Introduction to Geometric Programming Based Aircraft Design

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1 Introduction to Geometric Programming Based Aircraft Design Warren Hoburg University of California, Berkeley Electrical Engineering and Computer Science Department 8 th Research Consortium for Multidisciplinary System Design Workshop July 16, 2013

2 Specifications Baseline design Evaluate objective and constraints Change design No Is the design optimal? Yes Optimal Design [Adapted from Alonso 2012]

3 Specifications Challenge #1: Coupled Models Baseline design Evaluate objective and constraints Change design [Kroo 1994] No Is the design optimal? Yes Significant disciplinary separation Expensive function evaluations Optimal Design [Adapted from Alonso 2012]

4 Specifications Challenge #1: Coupled Models Baseline design Evaluate objective and constraints Challenge #2: Competing Objectives Change design No Is the design optimal? minimize w 1 ṁ fuel + w 2 V max + w 3 m pay fuel flow rate Yes Optimal Design V max m pay [Adapted from Alonso 2012]

5 Specifications Challenge #1: Coupled Models Baseline design Evaluate objective and constraints Challenge #2: Competing Objectives Change design No Is the design optimal? minimize w 1 ṁ fuel + w 2 V max + w 3 m pay fuel flow rate Yes Change specifications Optimal Design V max m pay [Adapted from Alonso 2012]

6 Specifications Challenge #1: Coupled Models Change design Baseline design Evaluate objective and constraints Challenge #2: Competing Objectives Want Pareto frontier must solve many times fuel flow rate V max m pay No Is the design optimal? Change specifications Yes Optimal Design [Adapted from Alonso 2012]

7 Specifications Challenge #1: Coupled Models Change specifications Change design No Baseline design Evaluate objective and constraints Is the design optimal? Yes Optimal Design Challenge #2: Competing Objectives Want Pareto frontier must solve many times Challenge #3: Solution Quality Local vs. global optima fuel flow rate V max m pay [Adapted from Alonso 2012]

8 Specifications Challenge #1: Coupled Models Change design Baseline design Evaluate objective and constraints Challenge #2: Competing Objectives Want Pareto frontier must solve many times fuel flow rate V max m pay Change specifications No Is the design optimal? Yes Optimal Design Challenge #3: Solution Quality Local vs. global optima Sensitivity to initial guess [Adapted from Alonso 2012]

9 Specifications Baseline design Evaluate objective and constraints Change design No Is the design optimal? Yes Change specifications Optimal Design [Adapted from Alonso 2012]

10 Specifications Baseline design Evaluate objective and constraints Insight Surprisingly, many relationships in engineering design have an underlying convex structure. Change design No Is the design optimal? Yes Change specifications Optimal Design [Adapted from Alonso 2012]

11 Specifications Exploit convex structure Change design Baseline design Evaluate objective and constraints Insight Surprisingly, many relationships in engineering design have an underlying convex structure. No Is the design optimal? Change specifications Yes Optimal Design [Adapted from Alonso 2012]

12 Specifications Exploit convex structure Change design No Baseline design Evaluate objective and constraints Is the design optimal? Insight Surprisingly, many relationships in engineering design have an underlying convex structure. Benefits Globally optimal solutions Change specifications Yes Optimal Design [Adapted from Alonso 2012]

13 Specifications Exploit convex structure Baseline design Insight Change specifications Change design No Evaluate objective and constraints Is the design optimal? Yes Optimal Design Surprisingly, many relationships in engineering design have an underlying convex structure. Benefits Globally optimal solutions Robust algorithms - no initial guesses; no parameters to tune [Adapted from Alonso 2012]

14 Specifications Exploit convex structure Baseline design Insight Change specifications Change design No Evaluate objective and constraints Is the design optimal? Yes Optimal Design Surprisingly, many relationships in engineering design have an underlying convex structure. Benefits Globally optimal solutions Robust algorithms - no initial guesses; no parameters to tune Extremely fast solutions, even for large problems [Adapted from Alonso 2012]

15 Today s Talk Approach Overview The Power of Lagrange Duality GP-compatible Modeling for Aircraft Design

16 Today s Talk Approach Overview The Power of Lagrange Duality GP-compatible Modeling for Aircraft Design

17 Optimization Compromizes least squares vs. simulated annealing

18 Optimization Compromizes least squares vs. simulated annealing fast reliable extremely specific still, widely used

19 Optimization Compromizes least squares vs. simulated annealing fast reliable extremely specific still, widely used Extremely general Slow convergence Requires hand-holding

20 Optimization Compromizes least squares simulated annealing

21 Optimization Compromizes convex programs {}}{ least squares LP GP SDP simulated annealing

22 Optimization Compromizes convex programs {}}{ least squares LP GP SDP SQP,NLP simulated annealing

23 General Nonlinear Program Decision Variables x R n v Objective Function f 0 (x) : R n v R Constraints 0 f i (x) : R n v R 0 = h i (x) : R n v R

24 General Nonlinear Program Decision Variables x R n v Objective Function f 0 (x) : R n v R Constraints 0 f i (x) : R n v R 0 = h i (x) : R n v R In general, extremely difficult to solve

25 General Nonlinear Program Convex Program Decision Variables x R n v Objective Function f 0 (x) : R n v R Constraints 0 f i (x) : R n v R 0 = h i (x) : R n v R Same as nonlinear program, except f i (x) must be convex h i (x) must be affine In general, extremely difficult to solve

26 General Nonlinear Program Convex Program Decision Variables x R n v Objective Function f 0 (x) : R n v R Constraints 0 f i (x) : R n v R 0 = h i (x) : R n v R Same as nonlinear program, except f i (x) must be convex h i (x) must be affine In general, extremely difficult to solve Very efficient to solve

27 Geometric Program: Definition

28 Geometric Program: Definition Monomial Function m(x) = c n i=1 x a i i, c > 0 (e.g., 1 2 ρv 2 C L S)

29 Geometric Program: Definition Monomial Function m(x) = c n i=1 x a i i, c > 0 (e.g., 1 2 ρv 2 C L S) Posynomial Function: sum of monomials K n p(x) = c k x a ik i, c k > 0 (e.g., C D0 + C L 2 ) πea k=1 i=1

30 Geometric Program: Definition Monomial Function m(x) = c n i=1 x a i i, c > 0 (e.g., 1 2 ρv 2 C L S) Posynomial Function: sum of monomials K n p(x) = c k x a ik i, c k > 0 (e.g., C D0 + C L 2 ) πea k=1 Geometric Program (GP) i=1 minimize p 0 (x) subject to p i (x) 1, i = 1,..., N p, m j (x) = 1, j = 1,..., N m with p i posynomial, m i monomial x = (x 1, x 2,..., x n ) > 0

31 variable change: y i := log x i Geometric Program: Convex Formulation

32 variable change: y i := log x i Geometric Program: Convex Formulation Monomials m(x) = c n i=1 x a i i : affine in y after log transform log m = b + a T y (b = log c)

33 variable change: y i := log x i Geometric Program: Convex Formulation Monomials m(x) = c n i=1 x a i i : affine in y after log transform log m = b + a T y (b = log c) Posynomials K k=1 c k n i=1 x a ik i : convex in y after log transform ( K ) log p = log e b k+a T k y k=1

34 variable change: y i := log x i Geometric Program: Convex Formulation Monomials m(x) = c n i=1 x a i i : affine in y after log transform log m = b + a T y (b = log c) Posynomials K k=1 c k n i=1 x a ik i : convex in y after log transform ( K ) log p = log e b k+a T k y k=1 GP in convex form ( K0 ) minimize log k=1 exp(b 0k + a0k T ( y) Ki ) subject to log k=1 exp(b ik + aiky) T 0, Gy + h = 0 i = 1,..., N p

35 Geometric Program: Convex Formulation monomials a < 0 a > 1 a == 1 0 < a < 1 a == 0

36 Geometric Program: Convex Formulation monomials monomials (log space) a < 0 a > 1 a == 1 a < 0 a > 1 a == 1 0 < a < 1 a == 0 0 < a < 1 a == 0

37 Geometric Program: Convex Formulation monomials monomials (log space) a < 0 a > 1 a == 1 a < 0 a > 1 a == 1 0 < a < 1 a == 0 0 < a < 1 a == 0 posynomials.2x x 2 +x x 4.03x 1 +.8x 0.2

