Lagrange Multipliers

Mathematical Modeling · Axiom Academy

LESSON Lagrange Multipliers - Mathematical Modeling Constrained Optimization in Mathematical Modeling In real-world modeling, we rarely have unlimited resources or freedom. Instead, we optimize subject to constraints: Economics: Maximize profit subject to a budget constraint Engineering: Minimize weight subject to strength requirements Physics: Minimize energy subject to conservation laws Biology: Maximize fitness subject to metabolic constraints The method of Lagrange multipliers provides a systematic approach to these constrained optimization problems. The Constrained Optimization Problem We want to find the extreme values (maxima or minima) of an objective function subject to one or more constraints of the form . The Lagrangian combines the objective function and constraint into a single function: Or equivalently, writing the constraint as g(x) - c = 0 : Here, is called the Lagrange multiplier . Lagrange Multiplier Conditions At a constrained optimum, the gradient of the Lagrangian equals zero: This gives us the system of equations: In component form for two variables: Geometric Interpretation: Gradients Are Parallel At a constrained optimum, the gradient of f must be parallel to the gradient of g : Why? Consider moving along the constraint curve (or surface) g(x) = c : The gradient is perpendicular to the constraint surface If had any component tangent to the constraint, we could move along the constraint and change f

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