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Optimization under Uncertainty in the Era of Big Data and Deep Learning: When Machine Learning Meets Mathematical Programming
Chao Ning, Fengqi You
TL;DR
Optimization under uncertainty must account for uncertain parameters that can undermine deterministic decisions. This paper reviews classical and data-driven mathematical programming, classifies data-driven uncertainty models, and examines research opportunities integrating machine learning with optimization. It concludes that data-driven mathematical programming frameworks are poised for rapid growth, while efficient online and scenario-based methods remain important challenges.
Problem
Uncertain parameters can make deterministic optimization solutions suboptimal or infeasible, motivating optimization methods that model uncertainty from data.
Method
The paper reviews classical uncertainty-hedging techniques and classifies data-driven optimization research by uncertainty modeling and optimization structure.
Results
The review analyzes advantages and disadvantages of different data-driven uncertainty models and concludes that data-driven mathematical programming frameworks are foreseeable to experience rapid growth.
Takeaways & Limitations
Promising research opportunities include closed-loop integration of machine learning and mathematical programming, deep-learning-enabled scenario optimization, and learning-while-optimizing multistage frameworks.
Takeaways & Limitations
Efficient algorithms for scenario programs with many discrete variables remain a research challenge, and scenario programs can become prohibitively large.
Abstract
from arXiv · showhide
This paper reviews recent advances in the field of optimization under uncertainty via a modern data lens, highlights key research challenges and promise of data-driven optimization that organically integrates machine learning and mathematical programming for decision-making under uncertainty, and identifies potential research opportunities. A brief review of classical mathematical programming techniques for hedging against uncertainty is first presented, along with their wide spectrum of applications in Process Systems Engineering. A comprehensive review and classification of the relevant publications on data-driven distributionally robust optimization, data-driven chance constrained program, data-driven robust optimization, and data-driven scenario-based optimization is then presented. This paper also identifies fertile avenues for future research that focuses on a closed-loop data-driven optimization framework, which allows the feedback from mathematical programming to machine learning, as well as scenario-based optimization leveraging the power of deep learning techniques. Perspectives on online learning-based data-driven multistage optimization with a learning-while-optimizing scheme is presented.
1. Introduction
Optimization under uncertainty addresses uncertain parameters that can make deterministic solutions suboptimal or infeasible, while data-driven methods formulate uncertainty models from observed data. This review surveys classical and data-driven approaches, classifies the literature, and identifies future research directions integrating machine learning with mathematical programming.
- Motivation: Uncertain parameters, including product demands, kinetic constants, and task durations, can make deterministic optimization solutions suboptimal or infeasible.Constraint violations can have disastrous consequences for solution quality.
- Motivation: Big data and deep learning create opportunities to extract useful information from process-industry data for decision-making, control, operations, and design.Process industries routinely collect and archive large volumes of data that are becoming increasingly important assets.
- Data-driven optimization: Data-driven optimization formulates uncertainty models from data, allowing uncertainty information to be harnessed automatically for data-driven decisions.The approach lets uncertainty data “speak” for themselves within the optimization algorithm.
- Review scope: The review summarizes and classifies research on data-driven optimization under uncertainty according to uncertainty modeling approaches and optimization structures.The paper also briefly reviews conventional mathematical programming techniques and their applications in Process Systems Engineering.
- Future directions: The paper identifies three future research directions for advancing optimization under uncertainty in the era of big data and deep learning.It highlights associated research challenges and potential methodologies, with the paper organized around background, literature review, future directions, and conclusions.
2. Background on optimization under uncertainty
Optimization under uncertainty uses stochastic programming, chance-constrained programming, and robust optimization to support decision making across uncertain process systems. These paradigms differ in how they represent uncertainty, respond to realizations, and manage computational difficulty.
- Stochastic programming: Stochastic programming models uncertain parameters with probability distributions and can represent decisions across multiple time stages, including corrective recourse after uncertainty is revealed.Two-stage models separate here-and-now decisions from wait-and-see decisions, with an objective combining first-stage costs and expected second-stage costs.
