Pattern Search for Mixed Variable Optimization Problems p.1/35
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1 Pattern Search for Mixed Variable Optimization Problems Mark A. Abramson (Air Force Institute of Technology) Charles Audet (École Polytechnique de Montréal) John Dennis, Jr. (Rice University) The views expressed here are those of the authors and do not reflect the official policy or position of the USAF, Department of Defense, US government, or our other sponsors: Boeing, ExxonMobil, CSRI, LASCI, NSF. Pattern Search for Mixed Variable Optimization Problems p.1/35
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3 Outline Mixed variable problem motivation and formulation Pattern Search for Mixed Variable Optimization Problems p.2/35
4 Outline Mixed variable problem motivation and formulation GPS for linearly constrained MVP problems Pattern Search for Mixed Variable Optimization Problems p.2/35
5 Outline Mixed variable problem motivation and formulation GPS for linearly constrained MVP problems Filter GPS for general constrained MVP problems Pattern Search for Mixed Variable Optimization Problems p.2/35
6 Outline Mixed variable problem motivation and formulation GPS for linearly constrained MVP problems Filter GPS for general constrained MVP problems Results for thermal insulation system design Pattern Search for Mixed Variable Optimization Problems p.2/35
7 Heat intercept insulation system L x i+1 i+1 x i i T i 1 T n+1 = T H T 0 = T C Pattern Search for Mixed Variable Optimization Problems p.3/35
8 Heat intercept insulation system L x i+1 i+1 x i i T i 1 min power(n, I, x, T ) T n+1 = T H T 0 = T C Pattern Search for Mixed Variable Optimization Problems p.3/35
9 Heat intercept insulation system L x i+1 i+1 x i i T i 1 T n+1 = T H T 0 = T C min power(n, I, x, T ) subject to n {1, 2,..., n max }, I I n+1 T i 1 T i T i+1, i = 1, 2,..., n n+1 x i = L, x i 0, i = 1, 2,..., n + 1 i=1 Pattern Search for Mixed Variable Optimization Problems p.3/35
10 The general MVP problem min x X f(x) subject to C(x) 0, Pattern Search for Mixed Variable Optimization Problems p.4/35
11 The general MVP problem where min x X f(x) subject to C(x) 0, x = (x c, x d ) R nc Z nd X = X c X d, where X c (x d ) = {x c R nc : l(x d ) A(x d )x c u(x d )} Pattern Search for Mixed Variable Optimization Problems p.4/35
12 The general MVP problem where min x X f(x) subject to C(x) 0, x = (x c, x d ) R nc Z nd X = X c X d, where X c (x d ) = {x c R nc : l(x d ) A(x d )x c u(x d )} f and C = (c 1, c 2,..., c p ) may be discontinuous, extended valued, costly Pattern Search for Mixed Variable Optimization Problems p.4/35
13 The general MVP problem where min x X f(x) subject to C(x) 0, x = (x c, x d ) R nc Z nd X = X c X d, where X c (x d ) = {x c R nc : l(x d ) A(x d )x c u(x d )} f and C = (c 1, c 2,..., c p ) may be discontinuous, extended valued, costly Pattern Search for Mixed Variable Optimization Problems p.4/35
14 Choices other than pattern search Some methods that come to mind are: MINLP methods: cannot handle categorical variables Pattern Search for Mixed Variable Optimization Problems p.5/35
15 Choices other than pattern search Some methods that come to mind are: MINLP methods: cannot handle categorical variables Search heuristics: huge numbers of evaluations and very limited convergence theory Pattern Search for Mixed Variable Optimization Problems p.5/35
16 Choices other than pattern search Some methods that come to mind are: MINLP methods: cannot handle categorical variables Search heuristics: huge numbers of evaluations and very limited convergence theory Simulated annealing Pattern Search for Mixed Variable Optimization Problems p.5/35
17 Choices other than pattern search Some methods that come to mind are: MINLP methods: cannot handle categorical variables Search heuristics: huge numbers of evaluations and very limited convergence theory Simulated annealing Tabu search Pattern Search for Mixed Variable Optimization Problems p.5/35
18 Choices other than pattern search Some methods that come to mind are: MINLP methods: cannot handle categorical variables Search heuristics: huge numbers of evaluations and very limited convergence theory Simulated annealing Tabu search Evolutionary algorithms Pattern Search for Mixed Variable Optimization Problems p.5/35
19 Choices other than pattern search Some methods that come to mind are: MINLP methods: cannot handle categorical variables Search heuristics: huge numbers of evaluations and very limited convergence theory Simulated annealing Tabu search Evolutionary algorithms Other methods: SQP/direct search with 1 categorical variable Pattern Search for Mixed Variable Optimization Problems p.5/35
20 Generalized pattern searches INITIALIZATION of directions and step size For k = 1, 2,... SEARCH a finite set of mesh points End Pattern Search for Mixed Variable Optimization Problems p.6/35
21 Generalized pattern searches INITIALIZATION of directions and step size For k = 1, 2,... SEARCH a finite set of mesh points End POLL neighboring mesh points Pattern Search for Mixed Variable Optimization Problems p.6/35
22 Generalized pattern searches INITIALIZATION of directions and step size For k = 1, 2,... SEARCH a finite set of mesh points POLL neighboring mesh points UPDATE parameters: End Pattern Search for Mixed Variable Optimization Problems p.6/35
23 Generalized pattern searches INITIALIZATION of directions and step size For k = 1, 2,... SEARCH a finite set of mesh points End POLL neighboring mesh points UPDATE parameters: Success: Accept new iterate Pattern Search for Mixed Variable Optimization Problems p.6/35
24 Generalized pattern searches INITIALIZATION of directions and step size For k = 1, 2,... End SEARCH a finite set of mesh points POLL neighboring mesh points UPDATE parameters: Success: Accept new iterate Failure: Refine mesh Pattern Search for Mixed Variable Optimization Problems p.6/35
25 Details of kth POLL step Mesh: M k = {p k + k Dz : z Z D + }, Poll set: P k = {p k + k d : d D k D}, Pattern Search for Mixed Variable Optimization Problems p.7/35
26 Details of kth POLL step Mesh: M k = {p k + k Dz : z Z D + }, Poll set: P k = {p k + k d : d D k D}, where pk is the current poll center k > 0 is the mesh size parameter D k, D are positive spanning sets Pattern Search for Mixed Variable Optimization Problems p.7/35
27 Details of kth POLL step Mesh: M k = {p k + k Dz : z Z D + }, Poll set: P k = {p k + k d : d D k D}, where pk is the current poll center k > 0 is the mesh size parameter D k, D are positive spanning sets Examples: D = [I, I] D = [I, e] One of these directions should be a descent direction. Derivative information can reduce poll set to a singleton Pattern Search for Mixed Variable Optimization Problems p.7/35
28 Details of kth POLL step Mesh: M k = {p k + k Dz : z Z D + }, Poll set: P k = {p k + k d : d D k D}, where pk is the current poll center k > 0 is the mesh size parameter D k, D are positive spanning sets Examples: D = [I, I] One of these directions should be a descent direction. Derivative information can reduce poll set to a singleton D = [I, e] p k Pattern Search for Mixed Variable Optimization Problems p.7/35
29 Definition of local optimality x = (x c, x d ) X is a local minimizer of f with respect to neighbors N (x) X if ɛ > 0 such that f(x) f(v) v X ( B(y c, ɛ) y d). y N (x) Pattern Search for Mixed Variable Optimization Problems p.8/35
30 Definition of local optimality x = (x c, x d ) X is a local minimizer of f with respect to neighbors N (x) X if ɛ > 0 such that f(x) f(v) v X ( B(y c, ɛ) y d). y N (x) X c x d x Pattern Search for Mixed Variable Optimization Problems p.8/35
31 Definition of local optimality x = (x c, x d ) X is a local minimizer of f with respect to neighbors N (x) X if ɛ > 0 such that f(x) f(v) v X ( B(y c, ɛ) y d). y N (x) X c x d x Pattern Search for Mixed Variable Optimization Problems p.8/35
32 Definition of local optimality x = (x c, x d ) X is a local minimizer of f with respect to neighbors N (x) X if ɛ > 0 such that f(x) f(v) v X ( B(y c, ɛ) y d). y N (x) y X c x d x Pattern Search for Mixed Variable Optimization Problems p.8/35
33 Definition of local optimality x = (x c, x d ) X is a local minimizer of f with respect to neighbors N (x) X if ɛ > 0 such that f(x) f(v) v X ( B(y c, ɛ) y d). y N (x) X c y d y X c x d x Pattern Search for Mixed Variable Optimization Problems p.8/35
34 Heatshield discrete neighbors Replace any single insulator with a different type Pattern Search for Mixed Variable Optimization Problems p.9/35
35 Heatshield discrete neighbors Replace any single insulator with a different type Remove any intercept with its left insulator, and increase the thickness of its right insulator to absorb the deficit Pattern Search for Mixed Variable Optimization Problems p.9/35
36 Heatshield discrete neighbors Replace any single insulator with a different type Remove any intercept with its left insulator, and increase the thickness of its right insulator to absorb the deficit Add an intercept at any position: Pattern Search for Mixed Variable Optimization Problems p.9/35
37 Heatshield discrete neighbors Replace any single insulator with a different type Remove any intercept with its left insulator, and increase the thickness of its right insulator to absorb the deficit Add an intercept at any position: The existing insulator is divided (rounded to the mesh) Pattern Search for Mixed Variable Optimization Problems p.9/35
38 Heatshield discrete neighbors Replace any single insulator with a different type Remove any intercept with its left insulator, and increase the thickness of its right insulator to absorb the deficit Add an intercept at any position: The existing insulator is divided (rounded to the mesh) The cooling temperature is set to the average of the two intercepts adjacent to it (rounded to the mesh) Pattern Search for Mixed Variable Optimization Problems p.9/35
39 Construction of the poll set x d P k = {u, v, w} x d k u x k v w x c 1 x c 2 Pattern Search for Mixed Variable Optimization Problems p.10/35
40 Construction of the poll set x d y1 0 P k = {u, v, w} N (x k ) = {x k, y 0 1, y 0 2} x d k u v x k w y2 0 y 0 1 N (x k ) satisfies f(x k ) < f(y 0 1) < f(x k ) + ξ x c 1 x c 2 Pattern Search for Mixed Variable Optimization Problems p.11/35
41 Construction of the poll set x d k x d a u b y 1 1 y1 0 c v x k w y2 0 P k = {u, v, w} N (x k ) = {x k, y 0 1, y 0 2} X k = {y 1 1} {a, b, c} y 0 1 N (x k ) satisfies f(x k ) < f(y 0 1) < f(x k ) + ξ x c 1 Poll Set: P k N (x k ) X k x c 2 Pattern Search for Mixed Variable Optimization Problems p.12/35
42 Filter GPS for nonlinear constraints f f F k (h I k,f k I) F k h max h h(x) = C(x) + 2 Pattern Search for Mixed Variable Optimization Problems p.13/35
