80 likes | 99 Views
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.
E N D
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: