Some Recent Advances in Mixed-Integer Nonlinear Programming
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1 Some Recent Advances in Mixed-Integer Nonlinear Programming Andreas Wächter IBM T.J. Watson Research Center Yorktown Heights, New York SIAM Conference on Optimization 2008 Boston, MA May 12, 2008 Andreas Wächter (IBM) MINLP SIOPT / 30
2 An MINLP Research Initiative CMU-IBM research collaboration, started in 2004 The Team: CMU Pietro Belotti Lorenz T. Biegler Gérard Cornuéjols Ignacio E. Grossmann Carl D. Laird (Texas A&M) François Margot Nick Sawaya Nick Sahinidis IBM Pierre Bonami (CNRS Marseilles) Andrew R. Conn Claudia D Ambrosio (U Bologna) John J. Forrest Joao Goncalves Oktay Günlük Laszlo Ladanyi Jon Lee Andrea Lodi (U Bologna) Andreas Wächter Andreas Wächter (IBM) MINLP SIOPT / 30
3 Mixed-Integer Nonlinear Programming (MINLP) min f (x, y) s.t. c(x, y) 0 y L y y U x {0, 1} n, y R p f, c sufficiently smooth (e.g., C 2 ) Often in practice: Simplify original problem to obtain NLP by relaxing integrality conditions (rounding) MILP by approximating nonlinearities (piece-wise linear) Andreas Wächter (IBM) MINLP SIOPT / 30
4 Mixed-Integer Nonlinear Programming (MINLP) min f (x, y) s.t. c(x, y) 0 y L y y U x {0, 1} n, y R p f, c sufficiently smooth (e.g., C 2 ) and convex Often in practice: Simplify original problem to obtain NLP by relaxing integrality conditions (rounding) MILP by approximating nonlinearities (piece-wise linear) Goal: Design exact algorithms In this talk: Convex MINLP (f, c convex) Andreas Wächter (IBM) MINLP SIOPT / 30
5 The Power Of MILP MILP has been extensively explored for decades Based on branch-and-bound [Dakin (1965)] Very powerful algorithms, techniques, and codes Can solve very large problems Used heavily in practice Andreas Wächter (IBM) MINLP SIOPT / 30
6 The Power Of MILP MILP has been extensively explored for decades Based on branch-and-bound [Dakin (1965)] Very powerful algorithms, techniques, and codes Can solve very large problems Used heavily in practice How can this be used for MINLP? Use MILP solvers directly: Piece-wise linear approximation (SOS constraints) Outer approximation Andreas Wächter (IBM) MINLP SIOPT / 30
7 The Power Of MILP MILP has been extensively explored for decades Based on branch-and-bound [Dakin (1965)] Very powerful algorithms, techniques, and codes Can solve very large problems Used heavily in practice How can this be used for MINLP? Use MILP solvers directly: Piece-wise linear approximation (SOS constraints) Outer approximation In a nonlinear branch-and-bound algorithm: Try to learn from MILP tricks Andreas Wächter (IBM) MINLP SIOPT / 30
8 The Power Of MILP MILP has been extensively explored for decades Based on branch-and-bound [Dakin (1965)] Very powerful algorithms, techniques, and codes Can solve very large problems Used heavily in practice How can this be used for MINLP? Use MILP solvers directly: Piece-wise linear approximation (SOS constraints) Outer approximation In a nonlinear branch-and-bound algorithm: Try to learn from MILP tricks Andreas Wächter (IBM) MINLP SIOPT / 30
9 Outer Approximation (Duran, Grossmann [1986]) min z (linear objective) s.t. f (x, y) z c(x, y) 0 x {0, 1} n, y R p, z R Andreas Wächter (IBM) MINLP SIOPT / 30
10 Outer Approximation (Duran, Grossmann [1986]) min z (linear objective) s.t. f (x, y) z c(x, y) 0 x {0, 1} n, y R p, z R Approximate by MILP (hyperplanes) min z ( ) s.t. f (x k, y k ) T x x k y y k + f (x k, y k ) z ( ) c(x k, y k ) T x x k y y k + c(x k, y k ) 0 x {0, 1} n, y R p, z R for all (x k, y k ) T T contains linearization points Andreas Wächter (IBM) MINLP SIOPT / 30
11 Outer Approximation (Duran, Grossmann [1986]) min z (linear objective) s.t. f (x, y) z c(x, y) 0 x {0, 1} n, y R p, z R Approximate by MILP (hyperplanes) min z ( ) s.t. f (x k, y k ) T x x k y y k + f (x k, y k ) z ( ) c(x k, y k ) T x x k y y k + c(x k, y k ) 0 x {0, 1} n, y R p, z R for all (x k, y k ) T T contains linearization points augmented during algorithm Algorithm: Repeat 1 solve current MILP (x l, ỹ l ) 2 solve NLP with x l fixed y l 3 add (x l, y l ) to T Andreas Wächter (IBM) MINLP SIOPT / 30
12 Outer Approximation Discussion Original algorithm: Alternatingly solve NLPs and MILPs Finite termination Advantage: Simple to implement; uses all MILP techniques Disadvantage: Solve every MILP from scratch Andreas Wächter (IBM) MINLP SIOPT / 30
13 Outer Approximation Discussion Original algorithm: Alternatingly solve NLPs and MILPs Finite termination Advantage: Simple to implement; uses all MILP techniques Disadvantage: Solve every MILP from scratch Improvement [Quesada, Grossmann (1992)]: Build only one MILP enumeration tree Andreas Wächter (IBM) MINLP SIOPT / 30
14 Quesada-Grossmann LP LB=4 x 2 =0 x 2 =1 LP LB=5 LP LB=6 x 3 =0 x 3 =1 x 1 =0 x 1 =1 integer feasible LP UB=7 LP infeasible LP LB=8 Andreas Wächter (IBM) MINLP SIOPT / 30
15 Quesada-Grossmann LP LB=4 x 2 =0 x 2 =1 LP LB=5 LP LB=6 x 3 =0 x 3 =1 x 1 =0 x 1 =1 integer feasible NLP UB=7.5 LP infeasible LP LB=8 Andreas Wächter (IBM) MINLP SIOPT / 30
16 Outer Approximation Discussion Original algorithm: Alternatingly solve NLPs and MILPs Finite termination Advantage: Simple to implement; uses all MILP techniques Disadvantage: Need to solve every MILP from scratch Improvement [Quesada, Grossmann (1992)]: Build only one MILP enumeration tree Solve NLP for every MILP integer feasible solution Add new outer approximation cuts to current MILP Andreas Wächter (IBM) MINLP SIOPT / 30
17 Outer Approximation Discussion Original algorithm: Alternatingly solve NLPs and MILPs Finite termination Advantage: Simple to implement; uses all MILP techniques Disadvantage: Need to solve every MILP from scratch Improvement [Quesada, Grossmann (1992)]: Build only one MILP enumeration tree Solve NLP for every MILP integer feasible solution Add new outer approximation cuts to current MILP Hybrid approach [Bonami et al. (2005)] Solve NLPs also at non-integer nodes For example, solve NLP in every 10th node + Includes information about nonlinear geometry more quickly Requires solution of more NLPs Abhishek, Leyffer, Linderoth (2007) (FilMINT code): Don t solve NLP, just add linearization (Extended cutting plane) Andreas Wächter (IBM) MINLP SIOPT / 30
18 Preliminary Numerical Experiments Software implementation Bonmin (Open source software on COIN-OR) Based on other COIN-OR projects (Cbc, Clp, Cgl, Ipopt,... ) - Essential for fast development: Availability of open source NLP solvers: FilterSQP [Fletcher, Leyffer] and Ipopt Andreas Wächter (IBM) MINLP SIOPT / 30
19 Preliminary Numerical Experiments Software implementation Bonmin (Open source software on COIN-OR) Based on other COIN-OR projects (Cbc, Clp, Cgl, Ipopt,... ) - Essential for fast development: Availability of open source NLP solvers: FilterSQP [Fletcher, Leyffer] and Ipopt Test problems Representative selection of 44 convex MINLPs from - CMU/IBM library - MacMinlp [Leyffer] Difficult, but mostly solvable within 3 hour time limit Problem statistics # total vars: (289.8); # discrete vars: (93.7) # constraints: (395.4) Andreas Wächter (IBM) MINLP SIOPT / 30
20 Bonmin with Ipopt (CPU) 100 Performance Hybrid_0.1.4 QG_0.1.4 OA_ % of problems not more than x times worse than best Andreas Wächter (IBM) MINLP SIOPT / 30
21 Developer Version with FilterSQP (CPU) 100 Performance Hybrid QG OA 80 % of problems not more than x times worse than best Andreas Wächter (IBM) MINLP SIOPT / 30
22 The Success Story Of MILP In: Bixby, Fenelon, Gu, Rothberg, Wunderling (2004) Mixed-Integer Programming: A Progress Report What lead to the dramatic improvement of MILP solvers? Andreas Wächter (IBM) MINLP SIOPT / 30
23 The Success Story Of MILP In: Bixby, Fenelon, Gu, Rothberg, Wunderling (2004) Mixed-Integer Programming: A Progress Report What lead to the dramatic improvement of MILP solvers? Very efficient node solvers Andreas Wächter (IBM) MINLP SIOPT / 30
24 The Success Story Of MILP In: Bixby, Fenelon, Gu, Rothberg, Wunderling (2004) Mixed-Integer Programming: A Progress Report What lead to the dramatic improvement of MILP solvers? Very efficient node solvers Variable/node selection Primal heuristics Presolve Cutting planes Andreas Wächter (IBM) MINLP SIOPT / 30
25 The Success Story Of MILP In: Bixby, Fenelon, Gu, Rothberg, Wunderling (2004) Mixed-Integer Programming: A Progress Report What lead to the dramatic improvement of MILP solvers? Very efficient node solvers Variable/node selection Primal heuristics Presolve Cutting planes What can we learn from this for a B&B-based method for MINLP? Andreas Wächter (IBM) MINLP SIOPT / 30
26 Branch-and-bound: Variable Selection LB=4 x 2 =0 x 2 =1 LB=5 LB=6 x 3 =0 x 3 =1 x 1 =0 x 1 =1 integer feasible UB=7 infeasible LB=8 Andreas Wächter (IBM) MINLP SIOPT / 30
27 Variable Selection Some possible options: Random Most-fractional (most integer-infeasible) - used in MINLP-BB [Fletcher, Leyffer] Andreas Wächter (IBM) MINLP SIOPT / 30
28 Variable Selection Some possible options: Random Most-fractional (most integer-infeasible) - used in MINLP-BB [Fletcher, Leyffer] Strong branching [Applegate et al. (1995)] Pseudo costs [Benichou et al. (1971), Forrest et al. (1974)] - optional in SBB [GAMS] Reliability branching [Achterberg et al. (2005)] Andreas Wächter (IBM) MINLP SIOPT / 30
29 Strong Branching Q: Which variable x i should be branched on? x? =0 x? =1 Andreas Wächter (IBM) MINLP SIOPT / 30
30 Strong Branching Q: Which variable x i should be branched on? Idea: Try some candidates x i1, x i1 =0 x i1 =1 LB 0 i 1 LB 1 i 1 Andreas Wächter (IBM) MINLP SIOPT / 30
31 Strong Branching Q: Which variable x i should be branched on? Idea: Try some candidates x i1, x i2,... x i2 =0 x i2 =1 LB 0 i 2 LB 1 i 2 Andreas Wächter (IBM) MINLP SIOPT / 30
32 Strong Branching Q: Which variable x i should be branched on? Idea: Try some candidates x i1, x i2,... Choose candidate with largest LBi 0 and LBi 1 x i2 =0 x i2 =1 LB 0 i 2 LB 1 i 2 Andreas Wächter (IBM) MINLP SIOPT / 30
33 Strong Branching Q: Which variable x i should be branched on? Idea: Try some candidates x i1, x i2,... Choose candidate with largest LBi 0 and LBi 1 If candidate s child infeasible: fix variable x i2 =0 x i2 =1 infeasible LB 1 i 2 Andreas Wächter (IBM) MINLP SIOPT / 30
34 Strong Branching Q: Which variable x i should be branched on? Idea: Try some candidates x i1, x i2,... Choose candidate with largest LBi 0 and LBi 1 If candidate s child infeasible: fix variable If LB 0/1 i > UB: fix variable x i2 =0 x i2 =1 LB 0 i 2 > UB LB 1 i 2 Andreas Wächter (IBM) MINLP SIOPT / 30
35 Strong Branching Q: Which variable x i should be branched on? Idea: Try some candidates x i1, x i2,... Choose candidate with largest LBi 0 and LBi 1 If candidate s child infeasible: fix variable If LB 0/1 i > UB: fix variable Requires to solve many relaxations x i2 =0 x i2 =1 LB 0 i 2 > UB LB 1 i 2 Andreas Wächter (IBM) MINLP SIOPT / 30
36 Strong Branching Improvements Approximate node solutions For MILP: Limit the number of simplex iterations Dual simplex algorithm gives valid bounds Andreas Wächter (IBM) MINLP SIOPT / 30
37 Strong Branching Improvements Approximate node solutions For MILP: Limit the number of simplex iterations Dual simplex algorithm gives valid bounds For MINLP: Solve approximation problem LP: Linearize functions at parent solution QP: Use QP from last SQP iteration (BQPD [Fletcher]) Andreas Wächter (IBM) MINLP SIOPT / 30
38 Strong Branching Improvements Approximate node solutions For MILP: Limit the number of simplex iterations Dual simplex algorithm gives valid bounds For MINLP: Solve approximation problem LP: Linearize functions at parent solution QP: Use QP from last SQP iteration (BQPD [Fletcher]) Can use hot-starts (reuse factorization) Only one bound changes Andreas Wächter (IBM) MINLP SIOPT / 30
39 Strong Branching Improvements Pseudo costs Idea: Collect statistical data about the effect of fixing each x i : Average change in LB 0 i and LB 1 i per unit change in x i (up and down change separately) Use to estimate LB 0 i and LB 1 i of child nodes Andreas Wächter (IBM) MINLP SIOPT / 30
40 Strong Branching Improvements Pseudo costs Idea: Collect statistical data about the effect of fixing each x i : Average change in LB 0 i and LB 1 i per unit change in x i (up and down change separately) Use to estimate LB 0 i and LB 1 i of child nodes Initialize with strong branching Update each time a node has been solved Andreas Wächter (IBM) MINLP SIOPT / 30
41 Strong Branching Improvements Pseudo costs Idea: Collect statistical data about the effect of fixing each x i : Average change in LB 0 i and LB 1 i per unit change in x i (up and down change separately) Use to estimate LB 0 i and LB 1 i of child nodes Initialize with strong branching Update each time a node has been solved Reliability branching Pseudo costs, but do strong-branching on non-trusted variables Limit the number of strong-branching solves Andreas Wächter (IBM) MINLP SIOPT / 30
42 Variable Selection Comparative experiments in literature: MILP Linderoth, Savelsbergh (1999): - Pseudo costs work very well Achterberg, Koch, Martin (2005): - Reliability branching best - Most-fractional about as good as Random Andreas Wächter (IBM) MINLP SIOPT / 30
43 Variable Selection Comparative experiments in literature: MILP Linderoth, Savelsbergh (1999): - Pseudo costs work very well Achterberg, Koch, Martin (2005): - Reliability branching best - Most-fractional about as good as Random MINLP Gupta, Ravindran (1985) - Most-fractional works best Andreas Wächter (IBM) MINLP SIOPT / 30
44 Branch-And-Bound Comparison (# Nodes) Performance Random MostFra StrongNLP StrongQP PseudoNLP PseudoQP % of problems not more than x times worse than best Andreas Wächter (IBM) MINLP SIOPT / 30
45 Branch-And-Bound Comparison (CPU time) Performance Random MostFra StrongNLP StrongQP PseudoNLP PseudoQP % of problems not more than x times worse than best Andreas Wächter (IBM) MINLP SIOPT / 30
46 B&B and Hybrid Comparison 100 Performance PseudoQP Hybrid 80 % of problems not more than x times worse than best Andreas Wächter (IBM) MINLP SIOPT / 30
47 Experiments Summary Strong-branching, pseudo-costs work for nonlinear B&B Hot-started QP approximations improve performance LP approximation not efficient In these experiments: Reliability branching not helpful Andreas Wächter (IBM) MINLP SIOPT / 30
48 Experiments Summary Strong-branching, pseudo-costs work for nonlinear B&B Hot-started QP approximations improve performance LP approximation not efficient In these experiments: Reliability branching not helpful B&B competitive to OA-based Hybrid method Methods should learn from each other - e.g., use nonlinear strong-branching in Hybrid approach Best choice depends on problem instance Need to identify relevant problem characteristics Andreas Wächter (IBM) MINLP SIOPT / 30
49 Experiments Summary Strong-branching, pseudo-costs work for nonlinear B&B Hot-started QP approximations improve performance LP approximation not efficient In these experiments: Reliability branching not helpful B&B competitive to OA-based Hybrid method Methods should learn from each other - e.g., use nonlinear strong-branching in Hybrid approach Best choice depends on problem instance Need to identify relevant problem characteristics Number of nodes for solved problems: Min Max GeoMean Hybrid StrongQP Andreas Wächter (IBM) MINLP SIOPT / 30
50 Node Solvers In MILP: Very efficient implementation of dual simplex Tailored to B&B: Changes in bounds; added cuts Hot-starts (reusing factorization) extremely efficient Andreas Wächter (IBM) MINLP SIOPT / 30
51 Node Solvers In MILP: Very efficient implementation of dual simplex Tailored to B&B: Changes in bounds; added cuts Hot-starts (reusing factorization) extremely efficient In MINLP: NLP solvers now much more robust and efficient than in the past For trimloss4: Solved >2,000,000 NLPs! (105 [85] var, 64 con) Andreas Wächter (IBM) MINLP SIOPT / 30
52 Node Solvers In MILP: Very efficient implementation of dual simplex Tailored to B&B: Changes in bounds; added cuts Hot-starts (reusing factorization) extremely efficient In MINLP: NLP solvers now much more robust and efficient than in the past For trimloss4: Solved >2,000,000 NLPs! (105 [85] var, 64 con) Large-scale problems: Large-scale active-set methods? Combine interior-point and active-set methods? Andreas Wächter (IBM) MINLP SIOPT / 30
53 Node Solvers In MILP: Very efficient implementation of dual simplex Tailored to B&B: Changes in bounds; added cuts Hot-starts (reusing factorization) extremely efficient In MINLP: NLP solvers now much more robust and efficient than in the past For trimloss4: Solved >2,000,000 NLPs! (105 [85] var, 64 con) Large-scale problems: Large-scale active-set methods? Combine interior-point and active-set methods? Hot-starts possible? Andreas Wächter (IBM) MINLP SIOPT / 30
54 Node Solvers In MILP: Very efficient implementation of dual simplex Tailored to B&B: Changes in bounds; added cuts Hot-starts (reusing factorization) extremely efficient In MINLP: NLP solvers now much more robust and efficient than in the past For trimloss4: Solved >2,000,000 NLPs! (105 [85] var, 64 con) Large-scale problems: Large-scale active-set methods? Combine interior-point and active-set methods? Hot-starts possible? Storing warm-start information more memory intensive In experiments: Use optimal solution of root node Andreas Wächter (IBM) MINLP SIOPT / 30
55 Node Solvers In MILP: Very efficient implementation of dual simplex Tailored to B&B: Changes in bounds; added cuts Hot-starts (reusing factorization) extremely efficient In MINLP: NLP solvers now much more robust and efficient than in the past For trimloss4: Solved >2,000,000 NLPs! (105 [85] var, 64 con) Large-scale problems: Large-scale active-set methods? Combine interior-point and active-set methods? Hot-starts possible? Storing warm-start information more memory intensive In experiments: Use optimal solution of root node Need fast detection of infeasibility Andreas Wächter (IBM) MINLP SIOPT / 30
56 Cuts Approximate convex hull of integer-feasible points Strengthen the relaxation Andreas Wächter (IBM) MINLP SIOPT / 30
57 Cuts Approximate convex hull of integer-feasible points Strengthen the relaxation MILP: (hot topic over past 30 years) Many cut generators available (many easy to compute) Andreas Wächter (IBM) MINLP SIOPT / 30
58 Cuts Approximate convex hull of integer-feasible points Strengthen the relaxation MILP: (hot topic over past 30 years) Many cut generators available (many easy to compute) MINLP: For linear parts, can use MILP machinery directly - Hybrid method works with linear formulation - B&B: could work with linearizations Andreas Wächter (IBM) MINLP SIOPT / 30
59 Cuts Approximate convex hull of integer-feasible points Strengthen the relaxation MILP: (hot topic over past 30 years) Many cut generators available (many easy to compute) MINLP: For linear parts, can use MILP machinery directly - Hybrid method works with linear formulation - B&B: could work with linearizations Some research specific for nonlinear case: - Stubbs, Mehrotra (1999, 2002) - Atamtürk, Narayanan (2007) -... Can also use nonlinear cuts Ideally: Need access to problem representation (expression tree) Andreas Wächter (IBM) MINLP SIOPT / 30
60 Other MILP techniques Primal heuristics (quickly finding good integer feasible points) Have answer when time limit exceeded Improve upper bounds (e.g, for strong branching) Andreas Wächter (IBM) MINLP SIOPT / 30
61 Other MILP techniques Primal heuristics (quickly finding good integer feasible points) Have answer when time limit exceeded Improve upper bounds (e.g, for strong branching) MILP: A dozen generic heuristics (root node and in tree) (hot topic over last 7 years) MINLP: Preliminary work, e.g., - Nonlinear feasibility pump [Bonami et al. (2006)] Andreas Wächter (IBM) MINLP SIOPT / 30
62 Other MILP techniques Node selection In experiments: Use best-bound (node with smallest LB) Diving - Quickly find integer solution - Allows hot-starts when proceeding to child nodes Andreas Wächter (IBM) MINLP SIOPT / 30
63 Other MILP techniques Node selection In experiments: Use best-bound (node with smallest LB) Diving - Quickly find integer solution - Allows hot-starts when proceeding to child nodes Presolve (tighten and simplify formulation) At root node and in search tree MILP: Just look at coefficients of linear functions MINLP: General nonlinear functions difficult to predict - Requires access to problem representation (e.g., expression tree) Andreas Wächter (IBM) MINLP SIOPT / 30
64 What is Good Modeling? Example: Uncapacitated facility location problem min n i=1 c ix i + n m i=1 j=1 d ijy ij s.t. m j=1 y ij = 1 (i = 1,...,n) Weak : n i=1 y ij n x i (j = 1,...,m) Strong : y ij x i (i = 1,...,n; j = 1,...,m) x {0, 1} n, y R m + MILP MINLP n = 30, m = 100 nodes time nodes time weak formulation 46, strong formulation Andreas Wächter (IBM) MINLP SIOPT / 30
65 What is Good Modeling? Example: Uncapacitated facility location problem min n i=1 c ix i + n m i=1 j=1 d ijyij 2 s.t. m j=1 y ij = 1 (i = 1,...,n) Weak : n i=1 y ij n x i (j = 1,...,m) Strong : y ij x i (i = 1,...,n; j = 1,...,m) x {0, 1} n, y R m + MILP MINLP n = 30, m = 100 nodes time nodes time weak formulation 46, , strong formulation , Andreas Wächter (IBM) MINLP SIOPT / 30
66 What is Good Modeling? Example: Uncapacitated facility location problem min n i=1 c ix i + n m i=1 j=1 d ijyij 2 s.t. m j=1 y ij = 1 (i = 1,...,n) Weak : n i=1 y ij n x i (j = 1,...,m) Strong : y ij x i (i = 1,...,n; j = 1,...,m) x {0, 1} n, y R m + MILP MINLP n = 30, m = 100 nodes time nodes time weak formulation 46, , strong formulation , weak with cuts/presolve Andreas Wächter (IBM) MINLP SIOPT / 30
67 The Nonconvex Case Global optimization already very difficult Spatial branch-and-bound with convex under-estimators Incorporation of discrete variables natural Several algorithms and codes: Alpha-BB [Adjiman et al.], BARON [Sahinidis, Tawarmalani], Couenne [Belotti et al.], LaGO, [Nowak, Vigerske],... Limitation in problem size Andreas Wächter (IBM) MINLP SIOPT / 30
