Constrained optimization Indirect methods Direct methods Indirect methods

Indirect methods • Sequential unconstrained optimization techniques (SUMT) • Exterior penalty function methods •

Exterior penalty function method • Minimize total objective function=objective function+penalty function • Penalty function:

Interior penalty function method • Minimize total objective function=objective function+penalty function • Penalty function:

Extended interior penalty function method • Incoprorates best features of interior and exterior penalty

Augmented Lagrange Multiplier (ALM) Method • Motivation: Other penalty function methods – total objective

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

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

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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

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 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, rp 0: