constrained optimization
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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
Constrained optimization
  • Indirect methods
  • Direct methods
indirect methods
Indirect methods
  • Sequential unconstrained optimization techniques (SUMT)
  • Exterior penalty function methods
  • Interior penalty function methods
  • Extended penalty function methods
  • Augmented Lagrange multiplier method
exterior penalty function method
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
interior penalty function method
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
extended interior penalty function method
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
augmented lagrange multiplier alm method
Augmented Lagrange Multiplier (ALM) Method
  • Motivation: Other penalty function methods – total objective function becomes ill conditioned for large values of the penalty multiplier
slide7
ALM method allows to find optimum without having to use extreme values of penalty multiplier
  • Takes advantage of K-T optimality conditions
slide8
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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