43 Filter GPS for nonlinear constraints f f F k (h I k,f k I) F k Poll center is either best feasible point or least infeasible point. h max h h(x) = C(x) + 2 Pattern Search for Mixed Variable Optimization Problems p.13/35
44 Filter GPS for nonlinear constraints f f F k (h I k,f k I) F k Poll center is either best feasible point or least infeasible point. For each trial point x, h(x) and f(x) are plotted on the bi-loss map. h max h h(x) = C(x) + 2 Pattern Search for Mixed Variable Optimization Problems p.13/35
45 Filter GPS for nonlinear constraints f f F k (h I k,f k I) F k Poll center is either best feasible point or least infeasible point. For each trial point x, h(x) and f(x) are plotted on the bi-loss map. h max h(x) = C(x) + 2 h If x is unfiltered, it is added to the filter; otherwise, the mesh is refined. Pattern Search for Mixed Variable Optimization Problems p.13/35
46 Construction of the poll set p d k x d a u b y 1 1 y1 0 c v p k w y2 0 P k = {u, v, w} N (p k ) = {p k, y 0 1, y 0 2} X k = {y 1 1} {a, b, c} y 0 1 N (p k ) satisfies the extended poll criteria x c 1 Poll Set: P k N (p k ) X k x c 2 Pattern Search for Mixed Variable Optimization Problems p.14/35
47 Local filter for extended polling f F k +ξf k f f F k * (h I k,f k I) F k h I k +ξh k h max h Main Filter Pattern Search for Mixed Variable Optimization Problems p.15/35
48 Local filter for extended polling f f F k +ξf k f F k * (h I k,f k I) F k f F L k h I k +ξh k h max h h I k +ξh k h Main Filter Local Filter Pattern Search for Mixed Variable Optimization Problems p.16/35
49 Filter GPS algorithm for MVP INITIALIZATION: Set 0, ξ > 0, and populate filter For k = 1, 2,..., do Update poll center p k and extended poll triggers ξ f k, ξh k Pattern Search for Mixed Variable Optimization Problems p.17/35
50 Filter GPS algorithm for MVP INITIALIZATION: Set 0, ξ > 0, and populate filter For k = 1, 2,..., do Update poll center p k and extended poll triggers ξ f k, ξh k Compute incumbent values f F k, f I k, hi k Pattern Search for Mixed Variable Optimization Problems p.17/35
51 Filter GPS algorithm for MVP INITIALIZATION: Set 0, ξ > 0, and populate filter For k = 1, 2,..., do Update poll center p k and extended poll triggers ξ f k, ξh k Compute incumbent values f F k, f I k, hi k While trial points are filtered do: Pattern Search for Mixed Variable Optimization Problems p.17/35
52 Filter GPS algorithm for MVP INITIALIZATION: Set 0, ξ > 0, and populate filter For k = 1, 2,..., do Update poll center p k and extended poll triggers ξ f k, ξh k Compute incumbent values f F k, f I k, hi k While trial points are filtered do: SEARCH: Any finite strategy on the mesh Pattern Search for Mixed Variable Optimization Problems p.17/35
53 Filter GPS algorithm for MVP INITIALIZATION: Set 0, ξ > 0, and populate filter For k = 1, 2,..., do Update poll center p k and extended poll triggers ξ f k, ξh k Compute incumbent values f F k, f I k, hi k While trial points are filtered do: SEARCH: Any finite strategy on the mesh POLL: Evaluate points in P k N (p k ) Pattern Search for Mixed Variable Optimization Problems p.17/35
54 Filter GPS algorithm for MVP INITIALIZATION: Set 0, ξ > 0, and populate filter For k = 1, 2,..., do Update poll center p k and extended poll triggers ξ f k, ξh k Compute incumbent values f F k, f I k, hi k While trial points are filtered do: SEARCH: Any finite strategy on the mesh POLL: Evaluate points in P k N (p k ) EXTENDED POLL: Evaluate points in X k (ξ f k, ξh k ) Pattern Search for Mixed Variable Optimization Problems p.17/35
