1 / 28

Engineering Optimization

Concepts and Applications WB 1440. Engineering Optimization. Fred van Keulen Matthijs Langelaar CLA H21.1 A.vanKeulen@tudelft.nl. f. x 2. h. Meaning:.  h.  f. x 1. Gradients parallel  tangents parallel  h tangent to isolines. Geometrical interpretation.

kaethe
Download Presentation

Engineering Optimization

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Concepts and Applications WB 1440 Engineering Optimization • Fred van Keulen • Matthijs Langelaar • CLA H21.1 • A.vanKeulen@tudelft.nl

  2. f x2 h Meaning: h f x1 Gradients parallel  tangents parallel h tangent to isolines Geometrical interpretation • For single equality constraint: simple geometrical interpretation of Lagrange optimality condition:

  3. f h x2 f h • Equivalent: stationary Lagrangian: x1 Summary • First order optimality condition for equality constrained problem: • Zero reduced gradient:

  4. Contents • Constrained Optimization: Optimality Criteria • Reduced gradient • Lagrangian • Sufficiency conditions • Inequality constraints • Karush-Kuhn-Tucker (KKT) conditions • Interpretation of Lagrange multipliers • Constrained Optimization: Algorithms

  5. h • Lagrange condition: f h h h h f h f f f h f f h f Sufficiency? • Until now, only stationary points considered. Does not guarantee minimum! maximum minimum minimum no extremum

  6. with obtained by differentiation of the constrained gradient, andsecond-order constraint perturbation: Constrained Hessian • Sufficiency conditions follow from 2nd order Taylor approximation • Second order information required:constrained Hessian:

  7. Lagrangian approach also yields: with Perturbations only in tangent subspace ofh! Sufficiency conditions • Via 2nd order Taylor approximation, it follows that at a minimum the following must hold: (Constrained Hessianpositive definite) and

  8. 2. Sufficient condition: minimum when (1) and: on tangent subspace. Summary • Optimality conditions for equality constrained problem: 1. Necessary condition: stationary point when:

  9. 1. Necessary condition: stationary point when Example x2 f h x1

  10. Contents • Constrained Optimization: Optimality Criteria • Reduced gradient • Lagrangian • Sufficiency conditions • Inequality constraints • Karush-Kuhn-Tucker (KKT) conditions • Interpretation of Lagrange multipliers • Constrained Optimization: Algorithms

  11. At optimum, only active constraints matter: Inequality constrained problems • Consider problem with only inequality constraints: • Optimality conditions similar to equality constrained problem

  12. Consider feasible local variation around optimum: (boundary optimum) (feasible perturbation) Inequality constraints • First order optimality:

  13. g2 x2 g1 f -f x1 • Interpretation: negative gradient (descent direction) lies in cone spanned by positive constraint gradients -f Optimality condition • Multipliers must be non-negative:

  14. Feasible cone • Descent direction: -f Optimality condition (2) g2 • Feasible direction: x2 g1 f x1 • Equivalent interpretation: no descent direction exists within the cone of feasible directions

  15. -f -f -f f f Examples f

  16. Formulation including all inequality constraints: Complementaritycondition and Optimality condition (3) • Active constraints:Inactive constraints:

  17. x1 x1 L m L x2 m x2 Example

  18. Mechanical application: contact • Lagrange multipliers also used in: • Contact in multibody dynamics • Contact in finite elements

  19. Contents • Constrained Optimization: Optimality Criteria • Reduced gradient • Lagrangian • Sufficiency conditions • Inequality constraints • Karush-Kuhn-Tucker (KKT) conditions • Interpretation of Lagrange multipliers • Constrained Optimization: Algorithms

  20. Lagrangian: (optimality) and (feasibility) (complementarity) Karush-Kuhn-Tucker conditions • Combining Lagrange conditions for equality and inequality constraints yields KKT conditions for general problem:

  21. on tangent subspace of h and active g. Sufficiency • KKT conditions are necessary conditions for local constrained minima • For sufficiency, consider the sufficiency conditions based on the active constraints: • Interpretation: objective and feasible domain locally convex

  22. Pitfall: • Sign conventions for Lagrange multipliers in KKT condition depend on standard form! • Presented theory valid for negative null form Additional remarks • Global optimality: • Globally convex objective function? • And convex feasible domain? Then KKT point gives global optimum

  23. Contents • Constrained Optimization: Optimality Criteria • Reduced gradient • Lagrangian • Sufficiency conditions • Inequality constraints • Karush-Kuhn-Tucker (KKT) conditions • Interpretation of Lagrange multipliers • Constrained Optimization: Algorithms

  24. KKT: Looking for: Significance of multipliers • Consider case where optimization problem depends on parameter a: Lagrangian:

  25. Significance of multipliers (2) Looking for: KKT:

  26. Multipliers give “price of raising the constraint” • Note, this makes it logical that at an optimum, multipliers of inequality constraints must be positive! Significance of multipliers (3) • Lagrange multipliers describe the sensitivity of the objective to changes in the constraints: • Similar equations can be derived for multiple constraints and inequalities

  27. Stress constraint: Example A, sy N Minimize mass (volume): l

  28. Stress constraint: Constraint sensitivity: Check: Example (2)

More Related