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Constrained Optimization: Jensen's Method of Lagrange Multipliers

Learn about constrained optimization using Jensen's Method of Lagrange Multipliers. Discover how to find the optimum by solving the stationary points of the Lagrangian. Explore examples and formulations in this mathematical approach.

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Constrained Optimization: Jensen's Method of Lagrange Multipliers

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  1. Constrained Optimization Ref: wikipedia; Jensen

  2. Method of Lagrange Multipliers • Consider the optimization problem Imagine walking along g=c; the value of f varies The maximum occurs where the contour line and g touch The gradient vectors of f and g are dependent at the maximum

  3. Lagrange Multiplier (cont) • Gradient vectors of f and g are dependent at the maximum • The new variable, l, is called the Lagrange multiplier • A new auxiliary function is introduced • The optimum can be obtained by solving the stationary points of the Lagrangian L

  4. Example

  5. General Formulation • Objective function: f(x) • The constraints: gk(x) = 0 • Lagrangian:

  6. Lagrange Multipliers in Action • Ex: a box of volumn V, with minimal surface area

  7. Closest Point of a Line to a Point Approach 1:

  8. Approach 2:

  9. Karush-Kuhn-Tucker Condition

  10. 1st iteration (all constraints inactive) (violate) Set g2 = 0 Active set method g2 g1 (4,4) g3 2nd iteration Solution (KKT):

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