38 Geometric Program: Convex Formulation monomials monomials (log space) a < 0 a > 1 a == 1 a < 0 a > 1 a == 1 0 < a < 1 a == 0 0 < a < 1 a == 0 posynomials posynomials (log space).2x x 2 +x x 4.2x x 1 +.8x x 2 +x x 4.03x 1 +.8x 0.2

39 Solution of Geometric Programs Interior-point methods u Benefits: Globally optimal solution, guaranteed Robust: no initial guesses or parameter tuning Off-the-shelf solvers c x* x*(10) Figures: [Boyd 2004]

40 Solution of Geometric Programs Interior-point methods u c x* x*(10) Benefits: Globally optimal solution, guaranteed Robust: no initial guesses or parameter tuning Off-the-shelf solvers Boyd GP benchmarks (2005) [1] dense GP: 1,000 variables; 10,000 constraints: less than 1 minute sparse GP: 10,000 variables; 1,000,000 constraints: minutes Figures: [Boyd 2004]

41 Running Example minimize A,S,C D,C L,W,W w,v subject to 1 2 ρv 2 C D S CD breakdown 1 (CDA 0) C D S CL definition 1 2W ρv 2 C L S + C D p C D + C 2 L C D πae weight breakdown 1 W 0 W + W w W wing weight S N lifta 3/2 W0 WS W w W w τ 2W stall speed 1 ρvmin 2 SC L,max

42 Running Example minimize A,S,C D,C L,W,W w,v subject to 1 2 ρv 2 C D S CD breakdown 1 (CDA 0) C D S CL definition 1 2W ρv 2 C L S + C D p C D + C 2 L C D πae weight breakdown 1 W 0 W + W w W wing weight S N lifta 3/2 W0 WS W w W w τ 2W stall speed 1 ρvmin 2 SC L,max CONSTANTS CDA CDp CLmax 1.5 Nult 3.8 Vmin 22 W e 0.95 rho 1.23 tau 0.12

43 GP Parameterization c Ā A map CDA 0 ρ C D p e W 0 N lift τ V min C L max A S C D C L W W w V 0 1/ /π e 5 1/ /2 1/2 1/

44 Globally Optimal Solution A S CD CL W 7495 Ww 2555 V OBJECTIVE VALUE:

45 Globally Optimal Solution A S CD CL W 7495 Ww 2555 V Solution is: Feasible (satisfies all constraints) OBJECTIVE VALUE:

46 Globally Optimal Solution A S CD CL W 7495 Ww 2555 V Solution is: Feasible (satisfies all constraints) Globally optimal (no other feasible solution has better objective value) OBJECTIVE VALUE:

47 Today s Talk Approach Overview The Power of Lagrange Duality GP-compatible Modeling for Aircraft Design

48 Primal problem (in convex form): Lagrange Dual of GP minimize subject to K 0 log exp(a T 0ky + b 0k ) k=1 K i log exp(a T iky + b ik ) 0, i = 1,..., m, (1) k=1

49 Primal problem (in convex form): Lagrange Dual of GP minimize subject to K 0 log exp(a T 0ky + b 0k ) k=1 K i log exp(a T iky + b ik ) 0, i = 1,..., m, (1) k=1 Lagrangian and dual function: K 0 L(y, z, λ, ν) = log exp z 0k + k=1 g(λ, ν) = inf L(y, z, λ, ν). y,z m K i λ i log exp z ik + i=1 k=1 m ν T i (A i y + b i z i ) i=0

50 Lagrange Dual of GP maximize subject to [ ] m K i ν T i b i ν ik log ν ik 1 T ν k=1 i m ν T i A i = 0 i=0 i=0 ν i 0, i = 0,..., m 1 T ν 0 = 1.

51 Lagrange Dual of GP maximize subject to [ ] m K i ν T i b i ν ik log ν ik 1 T ν k=1 i m ν T i A i = 0 i=0 i=0 ν i 0, i = 0,..., m 1 T ν 0 = 1. An equality-constrained entropy maximization

52 Lagrange Dual of GP maximize subject to [ ] m K i ν T i b i ν ik log ν ik 1 T ν k=1 i m ν T i A i = 0 i=0 i=0 ν i 0, i = 0,..., m 1 T ν 0 = 1. An equality-constrained entropy maximization (unnormalized) probability distributions ν i satisfy 1 T ν i = λ i

53 Constraint Sensitivities Consider perturbed GP: minimize subject to K 0 k=1 K i k=1 c 0k x a 0k c ik x a ik u i, i = 1,..., m.

