Engineering Optimization

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

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:

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

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

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

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:

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

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

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

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

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

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:

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

-f -f -f f f Examples f

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

x1 x1 L m L x2 m x2 Example

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

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

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

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

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

Significance of multipliers (2) Looking for: KKT:

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

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

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

Engineering Optimization