55 Filter GPS algorithm for MVP INITIALIZATION: Set 0, ξ > 0, and populate filter For k = 1, 2,..., do Update poll center p k and extended poll triggers ξ f k, ξh k Compute incumbent values f F k, f I k, hi k While trial points are filtered do: SEARCH: Any finite strategy on the mesh POLL: Evaluate points in P k N (p k ) EXTENDED POLL: Evaluate points in X k (ξ f k, ξh k ) Update: Pattern Search for Mixed Variable Optimization Problems p.17/35
56 Filter GPS algorithm for MVP INITIALIZATION: Set 0, ξ > 0, and populate filter For k = 1, 2,..., do Update poll center p k and extended poll triggers ξ f k, ξh k Compute incumbent values f F k, f I k, hi k While trial points are filtered do: SEARCH: Any finite strategy on the mesh POLL: Evaluate points in P k N (p k ) EXTENDED POLL: Evaluate points in X k (ξ f k, ξh k ) Update: If (found), set k+1 k and update filter Pattern Search for Mixed Variable Optimization Problems p.17/35
57 Filter GPS algorithm for MVP INITIALIZATION: Set 0, ξ > 0, and populate filter For k = 1, 2,..., do Update poll center p k and extended poll triggers ξ f k, ξh k Compute incumbent values f F k, f I k, hi k While trial points are filtered do: SEARCH: Any finite strategy on the mesh POLL: Evaluate points in P k N (p k ) EXTENDED POLL: Evaluate points in X k (ξ f k, ξh k ) Update: If (found), set k+1 k and update filter If (not found), set k+1 < k Pattern Search for Mixed Variable Optimization Problems p.17/35
58 Convergence theory assumptions All iterates lie in a compact set Pattern Search for Mixed Variable Optimization Problems p.18/35
59 Convergence theory assumptions All iterates lie in a compact set The linear constraint matrix A is rational Pattern Search for Mixed Variable Optimization Problems p.18/35
60 Convergence theory assumptions All iterates lie in a compact set The linear constraint matrix A is rational The mesh directions conform to the geometry of X c Pattern Search for Mixed Variable Optimization Problems p.18/35
61 Convergence theory assumptions All iterates lie in a compact set The linear constraint matrix A is rational The mesh directions conform to the geometry of X c All discrete neighbors lie on the current mesh Pattern Search for Mixed Variable Optimization Problems p.18/35
62 Convergence theory assumptions All iterates lie in a compact set The linear constraint matrix A is rational The mesh directions conform to the geometry of X c All discrete neighbors lie on the current mesh The set-valued neighborhood function N : X 2 X satisfies a notion of continuity. Pattern Search for Mixed Variable Optimization Problems p.18/35
63 Limit points of the algorithm subsequence K such that lim k K k = 0, with limit points: 1. ˆp = lim k K p k, where p k {p F k, pi k } 2. ŷ = lim k K y k, where y k N (p k ) and ŷ N (ˆp). 3. ẑ = lim k K z k, where z k are extended poll endpoints. y k p k Pattern Search for Mixed Variable Optimization Problems p.19/35
64 Limit points of the algorithm subsequence K such that lim k K k = 0, with limit points: 1. ˆp = lim k K p k, where p k {p F k, pi k } 2. ŷ = lim k K y k, where y k N (p k ) and ŷ N (ˆp). 3. ẑ = lim k K z k, where z k are extended poll endpoints. y k z k p k Pattern Search for Mixed Variable Optimization Problems p.20/35
65 Limit points of the algorithm subsequence K such that lim k K k = 0, with limit points: 1. ˆp = lim k K p k, where p k {p F k, pi k } 2. ŷ = lim k K y k, where y k N (p k ) and ŷ N (ˆp). 3. ẑ = lim k K z k, where z k are extended poll endpoints. y k z k p k 1 p k p k+1 ˆp Pattern Search for Mixed Variable Optimization Problems p.21/35