54 Constraint Sensitivities Consider perturbed GP: minimize subject to K 0 k=1 K i k=1 c 0k x a 0k c ik x a ik u i, i = 1,..., m. Define p (u) optimal objective value of perturbed GP

55 Constraint Sensitivities Consider perturbed GP: minimize subject to K 0 k=1 K i k=1 c 0k x a 0k c ik x a ik u i, i = 1,..., m. Define p (u) optimal objective value of perturbed GP Extremely useful fact: log p (u) log u i = u=1 ( p (u) p (1) ( u i 1 ) ) = λ i u=1

56 Constraint Sensitivities Consider perturbed GP: minimize subject to K 0 k=1 K i k=1 c 0k x a 0k c ik x a ik u i, i = 1,..., m. Define p (u) optimal objective value of perturbed GP Extremely useful fact: log p (u) log u i = u=1 ( p (u) p (1) ( u i 1 ) ) = λ i u=1 Best of all, modern solvers determine λ i s for free

57 Running Example Constraint Sensitivities OBJECTIVE --> D A S CD CL W 7495 Ww 2555 V weight breakdown % CD breakdown % CL definition % induced drag model 50.00% wing weight model 47.51% landing stall speed 22.71% fuselage drag model 8.14%

58 Running Example Constraint Sensitivities OBJECTIVE --> D DV A S CD CL W 7495 Ww 2555 V weight breakdown % CD breakdown % CL definition % induced drag model 50.00% wing weight model 47.51% landing stall speed 22.71% fuselage drag model 8.14% % % % 75.00% 77.41% 0.00% 2.41%

59 Running Example Constraint Sensitivities OBJECTIVE --> D DV D/V A S CD CL W 7495 Ww 2555 V weight breakdown % CD breakdown % CL definition % induced drag model 50.00% wing weight model 47.51% landing stall speed 22.71% fuselage drag model 8.14% % % % 75.00% 77.41% 0.00% 2.41% % % 50.00% 25.00% 28.34% 56.37% 13.63%

60 Constant Sensitivities GP with fixed variables x minimize subject to K 0 k=1 K i k=1 c 0k x ā0k x a 0k c ik x āik x a ik 1, i = 1,..., m, (2) Dual variables encode sensitivity of optimum to fixed variables: log p m = ν T i ā (j) i log x j i=0

61 Constant Sensitivities GP with fixed variables x minimize subject to K 0 k=1 K i k=1 c 0k x ā0k x a 0k c ik x āik x a ik 1, i = 1,..., m, (2) Dual variables encode sensitivity of optimum to fixed variables: log p m = ν T i ā (j) i log x j i=0 Running Example SNSTVTY CONST VALUE % W % e % Vmin % CDp % Nult % tau % CLmax % rho % CDA

62 Some Useful Bounds Dual Sensitivity Analysis Start with feasible solution p (u = 1)

63 Some Useful Bounds Dual Sensitivity Analysis Start with feasible solution p (u = 1) Perturb design constraints (via u)

64 Some Useful Bounds Dual Sensitivity Analysis Start with feasible solution p (u = 1) Perturb design constraints (via u) Performance bound: log p (u) log p (1) + λ T u

65 Some Useful Bounds Dual Sensitivity Analysis Start with feasible solution p (u = 1) Perturb design constraints (via u) Performance bound: log p (u) log p (1) + λ T u An optimistic estimate

66 Some Useful Bounds Design Averaging Dual Sensitivity Analysis Start with feasible solution p (u = 1) Perturb design constraints (via u) Performance bound: Consider two designs θ 1, θ 2, with objective values p 1, p 2 log p (u) log p (1) + λ T u An optimistic estimate

67 Some Useful Bounds Design Averaging Dual Sensitivity Analysis Start with feasible solution p (u = 1) Perturb design constraints (via u) Performance bound: log p (u) log p (1) + λ T u Consider two designs θ 1, θ 2, with objective values p1, p 2 Form geometric mean design θ (i) 3 = θ (i) 1 θ(i) 2 An optimistic estimate

