Global Optimization of Truss. Structure Design INFORMS J. N. Hooker. Tallys Yunes. Slide 1

Transcription

1 Slde 1 Global Optmzaton of Truss Structure Desgn J. N. Hooker Tallys Yunes INFORMS 2010

2 Truss Structure Desgn Select sze of each bar (possbly zero) to support the load whle mnmzng weght. Bar szes are dscrete. 10-bar cantlever truss 0 deg. freedom 2 deg. freedom Slde 2 Total 8 degrees of freedom Load

3 Truss Structure Desgn Notaton v = elongaton of bar s = force along bar h = length of bar d = node dsplacement A = cross-sectonal area of bar p = load along d.f. Slde 3

4 Truss Structure Desgn nonlnear mn } Mnmze total weght s. t. cos θ s = p, all cos θ d = v, all E Av = s, all h v v v, all d d d, all \/ h A ( A = A ) k k } Equlbrum } Compatblty } Hooke s law } Elongaton bounds } Dsplacement bounds } Logcal dsuncton Area must be one of several dscrete values A k Slde 4 Constrants can be mposed for multple loadng condtons

5 Truss Structure Desgn Introducng new varables lnearzes the problem but makes t much larger. MILP model Slde 5 mn k k k s. t. cos θ s = p, all E h cos θ d = v, all k k A v = s, all k k k v v v, all d d d, all k h A y y = 1, all k 0-1 varables ndcatng sze of bar Elongaton varable dsaggregated by bar sze Hooke s law becomes lnear

6 Soluton Approach Solve wth SIMPL, an ntegrated solver for optmzaton. SIMPL ntegrates MILP, constrant programmng, global optmzaton n a unfed approach. Slde 6

7 SIMPL Integraton Prncples Low-level ntegraton wth hgh-level modelng. Slde 7

8 SIMPL Integraton Prncples Low-level ntegraton wth hgh-level modelng. Succnct modelng wth meta-constrants. Model communcates problem structure to the solver. Slde 8

9 SIMPL Integraton Prncples Low-level ntegraton wth hgh-level modelng. Succnct modelng wth meta-constrants. Model communcates problem structure to the solver. General search-nfer-relax soluton algorthm. Enumerate problem restrctons. Branchng or logc-based Benders. Underlyng search/nference and search/relaxaton dualtes. Slde 9

10 SIMPL Integraton Prncples Low-level ntegraton wth hgh-level modelng. Succnct modelng wth meta-constrants. Model communcates problem structure to the solver. General search-nfer-relax soluton algorthm. Enumerate problem restrctons. Branchng or logc-based Benders. Underlyng search/nference and search/relaxaton dualtes Constrant-based control. Flterng, relaxaton, branchng. Slde 10

11 Classcal soluton methods CP solver Search: Branchng Inference: Fllterng Relaxaton: Doman store MILP solver Search: Branchng Inference: Cuttng planes, presolve, reduced cost varable fxng Relaxaton: LP Benders Search: Enumerate subproblems. Inference: Benders cuts Relaxaton: Master problem Slde 11

12 Classcal soluton methods Global optmzaton Search: Enumerate boxes Inference: Doman reducton, dual-based varable boundng Relaxaton: Convexfcaton SAT Search: Branchng Inference: Conflct clauses Relaxaton: Same as restrcton Local search Search: Enumerate neghborhoods. Inference: Tabu lst, etc. Relaxaton: Same as restrcton Slde 12

13 Truss Structure Desgn Integrated approach Use the orgnal model (don t ntroduce new varables) Branch by splttng the range of areas A No 0-1 or nteger varables! Generate a lnear quas-relaxaton, whch s much smaller than MILP model. Use logc cuts. Orgnal hand-coded method: Bollapragada, Ghattas, and JNH Slde 13

14 Branchng Dscrete bar szes A Value n soluton of current relaxaton

15 Branchng Slde 15 Dscrete bar szes A Value n soluton of current relaxaton A Branch by splttng nterval

16 Branchng Dscrete bar szes A Value n soluton of current relaxaton A Branch by splttng nterval Slde 16 Soluton of next relaxaton lkely to be at an endpont. Ths branchng ntellgence unavalable n 0-1 model.