66 Limit points of the algorithm subsequence K such that lim k K k = 0, with limit points: 1. ˆp = lim k K p k, where p k {p F k, pi k } 2. ŷ = lim k K y k, where y k N (p k ) and ŷ N (ˆp). 3. ẑ = lim k K z k, where z k are extended poll endpoints. y k+1 y y k k 1 ŷ z k p k 1 p k p k+1 ˆp Pattern Search for Mixed Variable Optimization Problems p.22/35
67 Limit points of the algorithm subsequence K such that lim k K k = 0, with limit points: 1. ˆp = lim k K p k, where p k {p F k, pi k } 2. ŷ = lim k K y k, where y k N (p k ) and ŷ N (ˆp) 3. ẑ = lim k K z k, where z k are extended poll endpoints y k+1 y y k k 1 ŷ z z k+1 k z k 1 ẑ p k 1 p k p k+1 ˆp Pattern Search for Mixed Variable Optimization Problems p.23/35
68 Filter convergence results Let D(ˆp) be the set of polling directions used i.o. h continuous at ˆp and ŷ h(ˆp) h(ŷ) f continuous at ˆp and ŷ and p k = p F k i. o. f(ˆp) f(ŷ) h Lipschitz near ˆp h (ˆp; (d, 0)) 0 d D(ˆp) f Lipschitz near ˆp and p k = p F k i. o. f (ˆp; (d, 0)) 0 d D(ˆp) h strictly differentiable at ˆp and h(ˆp) = 0 Similar results hold for certain ẑ Pattern Search for Mixed Variable Optimization Problems p.24/35
69 Filter convergence results f strictly differentiable at ˆp and p k = p F k i. o. c f(ˆp) Cd. Similar results hold for certain ẑ C d C d f with respect to the continuous variables Pattern Search for Mixed Variable Optimization Problems p.25/35
70 Filter convergence results f strictly differentiable at ˆp and p k = p F k i. o. c f(ˆp) Cd. Similar results hold for certain ẑ C d C d f with respect to the continuous variables Pattern Search for Mixed Variable Optimization Problems p.26/35
71 Heat intercept insulation system L x i+1 i+1 x i i T i 1 T n+1 = T H T 0 = T C Pattern Search for Mixed Variable Optimization Problems p.27/35
72 Heat intercept insulation system L x i+1 i+1 x i i T i 1 min power(n, I, x, T ) T n+1 = T H T 0 = T C Pattern Search for Mixed Variable Optimization Problems p.27/35
73 Heat intercept insulation system L x i+1 i+1 x i i T i 1 T n+1 = T H T 0 = T C min power(n, I, x, T ) subject to n {1, 2,..., n max }, I I n+1 T i 1 T i T i+1, i = 1, 2,..., n n+1 x i = L, x i 0, i = 1, 2,..., n + 1 i=1 Pattern Search for Mixed Variable Optimization Problems p.27/35
74 Previous heat shield studies Hilal & Boom: 1-3 intercepts, single insulator type, constant cross-sectional areas Pattern Search for Mixed Variable Optimization Problems p.28/35
75 Previous heat shield studies Hilal & Boom: 1-3 intercepts, single insulator type, constant cross-sectional areas Hilal & Eyssa: 1-3 intercepts, single insulator type, variable cross-sectional areas Pattern Search for Mixed Variable Optimization Problems p.28/35
76 Previous heat shield studies Hilal & Boom: 1-3 intercepts, single insulator type, constant cross-sectional areas Hilal & Eyssa: 1-3 intercepts, single insulator type, variable cross-sectional areas Kokkolaras, Audet & Dennis: Variable number of intercepts, multiple insulator types, constant cross-sectional areas Pattern Search for Mixed Variable Optimization Problems p.28/35
77 Previous heat shield studies Hilal & Boom: 1-3 intercepts, single insulator type, constant cross-sectional areas Hilal & Eyssa: 1-3 intercepts, single insulator type, variable cross-sectional areas Kokkolaras, Audet & Dennis: Variable number of intercepts, multiple insulator types, constant cross-sectional areas Current work: Variable number of intercepts, multiple insulator types, variable cross-sectional areas, load-bearing nonlinear constraints Pattern Search for Mixed Variable Optimization Problems p.28/35
