Constrained optimization Indirect methods Direct methods Indirect methods
![Constrained optimization • Indirect methods • Direct methods Constrained optimization • Indirect methods • Direct methods](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-1.jpg)
![Indirect methods • Sequential unconstrained optimization techniques (SUMT) • Exterior penalty function methods • Indirect methods • Sequential unconstrained optimization techniques (SUMT) • Exterior penalty function methods •](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-2.jpg)
![Exterior penalty function method • Minimize total objective function=objective function+penalty function • Penalty function: Exterior penalty function method • Minimize total objective function=objective function+penalty function • Penalty function:](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-3.jpg)
![Interior 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:](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-4.jpg)
![Extended interior penalty function method • Incoprorates best features of interior and exterior penalty Extended interior penalty function method • Incoprorates best features of interior and exterior penalty](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-5.jpg)
![Augmented Lagrange Multiplier (ALM) Method • Motivation: Other penalty function methods – total objective Augmented Lagrange Multiplier (ALM) Method • Motivation: Other penalty function methods – total objective](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-6.jpg)
![• ALM method allows to find optimum without having to use extreme values • ALM method allows to find optimum without having to use extreme values](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-7.jpg)
![• Equality contraints only: Total function: Lagrangian + penalty multiplier penalty function • • Equality contraints only: Total function: Lagrangian + penalty multiplier penalty function •](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-8.jpg)
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![Constrained optimization Indirect methods Direct methods Constrained optimization • Indirect methods • Direct methods](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-1.jpg)
Constrained optimization • Indirect methods • Direct methods
![Indirect methods Sequential unconstrained optimization techniques SUMT Exterior penalty function methods Indirect methods • Sequential unconstrained optimization techniques (SUMT) • Exterior penalty function methods •](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-2.jpg)
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 functionobjective functionpenalty function Penalty function Exterior penalty function method • Minimize total objective function=objective function+penalty function • Penalty function:](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-3.jpg)
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 functionobjective functionpenalty function Penalty function Interior penalty function method • Minimize total objective function=objective function+penalty function • Penalty function:](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-4.jpg)
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 Extended interior penalty function method • Incoprorates best features of interior and exterior penalty](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-5.jpg)
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 Augmented Lagrange Multiplier (ALM) Method • Motivation: Other penalty function methods – total objective](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-6.jpg)
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 • ALM method allows to find optimum without having to use extreme values](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-7.jpg)
• 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 • Equality contraints only: Total function: Lagrangian + penalty multiplier penalty function •](https://slidetodoc.com/presentation_image/13be13ba6eab97ce0ef140a00928d002/image-8.jpg)
• 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:
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