Mixed Integer Nonlinear Programs Theory, Algorithms & Applications

Transcription

1 Mixed Integer Nonlinear Programs Theory, Algorithms & Applications Mohit Tawarmalani Department of Mechanical and Industrial Engineering Nick Sahinidis Department of Chemical Engineering University Of Illinois Urbana-Champaign Acknowledgment: NSF CAREER award DMII

2 Problem Formulation (P) min f(x, y) Objective Function s.t. g(x, y) 0 Constraints x Z p Integrality Restrictions y R n Continuous Variables Challenges: Multimodal Objective Objective Integrality f(x) Feasible Space Nonconvex Constraints f(x) Convex Objective Projected Objective Nonconvex Feasible Space

3 The Pooling Problem f 11 f 21 3% S $6 1% S $16 Pool x 11 x 21 Blend X 2.5% S $9 X 100 f 12 2% S $10 x 22 x 12 Blend Y 1.5% S $15 Y 200 min cost {}}{ 6f f f 12 X-revenue {}}{ 9(x 11 + x 21 ) Y -revenue {}}{ 15(x 12 + x 22 ) s.t. q = 3f 11 + f 21 x 11 + x 12 Concentration Balance f 11 + f 21 = x 11 + x 12 f 12 = x 21 + x 22 Mass balance qx 11 +2x 21 x 11 + x qx 12 +2x 22 x 12 + x Quality Requirements x 11 + x x 12 + x Demands

4 Molecular Design Problems (Example: Refrigerant Design) UNIVERSE C Cl CH 3 N Cl CH CH 2 2 F CH N Cl CH 2 Cl F F Combinatorial Choice (Source of Discreteness) Freon (CCl 2 F 2 ) F F C Cl Cl Property Prediction (Source of Nonlinearity) Satisfies Thermodynamic requirements?

5 Discrete Location Problems Restaurants in Edmonton p-choice Utility = w j location: j customer: i U ij x j k U d i ikx k }{{} Utility Ratio U ij x j d i w j Utility of location j for customer i decision variable for locating facility j demand by customer i preferential weight of site j

6 Motivation for an Algorithm Applications: Combinatorial Chemistry and Molecular Biology Utility Maximization (Risk Minimization) Parameter Estimation (Design of Experiments) Engineering Design (Process Synthesis, CAD) Layout Design (Area Restrictions) Theoretical Challenges Important Subclasses: Mixed Integer Linear Programming Continuous Nonlinear Programming Nonlinear 0-1 Programming Search for Globality (Deterministic): Either optimality assured or a gap estimate Rigorous termination criteria Verifying local optimality is NP-hard (Murty and Kabadi 1987)

7 Deterministic Approaches Branch and Bound Bound the problem over successively refined partitions Convexification Outer-approximate with increasingly tighter convex programs Decomposition Project out some variables (move them to a subproblem) Our approach Branch and Bound + Convexification

8 Branch and Bound Algorithm Objective P Objective P U R R L Variable L Variable a. Lower Bounding b. Upper Bounding Objective P U L R1 R2 R R1 R R2 Fathom Variable Subdivide c. Domain Subdivision d. Search Tree

9 Illustrative Example 4 f = 5 x 2 x 1 x 1 x 2 f = Separable Reformulation : x 1 x 2 = 1 2 min x 1 x 2 s.t. x 1 x x x 2 4 { (x1 + x 2 ) 2 x 2 1 2} x2 Reformulation min x 1 x 2 s.t. (x 1 + x 2 ) 2 x 2 1 x x x 2 4 Relaxation (not best) 0 x 1 6 6x 1 36 x 2 1 min x 1 x 2 s.t. (x 1 + x 2 ) 2 6x 1 4x x x 2 4 Relaxation Solution: x 1 =6,x 2 =0.89, L = 6.89 Local Search (MINOS): x 1 =6,x 2 =0.67, L = 6.67 Relaxation Solution provides good starting point for local search

10 Challenges Standard Global Optimization Converges Slowly Algorithmic Issues: Tight Relaxations Domain Reduction Finiteness Issues Implementation Issues: Existing NLP solvers often unreliable General purpose (automatic) software Enable solution of industrially relevant problems

11 Factorable Functions (McCormick, 1976) Definition: The factorable functions are recursive compositions of sums and products of functions of single variables. Example: f(x, y) = exp(xy + z ln w)z 3 f { }} { x 5 {}}{ x 3 x 6 ( {}}{{}}{ exp( xy + z ln w }{{}}{{} ) z 3 ) 0.5 x 1 } {{ x 2 } } x 4 {{ } x 7 x 1 = xy x 2 =ln(w) x 3 = zx 2 x 4 = x 1 + x 3 x 5 =exp(x 4 ) x 6 = z 3 x 7 = x 5 x 6 f = x 7

