Learning Linear Programs from Optimal Decisions

We propose a flexible gradient-based framework for learning linear programs from optimal decisions. Linear programs are often specified by hand, using prior knowledge of relevant costs and constraints. In some applications, linear programs must instead be learned from observations of optimal decisions. Learning from optimal decisions is a particularly challenging bi-level problem, and much of the related inverse optimization literature is dedicated to special cases. We tackle the general problem, learning all parameters jointly while allowing flexible parametrizations of costs, constraints, and loss functions. We also address challenges specific to learning linear programs, such as empty feasible regions and non-unique optimal decisions. Experiments show that our method successfully learns synthetic linear programs and minimum-cost multi-commodity flow instances for which previous methods are not directly applicable. We also provide a fast batch-mode PyTorch implementation of the homogeneous interior point algorithm, which supports gradients by implicit differentiation or backpropagation.

[1]  M. Powell.  A Direct Search Optimization Method That Models the Objective and Constraint Functions by Linear Interpolation , 1994 .

[2]  J. Zico Kolter,et al.  OptNet: Differentiable Optimization as a Layer in Neural Networks , 2017, ICML.

[3]  Kimia Ghobadi,et al.  Inferring linear feasible regions using inverse optimization , 2021, Eur. J. Oper. Res..

[4]  Richard J. Caron.  Redundancy in Nonlinear Programs , 2009, Encyclopedia of Optimization.

[5]  Yoshua Bengio,et al.  Gradient-Based Optimization of Hyperparameters , 2000, Neural Computation.

[6]  Justin Domke,et al.  Generic Methods for Optimization-Based Modeling , 2012, AISTATS.

[7]  Marvin D. Troutt,et al.  Linear programming system identification , 2005, Eur. J. Oper. Res..

[8]  Wieslawa T. Obuchowska,et al.  Minimal representation of quadratically constrained convex feasible regions , 1995, Math. Program..

[9]  Gökhan BakIr,et al.  Predicting Structured Data , 2008 .

[10]  Timothy C.Y. Chan,et al.  Inverse optimization for the recovery of constraint parameters , 2018, Eur. J. Oper. Res..

[11]  Philippe L. Toint,et al.  On an instance of the inverse shortest paths problem , 1992, Math. Program..

[12]  Stephen P. Boyd,et al.  Differentiating Through a Conic Program , 2019 .

[13]  Zuo-Jun Max Shen,et al.  Inverse Optimization with Noisy Data , 2015, Oper. Res..

[14]  Andreas Krause,et al.  Differentiable Learning of Submodular Models , 2017, NIPS 2017.

[15]  Numerische,et al.  The Nonlinear Programming Method of Wilson , Han , and Powell with an Augmented Lagrangian Type Line Search Function Part 2 : An Efficient Implementation with Linear Least Squares Subproblems , 2005 .

[16]  Katta G. Murty,et al.  Infeasibility analysis for linear systems, a survey , 2000 .

[17]  Sebastian Nowozin,et al.  Advanced Structured Prediction , 2014 .

[18]  Stephen P. Boyd,et al.  Imputing a convex objective function , 2011, 2011 IEEE International Symposium on Intelligent Control.

[19]  Francis Bach,et al.  Learning with Differentiable Perturbed Optimizers , 2020, ArXiv.

[20]  Yiran Chen,et al.  Generalized Inverse Optimization through Online Learning , 2018, NeurIPS.

[21]  Jay E. Strum.  Note on "Two-Sided Shadow Prices" , 1969 .

[22]  J. Telgen.  Identifying Redundant Constraints and Implicit Equalities in Systems of Linear Constraints , 1983 .

[23]  Yi-Hao Kao,et al.  Directed Regression , 2009, NIPS.

[24]  Yinyu Ye,et al.  A simplified homogeneous and self-dual linear programming algorithm and its implementation , 1996, Ann. Oper. Res..

[25]  Ravindra K. Ahuja,et al.  Inverse Optimization , 2001, Oper. Res..

[26]  Priya L. Donti,et al.  Task-based End-to-end Model Learning in Stochastic Optimization , 2017, NIPS.

[27]  Veselin Stoyanov,et al.  Empirical Risk Minimization of Graphical Model Parameters Given Approximate Inference, Decoding, and Model Structure , 2011, AISTATS.

