: optimization is the mathematics of finding the best possible outcome — the maximum profit, minimum cost, or most efficient use of limited resources.
Key Ideas
1Objective Function. The quantity you're trying to maximize or minimize, such as profit, cost, or distance.
2Constraints. Limits or restrictions on the variables involved, such as a fixed budget or limited time.
3Unconstrained Optimization. Finding a maximum or minimum using calculus (setting the derivative to zero), with no restrictions on the variable's range.
4Constrained Optimization. Finding the best outcome while satisfying one or more constraints, often visualized as finding the best point within a feasible region.
5Linear Programming. A specific type of constrained optimization where both the objective function and constraints are linear; solutions occur at corners of the feasible region.
Worked Examples
A company's profit is P(x) = -2x^2 + 80x - 300, where x is units sold. Find the number of units that maximizes profit.
20 units maximizes profit
Using the previous example, find the maximum profit.
$500
A factory has a budget constraint: it can spend at most $1000 on materials at $50 each. What is the constraint's maximum quantity?
20 units maximum, given the budget constraint
Practice Yourself
What is the term for the quantity being maximized or minimized?
The objective function
What are 'constraints' in an optimization problem?
Limits or restrictions on the variables
For P(x)=-x^2+40x, find the x that maximizes P (Hint: P'(x)=-2x+40).
x=20
What type of optimization has both a linear objective function and linear constraints?
Linear programming