68 Some Useful Bounds Design Averaging Dual Sensitivity Analysis Start with feasible solution p (u = 1) Perturb design constraints (via u) Performance bound: log p (u) log p (1) + λ T u Consider two designs θ 1, θ 2, with objective values p1, p 2 Form geometric mean design θ (i) 3 = Performance bound: θ (i) 1 θ(i) 2 An optimistic estimate p 3 p 1 p 2

69 Some Useful Bounds Design Averaging Dual Sensitivity Analysis Start with feasible solution p (u = 1) Perturb design constraints (via u) Performance bound: log p (u) log p (1) + λ T u Consider two designs θ 1, θ 2, with objective values p1, p 2 Form geometric mean design θ (i) 3 = Performance bound: θ (i) 1 θ(i) 2 An optimistic estimate p 3 p 1 p 2 A pessimistic estimate

70 Feasibility Analysis When constraints cannot all be satisfied, GP solvers provide a mathematical certificate that no feasible point exists.

71 Feasibility Analysis When constraints cannot all be satisfied, GP solvers provide a mathematical certificate that no feasible point exists. In this case, look for closest feasible point: Original GP minimize p 0 (x) subject to p i (x) 1, i = 1,..., N p, m j (x) = 1, j = 1,..., N m Closest Feasible Point GP minimize s subject to p i (x) s, i = 1,..., N p, m j (x) = 1, j = 1,..., N m

72 Feasibility Analysis When constraints cannot all be satisfied, GP solvers provide a mathematical certificate that no feasible point exists. In this case, look for closest feasible point: Original GP minimize p 0 (x) subject to p i (x) 1, i = 1,..., N p, m j (x) = 1, j = 1,..., N m Closest Feasible Point GP minimize s subject to p i (x) s, i = 1,..., N p, m j (x) = 1, j = 1,..., N m The closest feasible point GP is always feasible, and its optimal point is within 100(s 1)% of satisfying the original inequality constraints.

73 Today s Talk Approach Overview The Power of Lagrange Duality GP-compatible Modeling for Aircraft Design

74 Conceptual Design Modeling Summary Fuselage Pressure Loads Fuselage Bending Loads Fuselage Weight Steady Level Flight Relations Wing Moments and Stresses Wing Weight Stability Tail Moments and Stresses Tail Weight Engine Weight Turbine Cycle Analysis Noise CG Envelope Active Gust Response Wing Profile Drag V-speeds and critical loading cases Wing Induced Drag Tail Drag Fuselage Drag Interference Drags Airfoil Shape Optimization Laminar Flow Control Compressibility Effects Propulsive Efficiency Blade Element Momentum Theory APU Sizing Hydraulic, Fuel, & Electrical System Weights Mission Breakdown and Fuel Burn Cruise Climb Loiter Performance/Endurance Takeoff Distance & 50 obstacle clearance Landing Distance Spoiler Sizing Climb Performance Engine-Out Operation Windmilling Drag Maneuverability High Lift System Sizing Control Surface Sizing Landing Gear Sizing Engine Ground Clearance Tail Strike Clearance Maintenance Costs Material Costs Manufacturability Assembly/Integration Time and Cost Fastener Count Supply Chain Dynamics

75 Conceptual Design Modeling Summary Fuselage Pressure Loads Fuselage Bending Loads Fuselage Weight Steady Level Flight Relations Wing Moments and Stresses Wing Weight Stability Tail Moments and Stresses Tail Weight Engine Weight Turbine Cycle Analysis Noise CG Envelope Active Gust Response Wing Profile Drag V-speeds and critical loading cases Wing Induced Drag Tail Drag Fuselage Drag Interference Drags Airfoil Shape Optimization Laminar Flow Control Compressibility Effects Propulsive Efficiency Blade Element Momentum Theory APU Sizing Hydraulic, Fuel, & Electrical System Weights Mission Breakdown and Fuel Burn Cruise Climb Loiter Performance/Endurance Takeoff Distance & 50 obstacle clearance Landing Distance Spoiler Sizing Climb Performance Engine-Out Operation Windmilling Drag Maneuverability High Lift System Sizing Control Surface Sizing Landing Gear Sizing Engine Ground Clearance Tail Strike Clearance Maintenance Costs Material Costs Manufacturability Assembly/Integration Time and Cost Fastener Count Supply Chain Dynamics