12 Challenges Via Example: y log(x) z 1 x 8 1 y 4 z y log(x) 1 x 8 1 y 4 z yl x l x log(x) 1 x 8 1 y 4 Convex Relaxation (McCormick style) z log(8)y +4 l r 4log(8) z 1 x 8 1 y 4 z l r l r log(8) (x 1) 7 Q: In general do we get the convex envelope? NO Q: In this case? YES (Convex Extensions Wed-9:00am) Q: Simplification? (log 8/7) max {7y +4x 32,x 1} - Non-differentiability -log(x) occurs elsewhere? (common subexpressions) Q: What if y = x? (Note:x log x is convex)

13 Ratio: Convex Extensions x 4 2 y 4 z x/y 2 x 4 2 y 4 Ratio: x/y Convex Extensions yield Convex Envelope y 4 2 x x/y -Envelope Factorable Relaxation is Weak 4 y p z p x L ( x U x x U x L ) 2 ) 2 ( x x y p (z z p ) y(z z p ) x U L x U x { L y p max y L xu x,y x } xl x U yu xl x U x { L } y p min y U xu x x xl,y x U yl xl x U x L z z p,z p y x 4 2 x/y -Factorable 4 z 0.5 +x/4 +y/8 z 2+x/2 y 2 x 4 2 y 4

14 RANGE REDUCTION Relaxed Value Function z U L x L x U x If a variable goes to its upper bound at the relaxed problem solution, this variable s lower bound can be improved

15 PROBING Q. A. What if a variable does not go to a bound? Use probing: temporarily fix variable at a bound. Relaxed Value Function z L U z U x L x* x U x

16 ILLUSTRATIVE EXAMPLE x 2 min f = x 1 x 2 s.t. x 1 x x x 2 4 (P) Reformulation: min f = x 1 x 2 s.t. x 2 3 x 2 1 x 2 8 x 3 = x 1 + x 2 0 x x x f = Relaxation: f = 6.67 min x 1 x 2 s.t. x 2 3 6x 1 4x 2 8 x 3 = x 1 + x 2 0 x x x 3 10 x 1 (R) Solution of R : x 1 = 6, x 2 = 0.89,L = 6.89,λ 1 = 0.2 Local Solution of P with MINOS : U = 6.67 Range reduction : x L 1 x U 1 (U L)/ λ 1 = 4.86 Probing (Solve R with x 2 0) : L = 6,λ 2 = 1 x L 2 = 0.67 Update R with x , x Solution is : L = 6.67 Proof of globality with NO Branching!!

17 Branch And Reduce Optimization Navigator BARON Preprocessor Data Organizer Range Reduction Solver Links I/O Handler Debugging Facilities USER MODULE Relaxation Local Search Range Reduction Heuristics Sparse Matrix Utilities Heuristics Core Module: expandable, application-independent Application Modules: factorable, linear multiplicative, bilinear, concave, fixed-charge, integer, fractional programming solve convex relaxations using CPLEX, OSL, MINOS, SNOPT, SDPA

18 Factorable NLP Module Formulation: min f(x, y) a g i (x, y) b i =1,...,m x L j x j x U j j =1,...,n y L j y j y U j j =1,...,p x j real y j integer Factorable Functions: f and g are recursive sums, products and ratios of the following terms: exponential, exp(x) logarithmic, log(x) monomial, x a Example: (Factorable Constraint) ( ) 1 x2 y 0.3 z x 2 p +exp xy 100 p 2 y

19 Pooling Problems in Literature Algorithm Foulds 92 Ben-Tal 94 GOP 96 BARON Computer CDC 4340 HP9000/730 RS6000/43P Linpack > Tolerance ** 10 6 Problem N tot T tot N tot N tot T tot N tot T tot Haverly Haverly Haverly Foulds Foulds Foulds Foulds Ben-Tal Ben-Tal Example Example Example Example * Blank indicates problem not reported or not solved ** 0.05% for Haverly 1, 2, 3, 0.05% for Ben-Tal 4 and 1% for Ben-Tal 5

20 Automotive Refrigerant Design (Joback and Stephanopoulos, 1984) Condenser T cnd Expansion Valve Tavg = (T cnd + Tevp)/2 Compressor Tevp Evaporator H ve max C pla s.t. H ve 18.4 C pla 32.2 P vpe 1.4 Property Prediction Constraints Structure Feasibility Constraints Nonnegativity Constraints Integrality Requirements High enthalpy of vaporization H ve reduces the amount of refrigerant Lower liquid heat capacity C pla reduces amount of vapor generated in expansion valve

21 Functional Groups Considered Acyclic Groups Cyclic Groups Halogen Groups Oxygen Groups Nitrogen Groups Sulfur Groups CH 3 r CH 2 r F OH NH 2 SH CH 2 r r > CH r Cl O > NH S > CH r > CH r Br r O r r r > NH r S r > C < r r > C < r r I > CO > N =CH 2 r > C < r r r r > CO =N =CH > C < r r CHO r =N r =C< r =CH r COOH CN =C= r =C< r r COO NO 2 CH r =C< r =O C =C< r r Number of Groups = 44 Maximum Selection Size = 15 Candidates = 39, 895, 566, 894, 524

