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Constrained Optimization

Constrained Optimization. Ref: wikipedia; Jensen. 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.

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Constrained Optimization

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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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