68 The Nonconvex Case Global optimization already very difficult Spatial branch-and-bound with convex under-estimators Incorporation of discrete variables natural Several algorithms and codes: Alpha-BB [Adjiman et al.], BARON [Sahinidis, Tawarmalani], Couenne [Belotti et al.], LaGO, [Nowak, Vigerske],... Limitation in problem size Heuristics based on convex MINLP algorithms Outer-approximation based (e.g., DICOPT [Grossmann et al.]) - use one side of equality constraints based on multipliers - allow penalized slack in OA cuts - delete violated OA cuts Andreas Wächter (IBM) MINLP SIOPT / 30
69 The Nonconvex Case Global optimization already very difficult Spatial branch-and-bound with convex under-estimators Incorporation of discrete variables natural Several algorithms and codes: Alpha-BB [Adjiman et al.], BARON [Sahinidis, Tawarmalani], Couenne [Belotti et al.], LaGO, [Nowak, Vigerske],... Limitation in problem size Heuristics based on convex MINLP algorithms Outer-approximation based (e.g., DICOPT [Grossmann et al.]) - use one side of equality constraints based on multipliers - allow penalized slack in OA cuts - delete violated OA cuts Nonlinear branch-and-bound - resolve NLPs from different starting points - do not trust lower bounds or infeasibilities Andreas Wächter (IBM) MINLP SIOPT / 30
70 Conclusions Encouraging progress New algorithms and implementations (e.g., Bonmin, FilMINT) Outer-approximation based algorithms - MILP framework with NLP solves Nonlinear branch-and-bound - Pseudo costs, QP-based strong branching Andreas Wächter (IBM) MINLP SIOPT / 30
71 Conclusions Encouraging progress New algorithms and implementations (e.g., Bonmin, FilMINT) Outer-approximation based algorithms - MILP framework with NLP solves Nonlinear branch-and-bound - Pseudo costs, QP-based strong branching Many open questions Can we repeat the success of MILP? - Further explore MILP techniques in the nonlinear case - Robust large-scale NLP solvers with hot starts? - Devise specific nonlinear techniques (e.g., cuts) Andreas Wächter (IBM) MINLP SIOPT / 30
72 Conclusions Encouraging progress New algorithms and implementations (e.g., Bonmin, FilMINT) Outer-approximation based algorithms - MILP framework with NLP solves Nonlinear branch-and-bound - Pseudo costs, QP-based strong branching Many open questions Can we repeat the success of MILP? - Further explore MILP techniques in the nonlinear case - Robust large-scale NLP solvers with hot starts? - Devise specific nonlinear techniques (e.g., cuts) Nonconvex problems Andreas Wächter (IBM) MINLP SIOPT / 30
73 Conclusions Encouraging progress New algorithms and implementations (e.g., Bonmin, FilMINT) Outer-approximation based algorithms - MILP framework with NLP solves Nonlinear branch-and-bound - Pseudo costs, QP-based strong branching Many open questions Can we repeat the success of MILP? - Further explore MILP techniques in the nonlinear case - Robust large-scale NLP solvers with hot starts? - Devise specific nonlinear techniques (e.g., cuts) Nonconvex problems Implementation - Collaboration essential (through open source?) - Accessible nonlinear problem representation - Parallel implementation Andreas Wächter (IBM) MINLP SIOPT / 30
74 Conclusions Encouraging progress New algorithms and implementations (e.g., Bonmin, FilMINT) Outer-approximation based algorithms - MILP framework with NLP solves Nonlinear branch-and-bound - Pseudo costs, QP-based strong branching Many open questions Can we repeat the success of MILP? - Further explore MILP techniques in the nonlinear case - Robust large-scale NLP solvers with hot starts? - Devise specific nonlinear techniques (e.g., cuts) Nonconvex problems Implementation - Collaboration essential (through open source?) - Accessible nonlinear problem representation - Parallel implementation Need representative real-world test problems Andreas Wächter (IBM) MINLP SIOPT / 30
75 Thank you! Andreas Wächter (IBM) MINLP SIOPT / 30
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