78 Nomenclature k(t ; I i ) = Thermal conductivity function for insulator i P i = Power applied to intercept i C i = Thermodynamic cycle coefficient at intercept i q i = Heat flow from intercept i to i 1 A i = Cross-sectional area of insulator i σ(t ; I i ) = maximum allowable stress function e(t ; I i ) = unit thermal expansion function ρ(i i ) = density of the insulator i material F = load (force) to be placed on the system m max = maximum allowable mass of the insulators δ = maximum allowable % thermal contraction Pattern Search for Mixed Variable Optimization Problems p.29/35
79 Heat shield objective and nonlinear constraints Minimize power: n P i = i=1 By Fourier s law: q i = A i x i n i=1 T i ( ) T H C i 1 (q i q i 1 ) T i T i 1 k(t ; I i )dt, i = 1, 2,..., n + 1 Pattern Search for Mixed Variable Optimization Problems p.30/35
80 Heat shield objective and nonlinear constraints Minimize power: n P i = i=1 By Fourier s law: q i = A i x i Stress: n i=1 T i ( ) T H C i 1 (q i q i 1 ) T i T i 1 k(t ; I i )dt, i = 1, 2,..., n + 1 F A i min{σ(t ; I i ) : T i 1 T T i } (binding) Pattern Search for Mixed Variable Optimization Problems p.30/35
81 Heat shield objective and nonlinear constraints Minimize power: n P i = i=1 By Fourier s law: q i = A i x i Stress: Mass: n i=1 T i ( ) T H C i 1 (q i q i 1 ) T i T i 1 k(t ; I i )dt, i = 1, 2,..., n + 1 F A i min{σ(t ; I i ) : T i 1 T T i } n+1 i=1 ρ(i i )A i x i m max (binding) Pattern Search for Mixed Variable Optimization Problems p.30/35
82 Heat shield objective and nonlinear constraints Minimize power: n P i = i=1 By Fourier s law: q i = A i x i Stress: Mass: n i=1 T i ( ) T H C i 1 (q i q i 1 ) T i T i 1 k(t ; I i )dt, i = 1, 2,..., n + 1 F A i min{σ(t ; I i ) : T i 1 T T i } n+1 i=1 Contraction: ρ(i i )A i x i m max n+1 i=1 T i T i 1 e(t ; I i )k(t ; I i )dt T i T i 1 k(t ; I i )dt ( x i L (binding) ) δ 100 Pattern Search for Mixed Variable Optimization Problems p.30/35
83 Heat shield implementation Materials: Nylon Teflon 6063-T5 Aluminum Fiberglass Epoxy (normal cloth) 304 Stainless Steel Fiberglass Epoxy (plane cloth) 1020 Low Carbon Steel Pattern Search for Mixed Variable Optimization Problems p.31/35
84 Heat shield implementation Materials: Nylon Teflon 6063-T5 Aluminum Fiberglass Epoxy (normal cloth) 304 Stainless Steel Fiberglass Epoxy (plane cloth) 1020 Low Carbon Steel Material Data from Lookup Tables or Graphs: Thermal Conductivity Unit Thermal Contraction Maximum Allowable Stress (Tensile Yield Strength) Pattern Search for Mixed Variable Optimization Problems p.31/35
85 Heat shield implementation Materials: Nylon Teflon 6063-T5 Aluminum Fiberglass Epoxy (normal cloth) 304 Stainless Steel Fiberglass Epoxy (plane cloth) 1020 Low Carbon Steel Material Data from Lookup Tables or Graphs: Thermal Conductivity Unit Thermal Contraction Maximum Allowable Stress (Tensile Yield Strength) Interpolation/Integration: Cubic splines, Simpson s rule Pattern Search for Mixed Variable Optimization Problems p.31/35
86 Heat shield implementation Materials: Nylon Teflon 6063-T5 Aluminum Fiberglass Epoxy (normal cloth) 304 Stainless Steel Fiberglass Epoxy (plane cloth) 1020 Low Carbon Steel Material Data from Lookup Tables or Graphs: Thermal Conductivity Unit Thermal Contraction Maximum Allowable Stress (Tensile Yield Strength) Interpolation/Integration: Cubic splines, Simpson s rule Search/Poll: No Search, Poll around p F k Pattern Search for Mixed Variable Optimization Problems p.31/35
87 Heat shield computational results Source P L A [ W cm] Insulators Hilal & Boom 68.6 E p E p E p Kokkolaras et al NNNNNNNEEET Kokkolaras rerun NNNNNNT EET T Hilal & Eyssa 53.2 E p E p E p with stress constraint EEEEEEEEEEE with all constraints EEEEEEEEEEE Pattern Search for Mixed Variable Optimization Problems p.32/35