76 Conceptual Design Modeling Summary Fuselage Pressure Loads Fuselage Bending Loads Fuselage Weight Steady Level Flight Relations Wing Moments and Stresses Wing Weight Stability Tail Moments and Stresses Tail Weight Engine Weight Turbine Cycle Analysis Noise CG Envelope Active Gust Response Wing Profile Drag V-speeds and critical loading cases Wing Induced Drag Tail Drag Fuselage Drag Interference Drags Airfoil Shape Optimization Laminar Flow Control Compressibility Effects Propulsive Efficiency Blade Element Momentum Theory APU Sizing Hydraulic, Fuel, & Electrical System Weights Mission Breakdown and Fuel Burn Cruise Climb Loiter Performance/Endurance Takeoff Distance & 50 obstacle clearance Landing Distance Spoiler Sizing Climb Performance Engine-Out Operation Windmilling Drag Maneuverability High Lift System Sizing Control Surface Sizing Landing Gear Sizing Engine Ground Clearance Tail Strike Clearance Maintenance Costs Material Costs Manufacturability Assembly/Integration Time and Cost Fastener Count Supply Chain Dynamics

77 Conceptual Design Modeling Summary Fuselage Pressure Loads Fuselage Bending Loads Fuselage Weight Steady Level Flight Relations Wing Moments and Stresses Wing Weight Stability Tail Moments and Stresses Tail Weight Engine Weight Turbine Cycle Analysis Noise CG Envelope Active Gust Response Wing Profile Drag V-speeds and critical loading cases Wing Induced Drag Tail Drag Fuselage Drag Interference Drags Airfoil Shape Optimization Laminar Flow Control Compressibility Effects Propulsive Efficiency Blade Element Momentum Theory APU Sizing Hydraulic, Fuel, & Electrical System Weights Mission Breakdown and Fuel Burn Cruise Climb Loiter Performance/Endurance Takeoff Distance & 50 obstacle clearance Landing Distance Spoiler Sizing Climb Performance Engine-Out Operation Windmilling Drag Maneuverability High Lift System Sizing Control Surface Sizing Landing Gear Sizing Engine Ground Clearance Tail Strike Clearance Maintenance Costs Material Costs Manufacturability Assembly/Integration Time and Cost Fastener Count Supply Chain Dynamics

78 Fitting Reduced-Order GP-compatible Models inputs disciplinary analysis output(s)

79 Fitting Reduced-Order GP-compatible Models inputs disciplinary analysis output(s) GP-compatible models can approximate any log-convex data [Boyd 2007]

80 Fitting Reduced-Order GP-compatible Models inputs disciplinary analysis output(s) GP-compatible models can approximate any log-convex data [Boyd 2007] Given set of data points (x 1, y 1 ),..., (x m, y m ) R n R

81 Fitting Reduced-Order GP-compatible Models inputs disciplinary analysis output(s) input data max affine: RMS error = softmax affine: RMS error = GP-compatible models can approximate any log-convex data [Boyd 2007] Given set of data points (x 1, y 1 ),..., (x m, y m ) R n R Minimize fitting error y f (x), subject to f F log z y = log x input data max affine: RMS error = softmax affine: RMS error = scaled softmax: RMS error = implicit softmax: RMS error = log z y = log x

82 Fitting Reduced-Order GP-compatible Models inputs disciplinary analysis output(s) input data max affine: RMS error = softmax affine: RMS error = GP-compatible models can approximate any log-convex data [Boyd 2007] Given set of data points (x 1, y 1 ),..., (x m, y m ) R n R Minimize fitting error y f (x), subject to f F Several choices for F, e.g. Max-affine functions [Magnani and Boyd 2008] Softmax affine functions [Hoburg et al. 2013] Implicit posynomials [Hoburg et al. 2013] log z log z y = log x input data max affine: RMS error = softmax affine: RMS error = scaled softmax: RMS error = implicit softmax: RMS error = y = log x

