Blackbox optimization with the MADS algorithm and the NOMAD software

Transcription

1 Blackbox optimization with the MADS algorithm and the NOMAD software Sébastien Le Digabel, Charles Audet, Viviane Rochon-Montplaisir, Christophe Tribes IREQ, NOMAD: Blackbox Optimization 1/41

2 Presentation outline Research team Blackbox optimization The MADS algorithm The NOMAD software package Conclusion NOMAD: Blackbox Optimization 2/41

3 Research team Blackbox optimization The MADS algorithm The NOMAD software package Conclusion NOMAD: Blackbox Optimization 3/41

4 Research team Blackbox optimization MADS NOMAD Conclusion Research team Professors Research associates Students Recently graduated NOMAD: Blackbox Optimization 4/41

5 Research team Blackbox optimization The MADS algorithm The NOMAD software package Conclusion NOMAD: Blackbox Optimization 5/41

6 Derivative-free & blackbox optimization Algorithms DFO / BBO Applications Theory NOMAD: Blackbox Optimization 6/41

7 Derivative-free & blackbox optimization Algorithms Tsunami detection buoys Fontan surgery Wing design MDO GMON positioning Hyperparameters optimization Applications Nonsmooth calculus Generalized gradient Clarkes derivatives Hypertangent cone Optimality conditions DFO / BBO Hidden constraints Stochasticity Categorical variables Biobjective problems Surrogates Theory NOMAD: Blackbox Optimization 6/41

8 Derivative-free & blackbox optimization Algorithms Tsunami detection buoys Fontan surgery Wing design MDO GMON positioning Hyperparameters optimization Applications Nonsmooth calculus Generalized gradient Clarkes derivatives Hypertangent cone Optimality conditions DFO / BBO Hidden constraints Stochasticity Categorical variables Biobjective problems Surrogates Theory NOMAD: Blackbox Optimization 6/41

9 Blackbox optimization We consider min x Ω f(x) where the evaluations of f and the functions defining Ω are the result of a computer simulation (a blackbox). x R n f(x) x Ω? NOMAD: Blackbox Optimization 7/41

10 Blackbox optimization We consider min x Ω f(x) where the evaluations of f and the functions defining Ω are the result of a computer simulation (a blackbox). x R n f(x) x Ω? Each call to the simulation may be expensive. The simulation can fail. Sometimes f(x) f(x). Derivatives are not available and cannot be approximated. NOMAD: Blackbox Optimization 7/41

11 Tsunamis detection buoys (2004) Source: ms1.html NOMAD: Blackbox Optimization 8/41

12 Variables The variables x may be a mix of Reals. Integers, binaries, granular, Categorical. Example: The number of heat shields is a variable. Types of materials are also variables. NOMAD: Blackbox Optimization 9/41

13 Constraints Domain: Ω = {x X : c j (x) 0, j J} R n X corresponds to unrelaxable constraints Cannot be violated; Example: x > 0 when log x is used inside the simulation. NOMAD: Blackbox Optimization 10/41

14 Constraints Domain: Ω = {x X : c j (x) 0, j J} R n X corresponds to unrelaxable constraints c j (x) 0: Relaxable and quantifiable constraints May be violated at intermediate designs; c j (x) measures the violation; Example: cost budget. NOMAD: Blackbox Optimization 10/41

15 Constraints Domain: Ω = {x X : c j (x) 0, j J} R n X corresponds to unrelaxable constraints c j (x) 0: Relaxable and quantifiable constraints Hidden constraints when the simulation fails, even for points in Ω; Example: Segmentation fault Bus error ERROR 42 DIVISION BY ZERO NOMAD: Blackbox Optimization 10/41

16 Constraints Domain: Ω = {x X : c j (x) 0, j J} R n X corresponds to unrelaxable constraints c j (x) 0: Relaxable and quantifiable constraints Hidden constraints Example: Chemical process: 7 variables, 4 constraints. The ASPEN software fails on 43% of the calls. NOMAD: Blackbox Optimization 10/41

17 Research team Blackbox optimization The MADS algorithm The NOMAD software package Conclusion NOMAD: Blackbox Optimization 11/41

18 General framework f(x) x Ω? Algorithm x NOMAD: Blackbox Optimization 12/41

19 Mesh Adaptive Direct Search (MADS) [Audet and Dennis, Jr., 2006]. Iterative algorithm that evaluates the blackbox at some trial points on a spatial discretization called the mesh. One iteration = search and poll. The search allows trial points generated anywhere on the mesh. The poll consists in generating a list of trial points constructed from poll directions. These directions grow dense. At the end of the iteration, the mesh size is reduced if no new success point is found. NOMAD: Blackbox Optimization 13/41

20 [0] Initializations (x 0, m 0 ) [1] Iteration k [1.1] Search select a finite number of mesh points evaluate candidates opportunistically [1.2] Poll (if the Search failed) construct poll set P k = {x k + m k d : d D k} sort(p k ) evaluate candidates opportunistically [2] Updates if success x k+1 success point increase m k else x k+1 x k decrease m k k k + 1, stop or go to [1] From the MADS original paper [Audet and Dennis, Jr., 2006]. NOMAD: Blackbox Optimization 14/41

21 Poll illustration (successive fails and mesh shrink) m k = 1 p 1 x k p 3 p 2 trial points={p 1, p 2, p 3 } NOMAD: Blackbox Optimization 15/41

22 Poll illustration (successive fails and mesh shrink) p 1 m k = 1 x k p 3 m k+1 = 1/4 x k p 4 p 5 p 6 p 2 trial points={p 1, p 2, p 3 } = {p 4, p 5, p 6 } NOMAD: Blackbox Optimization 15/41

23 Poll illustration (successive fails and mesh shrink) m k = 1 m k+1 = 1/4 m k+2 = 1/16 p 1 x k p 3 x k p 4 p 5 p 6 p 7 x k p 8 p 9 p 2 trial points={p 1, p 2, p 3 } = {p 4, p 5, p 6 } = {p 7, p 8, p 9 } NOMAD: Blackbox Optimization 15/41

24 Hierarchical convergence Iterates are bounded x k ˆx mesh min. k 0 NOMAD: Blackbox Optimization 16/41

25 Hierarchical convergence Iterates are bounded Lower semicontinous x k ˆx mesh min. k 0 f(ˆx) f(x k ) NOMAD: Blackbox Optimization 16/41

26 Hierarchical convergence Iterates are bounded Lower semicontinous Directional Lipschitz Custódio et Vicente 2012 x k ˆx mesh min. k 0 f(ˆx) f(x k ) f (ˆx; d) 0 NOMAD: Blackbox Optimization 16/41

27 Hierarchical convergence Iterates are bounded Lower semicontinous Directional Lipschitz Custódio et Vicente 2012 x k ˆx mesh min. k 0 f(ˆx) f(x k ) f (ˆx; d) 0 NOMAD: Blackbox Optimization 16/41

28 Hierarchical convergence Iterates are bounded Lower semicontinous Directional Lipschitz Custódio et Vicente 2012 Lipschitz x k ˆx mesh min. k 0 f(ˆx) f(x k ) f (ˆx; d) 0 f (ˆx; d) 0 NOMAD: Blackbox Optimization 16/41

29 Hierarchical convergence Iterates are bounded Lower semicontinous x k ˆx Directional Lipschitz mesh min. Custódio et Vicente 2012 k 0 Lipschitz f(ˆx) f(x k ) Regular f (ˆx; d) 0 f (ˆx; d) 0 f (ˆx; d) 0 NOMAD: Blackbox Optimization 16/41

30 Hierarchical convergence Iterates are bounded Lower semicontinous x k ˆx Directional Lipschitz mesh min. Custódio et Vicente 2012 k 0 Lipschitz f(ˆx) f(x k ) Regular f (ˆx; d) 0 Strictly f (ˆx; d) 0 differentiable f (ˆx; d) 0 KKT NOMAD: Blackbox Optimization 16/41

31 MADS extensions Constraints handling with the Progressive Barrier technique [Audet and Dennis, Jr., 2009]. Surrogates [Talgorn et al., 2015]. Categorical variables [Abramson, 2004]. Granular and discrete variables [Audet et al., 2018]. Global optimization [Audet et al., 2008a]. Parallelism [Le Digabel et al., 2010, Audet et al., 2008b]. Multiobjective optimization [Audet et al., 2008c]. Sensitivity analysis [Audet et al., 2012]. NOMAD: Blackbox Optimization 17/41

32 Surrogates Static surrogate: A cheaper model defined a priori by the user. It is used as a blackbox. Typically a simplified physics model. Variable precision is not yet considered. Dynamic surrogate: Model managed by the algorithm, based on past evaluations. It can be periodically updated. NOMAD: Blackbox Optimization 18/41

33 General framework with surrogates f(x) x Ω? Algorithm x R n NOMAD: Blackbox Optimization 19/41

34 General framework with surrogates x R n Surrogate Algorithm f(x) x Ω? f s (x) x Ω s? NOMAD: Blackbox Optimization 19/41

35 General framework with surrogates x R n Surrogate Update Algorithm f(x) x Ω? f s (x) x Ω s? NOMAD: Blackbox Optimization 19/41

36 Current projects funded by HQ and Rio Tinto Opportunism and ordering strategies [completed] Discrete and granular variables [completed] Hierarchical functions [to do] Important variables [to do] Robust optimization [in progress] Parallel decomposition [completed] Grey boxes [completed] Multi-precision surrogates [in progress] Development of NOMAD [in progress] NOMAD: Blackbox Optimization 20/41

37 Example of project: Parallel decomposition PSD-MADS algorithm (parallel-space decomposition of MADS). PhD student Nadir Amaioua. Latest version designed and tested at IREQ. Parallelism with MPI. Master-slaves paradigm. Runs on CASIR. Motivation: Problems with large numbers of variables (up to 4,000 in our latest tests). Idea: Slaves optimize on subspaces where large number of variables are fixed. Machine learning technique are used to build sensitivity matrices and decide the subspaces. NOMAD: Blackbox Optimization 21/41

38 Research team Blackbox optimization The MADS algorithm The NOMAD software package Conclusion NOMAD: Blackbox Optimization 22/41

39 NOMAD (Nonlinear Optimization with MADS) C++ implementation of MADS. Standard C++, no other package needed. Parallel versions with MPI. Runs on Linux, Mac OS X and Windows. MATLAB versions; Multiple interfaces (Python, Excel, etc.) Command-line and library interfaces. Distributed under the LGPL license. Complete user guide available in the package. Doxygen documentation available online. Download at NOMAD: Blackbox Optimization 23/41

40 NOMAD: History and team Developed since Current version: 3.8. Algorithm designers: M. Abramson, C. Audet, J. Dennis, S. Le Digabel, C. Tribes, V. Rochon-Montplaisir. Developers: Versions 1 and 2: G. Couture. Version 3 (2008): S. Le Digabel, C. Tribes. Version 4 (2019): C. Tribes, V. Rochon-Montplaisir. Support at nomad@gerad.ca. Related article in TOMS [Le Digabel, 2011]. NOMAD: Blackbox Optimization 24/41

41 More than 9,000 certified downloads since Softpedia Sourceforge Gerad Total ,800 1,600 1,400 1,200 1, NOMAD: Blackbox Optimization 25/41

42 Users from Sept to Sept (downloads from the GERAD website only) Industries: Hydro-Québec (IREQ), Rio Tinto, Bombardier, Airbus, Boeing, United Airlines, Siemens, Schneider Electric, GHD Toronto, Energen, Skyconseil, German Aerospace Center, Moscow Power Engineering Institute, Newmerical Technologies, Disney Research, US Federal Reserve Board, Softree, Aditazz, Essar Steel, Tatura Milk Industries, Westrock, Market Appeal, Corona Insights, The Climate Corporation. National Labs: Argonne, Sandia, Los Alamos. Universities. NOMAD: Blackbox Optimization 26/41

43 Main functionalities (1/2) Single or biobjective optimization. Variables: Continuous, integer, binary, categorical. Periodic. Fixed. Groups of variables. Searches: Latin-Hypercube (LH). Variable Neighborhood Search (VNS). Quadratic models. Statistical surrogates. User search. NOMAD: Blackbox Optimization 27/41

44 Main functionalities (2/2) Constraints treated with 4 different methods: Progressive Barrier (default). Extreme Barrier. Progressive-to-Extreme Barrier. Filter method. Several direction types: Coordinate directions. LT-MADS. OrthoMADS. Hybrid combinations. Sensitivity analysis. (all items correspond to published or submitted papers). NOMAD: Blackbox Optimization 28/41

45 NOMAD installation Pre-compiled executables are available for Windows and Mac. Installation programs copy these executables. On Unix/Linux, after download, launch an installation script. Two ways to use NOMAD: batch mode or library mode. NOMAD: Blackbox Optimization 29/41

46 Blackbox conception (batch mode) Command-line program that takes in argument a file containing x, and displays the values of f(x) and the c j (x) s. Can be coded in any language. Typically: > bb.exe x.txt displays f c1 c2 (objective and two constraints). NOMAD: Blackbox Optimization 30/41

47 Important parameters Necessary parameters: Blackbox characteristics (dimension n, number of constraints, etc.), starting point (x 0 ). All algorithmic parameters have default values. The most important are: Maximum number of blackbox evaluations. Starting point (more than one can be defined). Types of directions (more than one can be defined). Initial mesh size. Constraint types. Surrogate searches. Seeds. See the user guide for the description of all parameters, or use the nomad -h option. NOMAD: Blackbox Optimization 31/41