17 Quas-relaxaton Gven problem { f x } mn ( ) x S { f x } mn ( ) The problem s a quas-relaxaton f x S for any x S, there s an x S wth f (x ) f (x). A quas-relaxaton need not be a vald relaxaton. But ts optmal value s a vald lower bound on the optmal value of the orgnal problem. Slde 17

18 Quas-relaxaton Consder the problem mn f ( x ) g ( x, y ) 0, all n x R, y dscrete Slde 18

19 Quas-relaxaton Consder the problem mn f ( x ) g ( x, y ) 0, all n x R, y dscrete Each g s a vector of functons Slde 19

20 Quas-relaxaton Consder the problem mn f ( x ) g ( x, y ) 0, all n x R, y dscrete Each g s a vector of functons Slde 20 Each y s a scalar varable

21 Quas-relaxaton Consder the problem mn f ( x ) Each g s a vector of functons g ( x, y ) 0, all n x R, y dscrete Each y s a scalar varable Relaxng the problem by makng y contnuous could result n a nonconvex problem. Slde 21

22 Quas-relaxaton Consder the problem mn f ( x ) Each g s a vector of functons g ( x, y ) 0, all n x R, y dscrete Each y s a scalar varable Relaxng the problem by makng y contnuous could result n a nonconvex problem. But suppose the problem becomes convex when each y s fxed to a constant. Slde 22

23 Quas-relaxaton Consder the problem mn f ( x ) Each g s a vector of functons g ( x, y ) 0, all n x R, y dscrete Each y s a scalar varable Relaxng the problem by makng y contnuous could result n a nonconvex problem. But suppose the problem becomes convex when each y s fxed to a constant. Then we may be able to wrte a convex quas-relaxaton. Slde 23

24 Quas-relaxaton Consder the problem mn f ( x ) Theorem (JNH 2005) g ( x, y ) 0, all n x R, y dscrete If each g (x,y) s semhomogeneous n x and concave n scalar y, then the followng s a quas-relaxaton: mn f ( x ) 1 L 2 U g ( x, y ) + g ( x, y ) 0, all 1 α x x α x 2 (1 α ) x x (1 α ) x x = x + x α 1 2 [0,1] Slde 24

25 Quas-relaxaton Consder the problem mn f ( x ) Theorem (JNH 2005) g ( x, y ) 0, all n x R, y dscrete If each g (x,y) s semhomogeneous n x and concave n scalar y, then the followng s a quas-relaxaton: Slde 25 mn f ( x ) 1 L 2 U g ( x, y ) + g ( x, y ) 0, all 1 α x x α x 2 (1 α ) x x (1 α ) x x = x + x α 1 2 [0,1] g ( α x, y ) α g ( x, y ) for all x, y and α [0,1] g (0, y ) = 0 for all y

26 Quas-relaxaton Consder the problem mn f ( x ) Theorem (JNH 2005) g ( x, y ) 0, all n x R, y dscrete If each g (x,y) s semhomogeneous n x and concave n scalar y, then the followng s a quas-relaxaton: mn f ( x ) 1 L 2 U g ( x, y ) + g ( x, y ) 0, all 1 α x x α x 2 (1 α ) x x (1 α ) x x = x + x α 1 2 [0,1] Slde 26 Bounds on y

27 Quas-relaxaton Consder the problem mn f ( x ) Theorem (JNH 2005) g ( x, y ) 0, all n x R, y dscrete If each g (x,y) s semhomogeneous n x and concave n scalar y, then the followng s a quas-relaxaton: Slde 27 mn f ( x ) 1 L 2 U g ( x, y ) + g ( x, y ) 0, all 1 α x x α x 2 (1 α ) x x (1 α ) x x = x + x α 1 2 [0,1] Bounds on x

28 Quas-relaxaton Why? Take any feasble soluton x, y Slde 28 mn f ( x ) 1 L 2 U g ( x, y ) + g ( x, y ) 0, all 1 α x x α x 2 (1 α ) x x (1 α ) x x = x + x α 1 2 [0,1]

29 Quas-relaxaton Why? Take any feasble soluton Choose α so that Set x, y y = α y + (1 α ) y α α = = x x, x (1 ) x 1 2 Slde 29 mn f ( x ) 1 L 2 U g ( x, y ) + g ( x, y ) 0, all 1 α x x α x 2 (1 α ) x x (1 α ) x x = x + x α 1 2 [0,1]

