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

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• 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

• 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

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

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

• Takes advantage of K-T optimality conditions

• Equality contraints only: Total function:Lagrangian + penalty multiplierpenalty 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:

l

kkp

l

kkkp hrhFrA

1

2

1)()()(),,( xxxλx

),,( minimize to

Find

*prA λx

x