83 Fitting Reduced-Order GP-compatible Models inputs disciplinary analysis output(s) input data max affine: RMS error = softmax affine: RMS error = GP-compatible models can approximate any log-convex data [Boyd 2007] Given set of data points (x 1, y 1 ),..., (x m, y m ) R n R Minimize fitting error y f (x), subject to f F Several choices for F, e.g. Max-affine functions [Magnani and Boyd 2008] Softmax affine functions [Hoburg et al. 2013] Implicit posynomials [Hoburg et al. 2013] Fitting problem solved offline using trust region Newton methods log z log z y = log x input data max affine: RMS error = softmax affine: RMS error = scaled softmax: RMS error = implicit softmax: RMS error = y = log x

84 Fitting Reduced-Order GP-compatible Models inputs disciplinary analysis output(s) input data max affine: RMS error = softmax affine: RMS error = GP-compatible models can approximate any log-convex data [Boyd 2007] Given set of data points (x 1, y 1 ),..., (x m, y m ) R n R Minimize fitting error y f (x), subject to f F Several choices for F, e.g. Max-affine functions [Magnani and Boyd 2008] Softmax affine functions [Hoburg et al. 2013] Implicit posynomials [Hoburg et al. 2013] Fitting problem solved offline using trust region Newton methods Many extensions, e.g. conservative fitting, sparse fitting log z log z y = log x input data max affine: RMS error = softmax affine: RMS error = scaled softmax: RMS error = implicit softmax: RMS error = y = log x

85 Handling non-gp-compatible models Primary limitation of GP approach: models must be log-convex

86 Handling non-gp-compatible models Primary limitation of GP approach: models must be log-convex Can handle more general models using: Nonlinear change of variables Signomial programming Sequential convex programming Similar to solving a nonlinear program, but much work offloaded as GP

87 Handling non-gp-compatible models Primary limitation of GP approach: models must be log-convex Can handle more general models using: Nonlinear change of variables Signomial programming Sequential convex programming Similar to solving a nonlinear program, but much work offloaded as GP minimize p 0 (x) subject to p i (x) 1, i = 1,..., N p, m j (x) = 1, j = 1,..., N m q(x) 1 Sketch of sequential GP approach q(x) x 35

88 Handling non-gp-compatible models Primary limitation of GP approach: models must be log-convex Can handle more general models using: Nonlinear change of variables Signomial programming Sequential convex programming Similar to solving a nonlinear program, but much work offloaded as GP minimize p 0 (x) subject to p i (x) 1, i = 1,..., N p, m j (x) = 1, j = 1,..., N m q(x) 1 Sketch of sequential GP approach q(x) x 35

89 Handling non-gp-compatible models Primary limitation of GP approach: models must be log-convex Can handle more general models using: Nonlinear change of variables Signomial programming Sequential convex programming Similar to solving a nonlinear program, but much work offloaded as GP minimize p 0 (x) subject to p i (x) 1, i = 1,..., N p, m j (x) = 1, j = 1,..., N m q(x) 1 Sketch of sequential GP approach q(x) x 35

90 Handling non-gp-compatible models Primary limitation of GP approach: models must be log-convex Can handle more general models using: Nonlinear change of variables Signomial programming Sequential convex programming Similar to solving a nonlinear program, but much work offloaded as GP minimize p 0 (x) subject to p i (x) 1, i = 1,..., N p, m j (x) = 1, j = 1,..., N m q(x) 1 Sketch of sequential GP approach q(x) x 35

91 Handling non-gp-compatible models Primary limitation of GP approach: models must be log-convex Can handle more general models using: Nonlinear change of variables Signomial programming Sequential convex programming Similar to solving a nonlinear program, but much work offloaded as GP minimize p 0 (x) subject to p i (x) 1, i = 1,..., N p, m j (x) = 1, j = 1,..., N m q(x) 1 Sketch of sequential GP approach q(x) x 35

92 Handling non-gp-compatible models Primary limitation of GP approach: models must be log-convex Can handle more general models using: Nonlinear change of variables Signomial programming Sequential convex programming Similar to solving a nonlinear program, but much work offloaded as GP minimize p 0 (x) subject to p i (x) 1, i = 1,..., N p, m j (x) = 1, j = 1,..., N m q(x) 1 Sketch of sequential GP approach q(x) x 35