- Stochastic programming: Two-stage stochastic programs become computationally expensive as the number of scenarios grows, motivating decomposition methods such as Benders and Lagrangean decomposition.Integer recourse can make the expected recourse function nonconvex and even discontinuous, hindering conventional L-shaped methods.
- Chance-constrained optimization: Chance-constrained programming enforces constraint satisfaction with probability at least 1−ε, allowing decision makers to choose risk levels to trade objective performance against reliability.Individual and joint chance constraints correspond respectively to one constraint and multiple constraints.
- Chance-constrained optimization: Chance-constrained programs are generally computationally intractable because probability evaluation involves multivariate integration and the feasible region may be nonconvex.Sample average, sequential, and convex conservative approximation schemes address these computational challenges; tractable convex reformulations exist in some special cases.
- Robust optimization: Robust optimization hedges against the worst case within a specified uncertainty set rather than relying on probability distributions.The uncertainty set is a central modeling ingredient, and worst cases may correspond to the largest constraint violation or another adverse outcome.
- Robust optimization: Adaptive robust optimization typically generates less conservative solutions than static robust optimization by allowing later decisions to depend on realized uncertainty.Robust models can include first-stage decisions before uncertainty realization and second-stage decisions afterward.
3. Existing methods for data-driven optimization under uncertainty
The paper classifies data-driven optimization under uncertainty into four categories and reviews their formulations, advantages, assumptions, and applications. These methods use uncertainty data rather than assuming a perfectly known uncertainty model, with distributionally robust optimization providing worst-case guarantees over ambiguity sets.
- Recent data-driven optimization research is classified into data-driven stochastic, chance-constrained, robust, and scenario-based optimization.
- Unlike conventional techniques, these approaches infer uncertainty models from available uncertainty data rather than assuming them perfectly known a priori.
- Data-driven stochastic programming and DRO: Distributionally robust optimization constructs an ambiguity set of probability distributions from data and optimizes against its worst-case distribution.
- Data-driven stochastic programming and DRO: DRO offers performance guarantees over a family of distributions and can mitigate optimizer’s curse while improving out-of-sample performance.
- Data-driven stochastic programming and DRO: Ambiguity-set design must balance computational tractability, statistical meaning, and decision performance.
- Data-driven stochastic programming and DRO: Moment-based ambiguity sets use estimated support, mean, and covariance information, but basic formulations may not account for uncertainty in those estimates.
- Scenario optimization: Scenario-based chance-constrained optimization can provide probabilistic guarantees for unobserved uncertainty realizations using uncertainty data.
- Scenario optimization: For convex scenario approaches, the guarantee holds for all feasible solutions in contrast to approaches where it is valid only for the optimal solution.
4. Future research directions and opportunities
The paper identifies future research directions centered on closed-loop data-driven optimization, deep-learning integration with scenario-based optimization, and learning-while-optimizing schemes.
- Three promising directions are closed-loop data-driven optimization, deep-learning integration with scenario-based optimization, and learning-while-optimizing.
4.1. A “closed-loop” data-driven optimization framework with feedback from mathematical programming to machine learning
The paper proposes closing the loop between machine learning and mathematical programming by feeding optimization feedback back into the data-driven system. This could replace sequential open-loop interaction with a bidirectional framework, while introducing challenges in computational scale and feedback design.
- Data-driven optimization is described as a hybrid system combining machine learning for information extraction with mathematical programming for optimal decisions.
- Closed-loop framework: A closed-loop paradigm would send information from mathematical programming back to machine learning, enabling information flow in both directions.
- Open-loop limitations: Existing frameworks use a sequential open-loop scheme in which machine learning is trained without interaction with downstream mathematical programming.
- Closed-loop framework: A weighted loss combining the conventional machine-learning loss with the mathematical-programming objective is proposed as one possible feedback mechanism.
- Closed-loop framework: Iterative interaction between machine learning and mathematical programming is presented as another path for closing the loop.
