Graphical Method

Optimization · Axiom Academy

Learn to solve 2D linear programs by visualizing constraints and objectives Each constraint defines a boundary line. We plot these lines by finding their intercepts with the axes. For example, the constraint 2x + 3y = 12 crosses the x-axis at (6,0) and the y-axis at (0,4). 2. Identify Feasible Side of Each Constraint Each constraint inequality divides the plane into two half-planes. We shade the region that satisfies the inequality. For instance, if 2x + 3y ≤ 12, we shade the region toward the origin (test point (0,0)). 3. Find the Feasible Region (Intersection) The feasible region is the intersection of all constraint half-planes. It forms a convex polygon (or unbounded region). Only points within this region are valid candidate solutions. 4. Plot Objective Function Contours The objective function c'x = k forms a family of parallel lines for different values of k. Each contour line represents points with the same objective value. The gradient vector c points in the direction of increasing objective value. 5. Move Contour to Find Optimum We slide the objective contour line in the direction that improves the objective (increasing for max, decreasing for min) until it touches the feasible region at its last point. This extreme point is the optimal solution.

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