93 Handling non-gp-compatible models Primary limitation of GP approach: models must be log-convex Can handle more general models using: Nonlinear change of variables Signomial programming Sequential convex programming Similar to solving a nonlinear program, but much work offloaded as GP minimize p 0 (x) subject to p i (x) 1, i = 1,..., N p, m j (x) = 1, j = 1,..., N m q(x) 1 Sketch of sequential GP approach q(x) x 35

94 Handling non-gp-compatible models Primary limitation of GP approach: models must be log-convex Can handle more general models using: Nonlinear change of variables Signomial programming Sequential convex programming Similar to solving a nonlinear program, but much work offloaded as GP minimize p 0 (x) subject to p i (x) 1, i = 1,..., N p, m j (x) = 1, j = 1,..., N m q(x) 1 Sketch of sequential GP approach q(x) x 35

95 Handling non-gp-compatible models Primary limitation of GP approach: models must be log-convex Can handle more general models using: Nonlinear change of variables Signomial programming Sequential convex programming Similar to solving a nonlinear program, but much work offloaded as GP minimize p 0 (x) subject to p i (x) 1, i = 1,..., N p, m j (x) = 1, j = 1,..., N m q(x) 1 Sketch of sequential GP approach q(x) x 35

96 Handling non-gp-compatible models Primary limitation of GP approach: models must be log-convex Can handle more general models using: Nonlinear change of variables Signomial programming Sequential convex programming Similar to solving a nonlinear program, but much work offloaded as GP minimize p 0 (x) subject to p i (x) 1, i = 1,..., N p, m j (x) = 1, j = 1,..., N m q(x) 1 Sketch of sequential GP approach q(x) x 35

97 Take-aways Importance of mathematical structure

98 Take-aways Importance of mathematical structure Key to tractability:

99 Take-aways Importance of mathematical structure Key to tractability: convexity

100 Take-aways Importance of mathematical structure Key to tractability: convexity Result: reliable and efficient optimization that scales to large problems

101 Take-aways Importance of mathematical structure Key to tractability: convexity Result: reliable and efficient optimization that scales to large problems Looking Ahead Automatic identification of convexity in data

102 Take-aways Importance of mathematical structure Key to tractability: convexity Result: reliable and efficient optimization that scales to large problems Looking Ahead Automatic identification of convexity in data Understanding reparameterizations

Take-aways Importance of mathematical structure Key to tractability: convexity Result: reliable and efficient optimization that scales to large

103 Take-aways Importance of mathematical structure Key to tractability: convexity Result: reliable and efficient optimization that scales to large problems Looking Ahead Automatic identification of convexity in data Understanding reparameterizations (Community) development of standard model libraries

104 Thank you

105 Stephen Boyd, Seung-Jean Kim, Lieven Vandenberghe, and Arash Hassibi. A tutorial on geometric programming. Optimization and Engineering, Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, New York, NY, USA, D.J. Wilde. Globally optimal design. Wiley interscience publication. Wiley, Richard James Duffin, Elmor L Peterson, and Clarence Zener. Geometric programming: theory and application. Wiley New York, Charles S Beightler and Don T Phillips. Applied geometric programming, volume 150. Wiley New York, Mosek-ApS. References Mosek version Free academic license available at Yu Nesterov and A Nemirovsky. Interior-point polynomial methods in convex programming, volume 13 of studies in applied mathematics. SIAM, Philadelphia, PA, Sanjay Mehrotra. On the implementation of a primal-dual interior point method. SIAM Journal on optimization, 2(4): , T. Bui-Thanh, K. Willcox, and O. Ghattas.

Mechanics of Flight. Warren F. Phillips. John Wiley & Sons, Inc. Professor Mechanical and Aerospace Engineering Utah State University WILEY

Mechanics of Flight. Warren F. Phillips. John Wiley & Sons, Inc. Professor Mechanical and Aerospace Engineering Utah State University WILEY Mechanics of Flight Warren F. Phillips Professor Mechanical and Aerospace Engineering Utah State University WILEY John Wiley & Sons, Inc. CONTENTS Preface Acknowledgments xi xiii 1. Overview of Aerodynamics