48 Run NOMAD > nomad parameters.txt NOMAD: Blackbox Optimization 32/41

49 Advanced functionalities (library mode) No system calls: the code executes faster. The user can program a custom search strategy. The user can pre-process all evaluation points before they are evaluated. The user can decide the priority in which trial points are evaluated. The user can define callback functions that will be called at some specific events (new success, new iteration, new MADS run in bi-objective optimization) NOMAD: Blackbox Optimization 33/41

50 Examples included in the NOMAD package Simple examples: How to run NOMAD. Usage of parallellism. Categorical variables. MATLAB/Python interfaces. Advanced examples to illustrate additional possibilities: Multi-start from points generated with LH sampling. Problems used in library mode and coded as: A Windows DLL. A GAMS program. A CUTEst problem. A FORTRAN code. A GUI prototype in JAVA. NOMAD: Blackbox Optimization 34/41

51 Other MADS distributions MATLAB version within the Opti Toolbox package. Available in the MATLAB Optimization Toolbox. Old version, not maintained. NOMADM by M. Abramson. Old version, not maintained. Excel with the OpenSolver tool. (GPLv3). NOMAD: Blackbox Optimization 35/41

52 NOMAD 4 Motivated by the collaboration with Hydro-Québec and Rio Tinto. New Architecture. Emphasis on modularity and simplicity. Based on the book [Audet and Hare, 2017]. Avoid computation bottlenecks (continuous evaluations; processes do not wait on each other). Modify algorithm parameters while optimizing. Independent library of surrogates (SGTELIB). Usage of parallelism (detailed next slide). NOMAD: Blackbox Optimization 36/41

53 Parallelism in NOMAD 4 Blackbox level: Blackbox uses parallelism when available. Computer level: Launch multiple blackbox processes on different CPUs (OpenMP and MPI). Cluster level: Launch multiple blackbox processes on different machines (MPI). Blocks of trial points. Ability to feed a cluster in evaluations. Algorithmic improvements to produce more points for evaluation. Parallel space decomposition for problems of large dimension. NOMAD: Blackbox Optimization 37/41

54 Research team Blackbox optimization The MADS algorithm The NOMAD software package Conclusion NOMAD: Blackbox Optimization 38/41

55 Summary Blackbox optimization motivated by industrial applications. Algorithmic features backed by mathematical convergence analyses and published in optimization journals. NOMAD: Software package implementing MADS. Open source; LGPL license. Features: Constraints, biobjective, global optimization, surrogates, several types of variables, parallelism. Fast support at NOMAD can be customized through collaborations. NOMAD: Blackbox Optimization 39/41

56 References I Abramson, M. (2004). Mixed variable optimization of a Load-Bearing thermal insulation system using a filter pattern search algorithm. Optimization and Engineering, 5(2): Audet, C., Béchard, V., and Le Digabel, S. (2008a). Nonsmooth optimization through mesh adaptive direct search and variable neighborhood search. Journal of Global Optimization, 41(2): Audet, C. and Dennis, Jr., J. (2006). Mesh adaptive direct search algorithms for constrained optimization. SIAM Journal on Optimization, 17(1): Audet, C. and Dennis, Jr., J. (2009). A Progressive Barrier for Derivative-Free Nonlinear Programming. SIAM Journal on Optimization, 20(1): Audet, C., Dennis, Jr., J., and Le Digabel, S. (2008b). Parallel space decomposition of the mesh adaptive direct search algorithm. SIAM Journal on Optimization, 19(3): Audet, C., Dennis, Jr., J., and Le Digabel, S. (2012). Trade-off studies in blackbox optimization. Optimization Methods and Software, 27(4 5): NOMAD: Blackbox Optimization 40/41

57 References II Audet, C., Digabel, S. L., and Tribes, C. (2018). The mesh adaptive direct search algorithm for granular and discrete variables. Technical Report G , Les cahiers du GERAD. Audet, C. and Hare, W. (2017). Derivative-Free and Blackbox Optimization. Springer Series in Operations Research and Financial Engineering. Springer International Publishing. Audet, C., Savard, G., and Zghal, W. (2008c). Multiobjective optimization through a series of single-objective formulations. SIAM Journal on Optimization, 19(1): Le Digabel, S. (2011). Algorithm 909: NOMAD: Nonlinear Optimization with the MADS algorithm. ACM Transactions on Mathematical Software, 37(4):44:1 44:15. Le Digabel, S., Abramson, M., Audet, C., and Dennis, Jr., J. (2010). Parallel Versions of the MADS Algorithm for Black-Box Optimization. In Optimization days, Montreal. GERAD. Slides available at Talgorn, B., Le Digabel, S., and Kokkolaras, M. (2015). Statistical Surrogate Formulations for Simulation-Based Design Optimization. Journal of Mechanical Design, 137(2): NOMAD: Blackbox Optimization 41/41