This paper studies how to train machine-learning models that directly approximate the optimal solutions of constrained optimization problems. This is an empirical risk minimization under constraints, which is challenging as training must balance optimality and feasibility conditions. Supervised learning methods often approach this challenge by training the model on a large collection of pre-solved instances. This paper takes a different route and proposes the idea of Primal-Dual Learning (PDL), a self-supervised training method that does not require a set of pre-solved instances or an optimization solver for training and inference. Instead, PDL mimics the trajectory of an Augmented Lagrangian Method (ALM) and jointly trains primal and dual neural networks. Being a primal-dual method, PDL uses instance-specific penalties of the constraint terms in the loss function used to train the primal network. Experiments show that, on a set of nonlinear optimization benchmarks, PDL typically exhibits negligible constraint violations and minor optimality gaps, and is remarkably close to the ALM optimization. PDL also demonstrated improved or similar performance in terms of the optimality gaps, constraint violations, and training times compared to existing approaches.
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Learning Hard Optimization Problems: A Data Generation Perspective
Optimization problems are ubiquitous in our societies and are present in almost every segment of the economy. Most of these optimization problems are NP-hard and computationally demanding, often requiring approximate solutions for large-scale instances. Machine learning frameworks that learn to approximate solutions to such hard optimization problems are a potentially promising avenue to address these difficulties, particularly when many closely related problem instances must be solved repeatedly. Supervised learning frameworks can train a model using the outputs of pre-solved instances. However, when the outputs are themselves approximations, when the optimization problem has symmetric solutions, and/or when the solver uses randomization, solutions to closely related instances may exhibit large differences and the learning task can become inherently more difficult. This paper demonstrates this critical challenge, connects the volatility of the training data to the ability of a model to approximate it, and proposes a method for producing (exact or approximate) solutions to optimization problems that are more amenable to supervised learning tasks. The effectiveness of the method is tested on hard non-linear nonconvex and discrete combinatorial problems.
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- Award ID(s):
- 2007164
- PAR ID:
- 10337576
- Date Published:
- Journal Name:
- Advances in Neural Information Processing Systems 34 (NeurIPS 2021)
- Volume:
- 34
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Quantum Approximate Optimization algorithm (QAOA) aims to search for approximate solutions to discrete optimization problems with near-term quantum computers. As there are no algorithmic guarantee possible for QAOA to outperform classical computers, without a proof that bounded-error quantum polynomial time (BQP) ≠ nondeterministic polynomial time (NP), it is necessary to investigate the empirical advantages of QAOA. We identify a computational phase transition of QAOA when solving hard problems such as SAT—random instances are most difficult to train at a critical problem density. We connect the transition to the controllability and the complexity of QAOA circuits. Moreover, we find that the critical problem density in general deviates from the SAT-UNSAT phase transition, where the hardest instances for classical algorithms lies. Then, we show that the high problem density region, which limits QAOA’s performance in hard optimization problems (reachability deficits), is actually a good place to utilize QAOA: its approximation ratio has a much slower decay with the problem density, compared to classical approximate algorithms. Indeed, it is exactly in this region that quantum advantages of QAOA over classical approximate algorithms can be identified.
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Abstract We present a non-anticipative learning- and scenario-based prediction-optimization (ScenPredOpt) framework that combines deep learning, heuristics, and mathematical solvers for solving combinatorial problems under uncertainty. Specifically, we transform neural machine translation frameworks to predict the optimal solutions of scenario-based multi-stage stochastic programs. The learning models are trained efficiently using the input and solution data of the multi-stage single-scenario deterministic problems. Then our ScenPredOpt framework creates a mapping from the inputs used in training into an output of predictions that are close to optimal solutions. We present a Non-anticipative Encoder-Decoder with Attention (NEDA) approach, which ensures the non-anticipativity property of multi-stage stochastic programs and, thus, time consistency by calibrating the learned information based on the problem’s scenario tree and adjusting the hidden states of the neural network. In our ScenPredOpt framework, the percent predicted variables used for the solution are iteratively reduced through a relaxation of the problem to eliminate infeasibility. Then, a linear relaxation-based heuristic is performed to further reduce the solution time. Finally, a mathematical solver is used to generate the complete solution. We present the results on two NP-Hard sequential optimization problems under uncertainty: stochastic multi-item capacitated lot-sizing and stochastic multistage multidimensional knapsack. The results show that the solution time can be reduced by a factor of 599 with an optimality gap of only 0.08%. We compare the results of the ScenPredOpt framework with cutting-edge exact and heuristic solution algorithms for the problems studied and find that our framework is more effective. Additionally, the computational results demonstrate that ScenPredOpt can solve instances with a larger number of items and scenarios than the trained ones. Our non-anticipative learning-optimization approach can be beneficial for stochastic programming problems involving binary variables that are solved repeatedly with various types of dimensions and similar decisions at each period.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