[28]  Clemens Heuberger,et al.  Inverse Combinatorial Optimization: A Survey on Problems, Methods, and Results , 2004, J. Comb. Optim..

[29]  George Papandreou,et al.  Perturb-and-MAP random fields: Using discrete optimization to learn and sample from energy models , 2011, 2011 International Conference on Computer Vision.

[30]  Timothy C. Y. Chan,et al.  Generalized Inverse Multiobjective Optimization with Application to Cancer Therapy , 2014, Oper. Res..

[31]  Corinna Cortes,et al.  Support-Vector Networks , 1995, Machine Learning.

[32]  Justin Domke,et al.  Learning Graphical Model Parameters with Approximate Marginal Inference , 2013, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[33]  Ryan P. Adams,et al.  Gradient-based Hyperparameter Optimization through Reversible Learning , 2015, ICML.

[34]  Melvyn Sim,et al.  Adjustable Robust Optimization via Fourier-Motzkin Elimination , 2018, Oper. Res..

[35]  Sebastian Pokutta,et al.  An Online-Learning Approach to Inverse Optimization , 2018, ArXiv.

[36]  D. Aucamp,et al.  The Computation of Shadow Prices in Linear Programming , 1982 .

[37]  Daria Terekhov,et al.  Deep Inverse Optimization , 2018, CPAIOR.

[38]  Marvin D. Troutt,et al.  A maximum decisional efficiency estimation principle , 1995 .

[39]  Adam N. Elmachtoub,et al.  Smart "Predict, then Optimize" , 2017, Manag. Sci..

[40]  Samir Khuller,et al.  On Correcting Inputs: Inverse Optimization for Online Structured Prediction , 2015, FSTTCS.

[41]  Stephen Gould,et al.  Deep Declarative Networks: A New Hope , 2019, ArXiv.

[42]  Joel Nothman,et al.  SciPy 1.0-Fundamental Algorithms for Scientific Computing in Python , 2019, ArXiv.

[43]  Sebastian Pokutta,et al.  Emulating the Expert: Inverse Optimization through Online Learning , 2017, ICML.

[44]  Daniel Kuhn,et al.  Data-driven inverse optimization with imperfect information , 2015, Mathematical Programming.

[45]  Kimia Ghobadi,et al.  Multi-point Inverse Optimization of Constraint Parameters , 2020 .

[46]  Nguyen Dong Yen,et al.  Quadratic Programming and Affine Variational Inequalities: A Qualitative Study , 2005 .

[47]  Ben Taskar,et al.  Learning structured prediction models: a large margin approach , 2005, ICML.

[48]  Knud D. Andersen,et al.  The Mosek Interior Point Optimizer for Linear Programming: An Implementation of the Homogeneous Algorithm , 2000 .

[49]  Timothy C. Y. Chan,et al.  Inverse Optimization: Closed-Form Solutions, Geometry, and Goodness of Fit , 2015, Manag. Sci..

[50]  Juan M. Morales,et al.  Short-Term Forecasting of Price-Responsive Loads Using Inverse Optimization , 2016, IEEE Transactions on Smart Grid.

[51]  Timothy C. Y. Chan,et al.  A Unified Framework for Model Fitting and Evaluation in Inverse Linear Optimization , 2018 .

[52]  Donal O'Shea,et al.  Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.) , 1997, Undergraduate texts in mathematics.

[53]  Vishal Gupta,et al.  Data-driven estimation in equilibrium using inverse optimization , 2013, Mathematical Programming.

[54]  Gerard Sierksma,et al.  Balinski—Tucker simplex tableaus: Dimensions, degeneracy degrees, and interior points of optimal faces , 1998, Math. Program..

[55]  Clermont Dupuis,et al.  An Efficient Method for Computing Traffic Equilibria in Networks with Asymmetric Transportation Costs , 1984, Transp. Sci..

[56]  T. Chan,et al.  Goodness of Fit in Inverse Optimizaiton , 2015 .

[57]  T. Chan,et al.  Multiple Observations and Goodness of Fit in Generalized Inverse Optimization , 2018, 1804.04576.

[58]  Marvin D. Troutt,et al.  Linear programming system identification: The general nonnegative parameters case , 2008, Eur. J. Oper. Res..