- Research challenges: Large mathematical programs combined with big data may complicate machine-learning training, and designing effective feedback remains a key challenge.
- Additional research opportunities: Incorporating domain-specific prior knowledge, such as distributional structure, bounds, or correlations, could provide more reliable results with messy data.
4.2. Leveraging deep learning techniques for hedging against uncertainty in data-driven optimization
The paper surveys deep-learning architectures as tools for extracting nonlinear, spatial, and temporal structure from uncertainty data. It highlights deep generative models for scenario-based optimization and deep recurrent models for multistage uncertainty.
- Deep neural networks are proposed for uncovering useful patterns in complex, highly nonlinear uncertainty data for mathematical programming.
- Deep belief networks: Deep belief networks extract hierarchical latent features and can represent complicated nonlinear correlations among uncertain parameters.
- Convolutional neural networks: Convolutional neural networks are suited to spatially correlated uncertainty data and may provide probability weights for data-driven stochastic robust optimization.
- Recurrent neural networks: Recurrent neural networks model sequential data by storing historical inputs in state vectors, while LSTM and GRU variants address long-term-memory weaknesses.
- Multistage optimization: Deep RNNs and LSTM could capture temporal dynamics and trajectories of uncertainty across time stages in multistage optimization.
- Scenario-based optimization: Deep generative models are proposed to address limited uncertainty data in data-driven scenario-based optimization.
- Scenario-based optimization: Scenario-based optimization relies on sufficient uncertainty data for training generative models, but such data may be limited, expensive, or time-consuming to acquire.
4.3. Online learning-based data-driven optimization: a learning-while-optimizing paradigm for addressing uncertainty
Existing data-driven optimization frameworks commonly learn uncertainty models from fixed batches, leaving them unable to accommodate evolving real-time data. The paper proposes online learning-based optimization in which learning and sequential decision-making are integrated, while identifying computational and theoretical challenges.
- Current limitations: Most existing studies rely on batch machine learning and therefore fail to account for real-time uncertainty data.The limitation is especially relevant when uncertainty data are collected sequentially in practical systems.
- Current limitations: Batch-based data-driven optimization typically learns uncertainty sets from existing data and leaves them fixed rather than updating or refining them.This can make the data-driven system outdated when uncertainty distributions or support sets evolve over time.
- Current limitations: Previously proposed online uncertainty-set learning often retrains the data-driven system from scratch after new data arrive, limiting its suitability to slow-dynamics systems.Few studies address real-time data analytics for fast-dynamics systems such as chemical processes.
- Learning-while-optimizing paradigm: Online-learning-based data-driven optimization would update learning iteratively to incorporate real-time data while decisions are made sequentially under uncertainty.After decisions are made, realized uncertainties can be collected and added to the database.
- Learning-while-optimizing paradigm: A learning-while-optimizing scheme could use deep reinforcement learning, but online updating and efficient solution algorithms remain key research challenges.Further theoretical questions include solution convergence under distribution shift and bounds on computational complexity, required memory, and data.
5. Conclusions
The paper reviews mathematical programming for uncertainty and classifies data-driven approaches by uncertainty modeling, machine-learning integration, and computational algorithms. It concludes that closed-loop feedback, deep-learning-based scenario optimization, and online learning are promising directions.
- Conclusions: Conventional stochastic programming, robust optimization, and chance-constrained optimization are recognized paradigms for hedging against uncertainty.
- Conclusions: The review organizes data-driven mathematical programming under uncertainty around systematic uncertainty modeling, organic machine-learning integration, and efficient computational algorithms.
- Conclusions: The paper analyzes the advantages and disadvantages of different data-driven uncertainty models in detail.
- Future directions: Future research could close the loop by introducing feedback from the model-based mathematical-programming system to the data-driven machine-learning system.
- Future directions: Additional opportunities include deep-generative-model-based data-driven scenario optimization and online-learning mathematical-programming frameworks for real-time data.