30 Quas-relaxaton Why? Take any feasble soluton Choose α so that Set x, y y = α y + (1 α ) y α α = = x x, x (1 ) x 1 2 Then for each component of g we have mn f ( x ) 1 α x x α x ( ) ( ) α α + = + 2 (1 α ) x x (1 α ) x g x y g x y g x y g x y 1 2 (, L ) (, U ), L (1 ), U 1 L 2 U g ( x, y ) + g ( x, y ) 0, all x = x + x L U = α g ( x, y ) + (1 α ) g ( x, y ) g ( x, α y + (1 α ) y ) = g ( x, y ) α 1 2 [0,1] homogenety concavty Slde 30

31 Quas-relaxaton mn f ( x ) g ( x, y ) 0, all n x R, y dscrete So we have a feasble soluton of the quas-relaxaton wth value that s less than or equal to (n fact equal to) that of the orgnal problem. satsfed, by constructon Slde 31 mn f ( x ) 1 L 2 U g ( x, y ) + g ( x, y ) 0, all 1 α x x α x 2 (1 α ) x x (1 α ) x x = x + x α 1 2 [0,1] satsfed, by above argument

32 Quas-relaxaton mn f ( x ) g ( x, y ) 0, all n x R, y dscrete E Av = s h has the form g(x,y) = 0 wth g semhomogenous n x and concave (lnear) n y because we can wrte t E Av s h wth x = (v,s ), y = A. = 0 Slde 32

33 Truss Structure Desgn So we have a quas-relaxaton of the truss problem: Slde 33 mn h [ A y + A (1 y )] s. t. cos θ s = p, all cos θ d = v + v, all 0 1 E ( A v 0 + A v 1 ) = s, all h v y v v y, all 0 1 v (1 y ) v v (1 y ), all d d d, all 0 y 1, all Hooke s law s lnearzed Elongaton bounds splt nto 2 sets of bounds

34 Truss Structure Desgn Logc cuts v 0 and v 1 must have same sgn n a feasble soluton. If not, we branch by addng logc cuts v 0, v 1 0, v 0, v 1 0 Slde 34

35 Truss Structure Desgn In general, we can have a metaconstrant to represent the semhomogeneous constrant g(x,y) 0. Ths tells the solver to generate a quas-relaxaton. Slde 35

36 Truss Structure Desgn In general, we can have a metaconstrant to represent the semhomogeneous constrant g(x,y) 0. Ths tells the solver to generate a quas-relaxaton. Snce a blnear constrant xy = α s always semhomogeneous, we wll use a blnear metaconstrant wth a quas-relaxaton opton. Slde 36

37 Truss Structure Desgn SIMPL model Recognzed as lnear systems Slde 37

38 Truss Structure Desgn SIMPL model Recognzed as blnear system Slde 38

39 Truss Structure Desgn SIMPL model Generate quasrelaxaton for semhomogenous functon Slde 39

40 Truss Structure Desgn SIMPL model Branch frst on volated logc cuts for quasrelaxaton Slde 40

41 Truss Structure Desgn SIMPL model Then branch on A n-doman constrant. Volated when A s not one of the dscrete bar szes. Take upper branch frst. Slde 41

42 Truss Structure Desgn 10-bar cantlever truss Slde 42 Load

43 Truss Structure Desgn Computatonal results (seconds) Hand-coded ntegrated method No. bars Loads BARON CPLEX 11 Hand coded SIMPL * * Slde 43 *plus dsplacement bounds

44 Truss Structure Desgn 25-bar problem Slde 44

45 Truss Structure Desgn 72-bar problem Slde 45

46 Truss Structure Desgn Computatonal results (seconds) Hand-coded ntegrated method No. bars Loads BARON CPLEX 11 Hand coded SIMPL , , , > 24 hr* > 24 hr* > 24 hr* > 24 hr** > 24 hr*** * no feasble soluton found ** best feasble soluton has cost 32,748 Slde 46 *** best feasble soluton has cost 32,700

47 Current Verson of SIMPL To download: Clck the lnk to SIMPL on John Hooker s webste. See readme fle for complete nstructons. Download executable and assocated fles Operatonal on GNU/Lnux only Requres subsdary solvers CPLEX Eclpse (free download) Download problem nstances Includng all reported n ths talk. Slde 47