Surrogate models and the optimization of systems in the absence of algebraic models

Transcription

1 Surrogate models and the optimization of systems in the absence of algebraic models Nick Sahinidis Acknowledgments: Alison Cozad, David Miller, Zach Wilson Atharv Bhosekar, Luis Miguel Rios, Hua Zheng

2 SIMULATION OPTIMIZATION Pulverized coal plant Aspen Plus simulation provided by the National Energy Technology Laboratory 2

3 EXPERIMENT OPTIMIZATION Extraction Fermentation (twice) Polymerization Extrusion Distillation Process Systems Engineering Six functional units Maximize PTT yield Experimental system available in the Unit Operations Lab Several degrees of freedom Fermenter ph, T, NaCl concentration, O2 sparging conditions Extruder input concentrations, spinning conditions (discrete) 3

4 CHALLENGES No algebraic model Complex process alternatives OPTIMIZER Cost $ 1 reactor 2 reactors 3 reactors Reactor size SIMULATOR Costly simulations minutes seconds hours Scarcity of fully robust simulations 4

5 OPTIMIZATION TECHNOLOGY Derivative free optimization (DFO) Optimizes in the absence of algebraic models Derivatives are symbolically unavailable Derivatives may be time consuming to compute numerically Rios and Sahinidis (JOGO, 2013) Reviews algorithms and compares 20+ codes on 500+ problems Simulation optimization (SO) Addresses noise in function values Amaran, Sahinidis, Sharda and Bury (AOR, 2015) Reviews algorithms and applications 5

6 DFO ALGORITHMS LOCAL SEARCH METHODS Direct local search Nelder Mead simplex algorithm Generalized pattern search and generating search set Based on surrogate models Trust region methods Implicit filtering GLOBAL SEARCH METHODS Deterministic global search Lipschitzian based partitioning Multilevel coordinate search Stochastic global optimization Hit and run Simulated annealing Genetic algorithms Particle swarm Based on surrogate models Response surface methods Surrogate management framework Branch and fit 6

7 DFO SOFTWARE LOCAL SEARCH FMINSEARCH (Nelder-Mead) DAKOTA PATTERN (PPS) HOPSPACK (PPS) SID-PSM (Simplex gradient PPS) NOMAD (MADS) DFO (Trust region, quadratic model) IMFIL (Implicit Filtering) BOBYQA (Trust region, quadratic model) NEWUOA (Trust region, quadratic model) GLOBAL SEARCH DAKOTA SOLIS-WETS (DIRECT) DAKOTA DIRECT (DIRECT) TOMLAB GLBSOLVE (DIRECT) TOMLAB GLCSOLVE (DIRECT) MCS (Multilevel coordinate search) TOMLAB EGO (RSM using Kriging) TOMLAB RBF (RSM using RBF) SNOBFIT (Branch-and-Fit) TOMLAB LGO (LGO algorithm) STOCHASTIC ASA (Simulated annealing) CMA-ES (Evolutionary algorithm) DAKOTA EA (Evolutionary algorithm) GLOBAL (Clustering Multistart) PSWARM (Particle swarm) 7

8 PROBLEMS SOLVED 8

9 PROCESS DISAGGREGATION Block 1: Simulator Model generation Block 2: Simulator Model generation Block 3: Simulator Model generation Process Simulation Disaggregate process into process blocks Surrogate Models Build simple and accurate models with a functional form tailored for an optimization framework Optimization Model Add algebraic constraints design specs, heat/mass balances, and logic constraints 9

10 LEARNING PROBLEM Build a model of output variables as a function of input variables x over a specified interval Process simulation or Experiment Independent variables: Operating conditions, inlet flow properties, unit geometry Dependent variables: Efficiency, outlet flow conditions, conversions, heat flow, etc. 10

11 DESIRED MODEL ATTRIBUTES We aim to build surrogate models that are Accurate We want to reflect the true nature of the simulation Simple Interpretable; tailored for algebraic optimization Generated from a minimal data set Reduce experimental and simulation requirements 11

12 ALAMO Automated Learning of Algebraic Models using Optimization Start Initial sampling Build surrogate model Update training data set false Adaptive sampling Model converged? true Stop Model error Black-box function New model Current model 12

13 MODEL COMPLEXITY TRADEOFF Kriging [Krige, 63] Neural nets [McCulloch-Pitts, 43] Radial basis functions [Buhman, 00] Model accuracy Preferred region Linear response surface Model complexity 13

14 MODEL IDENTIFICATION Goal: Identify the functional form and complexity of the surrogate models Functional form: General functional form is unknown: Our method will identify models with combinations of simple basis functions 14

15 OVERFITTING AND TRUE ERROR Step 1: Define a large set of potential basis functions Step 2: Model reduction Error Ideal Model True error Empirical error Complexity Underfitting Overfitting 15

16 MODEL REDUCTION TECHNIQUES Qualitative tradeoffs of model reduction methods Stepwise regression [Efroymson, 60] Backward elimination [Oosterhof, 63] Forward selection [Hamaker, 62] Best subset methods Enumerate all possible subsets Regularized regression techniques Penalize the least squares objective using the magnitude of the regressors [Tibshirani, 95] 16

17 MODEL SIZING Goodness of fit measure Some measure of error that is sensitive to overfitting (AICc, BIC, Cp) Solve for the best one term model Solve for the best two term model 6 th term was not worth the added complexity Final model includes 5 terms Complexity = number of terms allowed in the model 17

18 MODEL SELECTION CRITERIA Balance fit (sum of square errors) with model complexity (number of terms in the model; denoted by p) Corrected Akaike Information Criterion log 1 2 Mallows Cp 2 Hannan Quinn Information Criterion log 1 Bayes Information Criterion Mean Squared Error log log log 1 18

19 CPU TIME COMPARISON Eight benchmarks from the UCI and CMU data sets Seventy noisy data sets were generated with multicolinearity and increasing problem size (number of bases) Cp BIC AIC, MSE, HQC CPU time (s) Problem Size BIC solves more than two orders of magnitude faster than AIC, MSE and HQC Optimized directly via a single mixed integer convex quadratic model 19

20 MODEL QUALITY COMPARISON BIC leads to smaller, more accurate models Larger penalty for model complexity Spurious Variables Included Cp BIC AIC HQC MSE Problem Size % Deviation from True Model Problem Size 20

21 ALAMO Automated Learning of Algebraic Models using Optimization Start Initial sampling Build surrogate model Update training data set Adaptive sampling Model error false Model converged? true Stop Error maximization sampling 21

22 ERROR MAXIMIZATION SAMPLING Search the problem space for areas of model inconsistency or model mismatch Find points that maximize the model error with respect to the independent variables Surrogate model Derivative free solvers work well in low dimensional spaces [Rios and Sahinidis, 12] Optimized using a black box or derivative free solver (SNOBFIT) [Huyer and Neumaier, 08] 22

23 COMPUTATIONAL RESULTS Goal Compare methods on three target metrics 1 Model accuracy 2 Data efficiency 3 Model simplicity Modeling methods compared ALAMO modeler Proposed best subset methodology The LASSO The lasso regularization Ordinary regression Ordinary least squares regression Sampling methods compared (over the same data set size) ALAMO sampler Proposed error maximization technique Single LH Single Latin hypercube (no feedback) 23

24 1 Model accuracy 2 Data efficiency 3 Model simplicity 70% of problems solved exactly Fraction of problems solved (0.005, 0.80) 80% of the problems had 0.5% error error maximization sampling ALAMO modeler the lasso Ordinary regression Normalized test error ALAMO modeler Ordinary regression the lasso single Latin hypercube Normalized test error 24

25 1 Model accuracy 2 Data efficiency 3 Model simplicity Fraction of problems solved ALAMO sampler Single LH ALAMO modeler ALAMO sampler Single LH the lasso ALAMO sampler Single LH Ordinary regression Normalized test error 25

26 1 Model accuracy 2 Data efficiency 3 Model simplicity Modeling type, Median more complexity than required Results over a test set of 45 known functions treated as black boxes with bases that are available to all modeling methods. 26

27 THEORY UTILIZATION empirical data non empirical information Use freely available system knowledge to strengthen model Physical limits First principles knowledge Intuition Non empirical restrictions can be applied to general regression problems 27

28 CONSTRAINED REGRESSION Challenging due to the semi infinite nature of the regression constraints Standard regression easy Surrogate model tough 28

29 IMPLIED PARAMETER RESTRICTIONS Step 1: Formulate constraint in z and x space Step 2: Identify a sufficient set of β space constraints 1 parametric constraint 4 β constraints Global optimization problems solved with BARON: archimedes.cheme.cmu.edu/?q=baron and 29

30 TYPES OF RESTRICTIONS Response bounds Individual responses Multiple responses pressure, temperature, compositions mass and energy balances, physical limitations mass balances, sum toone, state variables Response derivatives Alternative domains Boundary conditions Extrapolation zone velocity profile model monotonicity, numerical properties, convexity Problem Space safe extrapolation, boundary conditions Add no slip 30

31 CARBON CAPTURE SYSTEM DESIGN Surrogate models for each reactor and technology used Discrete decisions: How many units? Parallel trains? What technology used for each reactor? Continuous decisions: Unit geometries Operating conditions: Vessel temperature and pressure, flow rates, compositions 31

32 SUPERSTRUCTURE OPTIMIZATION Mixed-integer nonlinear programming model Economic model Process model Material balances Hydrodynamic/Energy balances Reactor surrogate models Link between economic model and process model Binary variable constraints Bounds for variables MINLP solved with BARON 32

33 ALAMO REMARKS Surrogate model New surrogate model Expanding the scope of algebraic optimization Using low complexity surrogate models to strike a balance between optimal decision making and model fidelity Surrogate model identification Simple, accurate model identification MINLP Error maximization sampling More information found per simulated data point DFO 33

34 NEW DFO ALGORITHMS Rios Sahinidis (2013) conclusions: Deterministic solvers outperform stochastic ones Surrogate model based algorithms have an edge Solve auxiliary algebraic models to global optimality to expedite search Decide where to sample objective function Guarantee geometry Construct surrogate models Higher quality surrogates Solve trust region subproblems Escape local minima; guarantee convergence BARON is highly efficient for problems below 100 variables Model and search (MAS) local search algorithm Branch and model (BAM) global search algorithm 34

35 MAS LOCAL SEARCH ALGORITHM x 5 x 3 x 1 Collect points around current iterate r 1 r 2 x 2 x 4 Add points: n linearly independent points r 1 r 2 Build and optimize linear or quadratic interpolating model Check for positive basis. Add points if necessary 35

36 PRELIMINARY RESULTS WITH MAS MAS 359 test problems 36

37 BAM GLOBAL SEARCH ALGORITHM Partition the space in a collection of hypercubes Eliminate potentially suboptimal hypercubes For remaining hypercubes, FIT models Sort hypercubes by size. Explore the largestt Sort hypercubes by predicted optimal values. Explore the best Optimize surrogate models 37

38 PRELIMINARY RESULTS WITH BAM BAM 38

39 CONCLUSIONS ALAMO provides algebraic models that are Accurate and simple Generated from a minimal number of function evaluations ALAMO s constrained regression facility allows modeling of Bounds on response variables Convexity/monotonicity of response variables Extensive results with MAS and BAM will be presented at the upcoming INFORMS and AIChE meetings ALAMO site archimedes.cheme.cmu.edu/?q=alamo DFO/SO site archimedes.cheme.cmu.edu/?q=bbo Test sets 39