22 Property Prediction Constraints T b = N n i T bi i=1 β = T br ln(T br ) T 6 br T b T c = N i=1 n it ci ( N i=1 n it ci ) 2 1 P c = ( N i=1 n ia i N i=1 n ip ci ) ( 2 N N ) C p0a = n i C p0ai n i C p0bi i=1 i=1 ( N ) + n i C p0ci T 2 avg T br = T b T c T avgr = T avg T c T cndr = T cnd T c T evpr = T evp T c i=1 ( N ) + n i C p0di T 3 avg i=1 α = ln T 6 br ( Pc ) T avg T br ln(T br ) ω = α β C pla = 1 { [ C p0a ω T avgr ( (1 T avgr) 1/ )]} T avgr 1 T avgr N H vb = n i H vbi i=1 ( 1 Tevp /T c H ve = H vb 1 T b /T c h = T br ln(p c /1.013) 1 T br G = h k = h/g (1 + T br) (3 + T br )(1 T br ) 2 ) 0.38 ln P vpcr = G T cndr [ 1 T 2 cndr + k(3 + T cndr)(1 T cndr ) 3] ln P vper = G T evpr [ 1 T 2 evpr + k(3 + T evpr)(1 T evpr ) 3] n i integer

23 Structure Feasibility Constraints N n i 2 i=1 Y A n i N max Y A A i A Y C n i N max Y C C i C Y M n i N max Y M M i M Y A + Y C 1 Y M Y A + Y C 3Y R n i N max Y R R i R N N n i b i 2( n i 1); i=1 i=1 ( N N )( N ) n i b i n i n i 1 i=1 n i 1if i=1 i=1 n i 1and i SD i S/D i D/S n i 1 n i 1if n i 1and n i 1 i ST i S/T i T /S n i 1if n i 1and n i 1 i SSR i S/SR i SR/S n i 1if n i 1and n i 1 i SDR i S/DR i DR/S n i 1if n i 1and n i 1 i DSR i D/SR i SR/D n i =2Z S ; n i =2Z SR n i =2Z B ; i B i S n i =2Z D i D n i =2Z DR i DR n i =2Z T i T n i (2 b i )=2m i N n i n j (b j 1) + 2 i=1 i SR j =1,...,N n i n i S ai if n i 0 i O, i H i H single-bonded n i n i D ai if n i 0 i O, i H i H double-bonded n i n i T ai if n i 0 i O, i H i H triple-bonded i H n i (S ai + D ai + T ai ) i O n i 2( i H n i 1)

24 Molecular Structures Molecular Structure H ve C pla Molecular Structure H ve C pla ClNO (Cl )( N =)(= O) IF (I )( F) O 3 ( r O r ) FNO (F )( N =)(= O) CHClO (Cl )( CH =)(= O) FSH (F )( SH) CH 3 Cl (CH 3 )( Cl) C 2 HClO 2 (= C <)(= CH )( Cl)(= O) All 44 feasible molecules identified CH 2 FN (F )( N =)(= CH 2 ) ClF (Cl )( F) CClFO (O =)(= C <)( Cl)( F) ClFO (Cl )( O )( F) C 3 H 4 (CH 3 )( C )( CH) C 2 F 2 ( C ) 2 ( F) C 3 H 6 O ( CH 3 ) 2 (= C <)(= O) C 3 H 3 FO (O =)(= CH ) 3 ( F) CF 2 NHO (O =)(= C <)( F ) 2 (> NH) C 2 H 6 ( CH 3 ) CHFO 3 (O =)(= CH )( O ) 2 ( F) C 3 H 3 F 3 ( r > CH r ) 3 ( F) For CCl 2 F 2, H ve /C pla 0.57

25 Located Restaurants IBM SP/2 Single Processor Time: 5.5 hours Relaxations take 95% time Iterations: 79 Nodes in Memory: 9

26 Miscellaneous Problems Problem m n N tot N mem T tot Bilinear Problem Design of Water Pump Alkylation Process Design Design of Insulated Tank HENS Chemical Equilibrium Pooling Problem Quadratic Bilinear Nonlinear Equality Economies of Scale Two-Stage Process Systems MINLP, Process Synthesis MINLP, Process Synthesis MINLP, Process Synthesis HENS Reinforced Concrete Beam Quadratic Constraints QCQP Reactor Network Design Two-Stage Process System Problem m n N tot N mem T tot Concave QP Biconvex Program Linearly Constrained QP Linear Multiplicative Concave QP Nonlinear Fixed Charges QCQP Reliability Fixture Design Fixture Design Molecular Design Bilinearly Constrained Bilinearly Constrained HENS MINLP Parameter Estimation Pressure Vessel Design Shekel Function Truss Design Reliability Design of Experiments

27 Convexify Reduce Isolate x* BRANCH-AND-REDUCE Facility Location Molecular design Supply chain operations