Constrained Problems

Optimization · Axiom Academy

INTRO Constrained Optimization What happens when you can't choose freely? Step 1: Constraints Are Everywhere Before we dive into the math, let's explore where constraints naturally appear in the real world. Click on each scenario to reveal its constraint. Move the sliders to see how constraints create a "feasible region" - the area where all constraints are satisfied. Notice how the region changes! Step 3: Where's the Best Solution? Now let's add an objective function we want to maximize: f(x,y) = 2x + 3y. Adjust the target value to see different level curves. Where does the maximum occur? Step 4: Why Constraints Make Things Harder Let's compare: drag the red point to find where you think the minimum is for each problem. Constraints define a feasible region where solutions must live. This fundamentally changes the optimization landscape. Unlike unconstrained problems, constrained optima often occur at constraint boundaries or corners, not at gradient-zero points. We need specialized techniques (Lagrange multipliers, KKT conditions) to handle constraints mathematically. Nearly all practical optimization problems have constraints - budgets, physics laws, resource limits, safety requirements.

This is the written version of the interactive lesson above. See the full Optimization course.