88 Heat shield computational results Source P L A [ W cm] Insulators Hilal & Boom 68.6 E p E p E p Kokkolaras et al NNNNNNNEEET Kokkolaras rerun NNNNNNT EET T Hilal & Eyssa 53.2 E p E p E p with stress constraint EEEEEEEEEEE with all constraints EEEEEEEEEEE Parameters: T H = 300 K, T C = 4.2 K, L = 100 cm, n max = 10 F = 250 kn, m max = 10 kg, δ = 5% Pattern Search for Mixed Variable Optimization Problems p.32/35
89 Heat shield computational results Source P L A [ W cm] Insulators Hilal & Boom 68.6 E p E p E p Kokkolaras et al NNNNNNNEEET Kokkolaras rerun NNNNNNT EET T Hilal & Eyssa 53.2 E p E p E p with stress constraint EEEEEEEEEEE with all constraints EEEEEEEEEEE Parameters: T H = 300 K, T C = 4.2 K, L = 100 cm, n max = 10 F = 250 kn, m max = 10 kg, δ = 5% Termination: k Pattern Search for Mixed Variable Optimization Problems p.32/35
90 Profile of heat shield run 12 x 104 FMGPS Performance 10 Power Required Number of function evaluations Power Required Number of function evaluations Pattern Search for Mixed Variable Optimization Problems p.33/35
91 Heat shield filter progress 2500 Filter after 150 evaluations 600 Zoom of filter on left f f Filter after 200 evaluations 600 Zoom of filter on left f f Filter after 500 evaluations 600 Zoom of filter on left f f h h Pattern Search for Mixed Variable Optimization Problems p.34/35
92 Conclusions We have found an effective method for mixed variable problems... BUT... Pattern Search for Mixed Variable Optimization Problems p.35/35
93 Conclusions We have found an effective method for mixed variable problems... BUT... It requires serious thinking by the user as to what constitutes an acceptable solution for each problem Pattern Search for Mixed Variable Optimization Problems p.35/35
94 Conclusions We have found an effective method for mixed variable problems... BUT... It requires serious thinking by the user as to what constitutes an acceptable solution for each problem It is not easy to incorporate surrogates with categorical variables Pattern Search for Mixed Variable Optimization Problems p.35/35
95 Conclusions We have found an effective method for mixed variable problems... BUT... It requires serious thinking by the user as to what constitutes an acceptable solution for each problem It is not easy to incorporate surrogates with categorical variables It is unlikely to be effective for many categorical variables Pattern Search for Mixed Variable Optimization Problems p.35/35
96 Conclusions We have found an effective method for mixed variable problems... BUT... It requires serious thinking by the user as to what constitutes an acceptable solution for each problem It is not easy to incorporate surrogates with categorical variables It is unlikely to be effective for many categorical variables We have analyzed the method, BUT.., better results depend on the filter method for continuous variables Pattern Search for Mixed Variable Optimization Problems p.35/35
97 Conclusions We have found an effective method for mixed variable problems... BUT... It requires serious thinking by the user as to what constitutes an acceptable solution for each problem It is not easy to incorporate surrogates with categorical variables It is unlikely to be effective for many categorical variables We have analyzed the method, BUT.., better results depend on the filter method for continuous variables Pattern Search for Mixed Variable Optimization Problems p.35/35
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