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Constrained optimization. Indirect methods Direct methods. Indirect methods. Sequential unconstrained optimization techniques (SUMT) Exterior penalty function methods Interior penalty function methods Extended penalty function methods Augmented Lagrange multiplier method.

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Constrained optimization

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Constrained optimization l.jpg

Constrained optimization

  • Indirect methods

  • Direct methods


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Indirect methods

  • Sequential unconstrained optimization techniques (SUMT)

  • Exterior penalty function methods

  • Interior penalty function methods

  • Extended penalty function methods

  • Augmented Lagrange multiplier method


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Exterior penalty function method

  • Minimize total objective function=objective function+penalty function

  • Penalty function: penalizes for violating constraints

  • Penalty multiplier

    • Small in first iterations, large in final iterations

  • Sequence of infeasible designs approaching optimum


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Interior penalty function method

  • Minimize total objective function=objective function+penalty function

  • Penalty function: penalizes for being too close to constraint boundary

  • Penalty multiplier

    • Large in first iterations, small in final iterations

  • Sequence of feasible designs approaching optimum

  • Needs feasible initial design

  • Total objective function discontinuous on constraint boundaries


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Extended interior penalty function method

  • Incoprorates best features of interior and exterior penalty function methods

    • Approaches optimum from feasible region

    • Does not need a feasible initial guess

    • Composite penalty function:

      • Penalty for being too close to the boundary from inside feasible region

      • Penatly for violating constraints

  • Disadvantages

    • Need to specify many paramenters

    • Total objective function becomes ill conditioned for large values of the penalty multiplier


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Augmented Lagrange Multiplier (ALM) Method

  • Motivation: Other penalty function methods – total objective function becomes ill conditioned for large values of the penalty multiplier


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  • ALM method allows to find optimum without having to use extreme values of penalty multiplier

  • Takes advantage of K-T optimality conditions


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  • Equality contraints only: Total function:

    Lagrangian + penalty multiplierpenalty function

  • If we knew the values of the Lagrange multipliers for the optimum, *, then we could find the optimum solution in one unconstrained minimizatio for any value of the penalty coefficient greater than a minimum threshold, rp0:


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