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Engineering Optimization – Concepts and Applications Engineering Optimization Concepts and Applications Fred van Keulen Matthijs Langelaar CLA H21.1

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Engineering Optimization – Concepts and Applications Covered so far … 1. Introduction: – Negative null form – Applications 1. Introduction: – Negative null form – Applications 2. Optimization problem: – Definition – Characteristics 2. Optimization problem: – Definition – Characteristics 3. Problem checking: – Boundedness – Monotonicity analysis 3. Problem checking: – Boundedness – Monotonicity analysis 4. Optimization model: – Model simplification – Approximation 4. Optimization model: – Model simplification – Approximation

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Engineering Optimization – Concepts and Applications Upcoming topics Optimization problem Definition Checking Negative null formModel Special topics Sensitivity analysis Topology optimization Linear / convex problems Solution methods Optimality criteria Optimization algorithms Unconstrained problems Optimality criteria Optimization algorithms Constrained problems

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Engineering Optimization – Concepts and Applications Contents ● Unconstrained problems – Transformation methods – Existence of solutions, optimality conditions – Nature of stationary points – Global optimality

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Engineering Optimization – Concepts and Applications Unconstrained Optimization ● Why? – Elimination of active constraints unconstrained problem – Develop basic understanding useful for constrained optimization – Transformation of constrained problems into unconstrained problems – Relevant engineering problems (potential energy minimization) Transformation of constrained problems into unconstrained problems

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Engineering Optimization – Concepts and Applications Transforming constrained problem ● Reformulation through barrier functions: g f x Transformation:

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Engineering Optimization – Concepts and Applications Transformed problem g f

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Engineering Optimization – Concepts and Applications Transformed problem ● Barrier functions result in feasible, interior optimum: r = 100 r = 200 r = 400 r = 800

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Engineering Optimization – Concepts and Applications Penalization g f Transformation: ● Alternative reformulation: penalty functions

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Engineering Optimization – Concepts and Applications Penalization (2) ● Penalty functions result in infeasible, exerior optimum: p = 4 p = 10 p = 20 p = 40

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Engineering Optimization – Concepts and Applications Problem transformation summary Barrier functionPenalty function Need feasibleYesNo starting point? Nature of optimumInteriorExterior (feasible)(infeasible) Type of constraints g g, h

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Engineering Optimization – Concepts and Applications Unconstrained Optimization ● Why? – Elimination of active constraints unconstrained problem – Develop basic understanding useful for constrained optimization – Transformation of constrained problems into unconstrained problems – Relevant engineering problems (potential energy minimization)

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Engineering Optimization – Concepts and Applications Unconstrained engineering problem ● Example: displacement of loaded structure ● Equilibrium: minimum potential energy F x = 5N F y = 5N k 1 = 8N/cm k 2 = 1N/cm 10 cm x1x1 x2x2

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Engineering Optimization – Concepts and Applications Unconstrained engineering problem ● Potential energy: x1x1 x2x2 ● Equilibrium: k1k1 k2k2 FyFy FxFx

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Engineering Optimization – Concepts and Applications Unconstrained engineering problem x2x2 x1x1 x1x1 x2x2

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Engineering Optimization – Concepts and Applications Contents ● Unconstrained problems – Transformation methods – Existence of solutions, optimality conditions – Nature of stationary points – Global optimality

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Engineering Optimization – Concepts and Applications Theory for solving unc. problems ● Assumptions: – Objective continuous and differentiable (C 1 ) – Domain closed and bounded (compact) f ]a, b[

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Engineering Optimization – Concepts and Applications Existence of minima ● Weierstrass Theorem: “A continuous function on a compact set has a maximum and a minimum in that set” ● Sufficient condition for existence! ● Interior optima only exist for non-monotonic functions

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Engineering Optimization – Concepts and Applications One-dimensional case ● Calculus: conditions for minimum of f – Derivative zero: – Second derivative positive: ● Interpretation through Taylor series: f x +h -h (necessary) (sufficient)

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Engineering Optimization – Concepts and Applications One-dimensional case (2) ● Condition for minimum: f”(x) > 0 ● Other possibilities: f”(x) < 0 maximum f”(x) = 0 ? Check higher order derivatives

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Engineering Optimization – Concepts and Applications Geometrical interpretation ● Positive locally increasing f x

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Engineering Optimization – Concepts and Applications One-dimensional case (3) ● Possible situations for stationary points ( f’ = 0 ): f x

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Engineering Optimization – Concepts and Applications Example ● Aspirin pill revisited: worst pill ever! – Maximize dissolving time minimize surface area r h ● Equality constraint active eliminate h

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Engineering Optimization – Concepts and Applications Example (2)

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Engineering Optimization – Concepts and Applications Multidimensional case ● Local approximation to multidimensional minimum by multidimensional Taylor series: GradientCondition for minimum:

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Engineering Optimization – Concepts and Applications Multidimensional case (2) ● For minimum, consider second-order approximation: Hessian

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Engineering Optimization – Concepts and Applications Multidimensional case (3) ● Second order approximation: ● Local minimum: – First Order Necessity Condition: – Second Order Sufficiency Condition:

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Engineering Optimization – Concepts and Applications Positive definite ● Hessian positive definite ● Test for positive definiteness: – Evaluate y T Hy for all y (impractical) – All eigenvalues i of H positive – Sylvester’s rule: all determinants of H and its principal submatrices are positive

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Engineering Optimization – Concepts and Applications Example ● Another loaded structure: FxFx FyFy k1k1 k2k2 uxux uyuy ● Equilibrium: ● First order necessity condition:

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Engineering Optimization – Concepts and Applications Example (2) ● Second order sufficiency: H positive definite? ● Note: H diagonal constant matrix. Separable objective function:

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Engineering Optimization – Concepts and Applications Quadratic functions ● Polynomial terms up to 2 nd order: ● General form: ● Note: 2 nd order Taylor series is exact

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Engineering Optimization – Concepts and Applications Quadratic functions (2) ● Transformation to symmetric matrix: ● Symmetric, since:

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Engineering Optimization – Concepts and Applications Quadratic functions (2) ● Optimality conditions: – Gradient: – Hessian: ● Stationary points:

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Engineering Optimization – Concepts and Applications Example ● Consider Hessian positive definite Stationary point

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Engineering Optimization – Concepts and Applications Example (2) ● Result: minimum at (1.2, -2.6): f x1x1 x2x2

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Engineering Optimization – Concepts and Applications Example: least squares ● Least squares fitting: unconstrained optimization problem x f ● Stationary point:

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Engineering Optimization – Concepts and Applications Contents ● Unconstrained problems – Transformation methods – Existence of solutions, optimality conditions – Nature of stationary points – Global optimality

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Engineering Optimization – Concepts and Applications ● Local nature: minimum Nature of stationary points ● Hessian H positive definite: – Quadratic form – Eigenvalues

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Engineering Optimization – Concepts and Applications Nature of stationary points (2) ● Hessian H negative definite: – Quadratic form – Eigenvalues ● Local nature: maximum

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Engineering Optimization – Concepts and Applications Nature of stationary points (3) ● Hessian H indefinite: – Quadratic form – Eigenvalues ● Local nature: saddle point

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Engineering Optimization – Concepts and Applications Nature of stationary points (4) ● Hessian H positive semi-definite: – Quadratic form – Eigenvalues ● Local nature: valley H singular!

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Engineering Optimization – Concepts and Applications Nature of stationary points (5) ● Hessian H negative semi-definite: – Quadratic form – Eigenvalues ● Local nature: ridge H singular!

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Engineering Optimization – Concepts and Applications Stationary point nature summary Definiteness H Nature x* Positive d.Minimum Positive semi-d.Valley IndefiniteSaddlepoint Negative semi-d.Ridge Negative d.Maximum

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Engineering Optimization – Concepts and Applications Structural example ● Compressed pin-jointed truss with rotational springs: F k1k1 k2k2 l Maximum (unstable) Saddles (unstable) Minima (stable)

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Engineering Optimization – Concepts and Applications Contents ● Unconstrained problems – Transformation methods – Existence of solutions, optimality conditions – Nature of stationary points – Global optimality

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Engineering Optimization – Concepts and Applications Global optimality ● Optimality conditions for unconstrained problem: – First order necessity:(stationary point) – Second order sufficiency: H positive definite at x* ● Optimality conditions only valid locally: local minimum ● When can we be sure x * is a global minimum?

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Engineering Optimization – Concepts and Applications Convex functions ● Convex function: any line connecting any 2 points on the graph lies above it (or on it): ● H positive (semi-)definite f locally convex (proof by Taylor approximation) ● Equivalent: tangent lines/planes stay below the graph

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Engineering Optimization – Concepts and Applications Convex domains ● Convex set: “A set S is convex if for every two points x 1, x 2 in S, the connecting line also lies completely inside S ”

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Engineering Optimization – Concepts and Applications Convexity and global optimality If: ● Objective f = (strictly) convex function ● Feasible domain = convex set Stationary point = (unique) global minimum ● Special case: f, g, h all linear linear optimization programming ● More general class: convex optimization (Ok for unconstrained optimization)

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Engineering Optimization – Concepts and Applications Example ● Quadratic functions with A positive definite are strictly convex: f x1x1 x2x2 Stationary point (1.2, -2.6) must be unique global optimum

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Engineering Optimization – Concepts and Applications Summary optimality conditions ● Conditions for local minimum of unconstrained problem: – First Order Necessity Condition: – Second Order Sufficiency Condition: H positive definite ● For convex f in convex feasible domain: condition for global minimum: – Sufficiency Condition:

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