1. Constrained optimization Indirect methods Direct methods

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

3. 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

4. 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

5. 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

6. Augmented Lagrange Multiplier (ALM) Method Motivation: Other penalty function methods – total objective function becomes ill conditioned for large values of the penalty multiplier

7. ALM method allows to find optimum without having to use extreme values of penalty multiplier Takes advantage of K-T optimality conditions

8. 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, r p0 :