This paper proposes a new class of real-time optimization schemes to overcome system-model mismatch of uncertain processes. This work's novelty lies in integrating derivative-free optimization schemes and multi-fidelity Gaussian processes within a Bayesian optimization framework. The proposed scheme uses two Gaussian processes for the stochastic system, one emulates the (known) process model, and another, the true system through measurements. In this way, low fidelity samples can be obtained via a model, while high fidelity samples are obtained through measurements of the system. This framework captures the system's behavior in a non-parametric fashion while driving exploration through acquisition functions. The benefit of using a Gaussian process to represent the system is the ability to perform uncertainty quantification in real-time and allow for chance constraints to be satisfied with high confidence. This results in a practical approach that is illustrated in numerical case studies, including a semi-batch photobioreactor optimization problem.
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We investigate a machine learning approach to option Greeks approximation based on Gaussian process (GP) surrogates. The method takes in noisily observed option prices, fits a nonparametric input-output map and then analytically differentiates the latter to obtain the various price sensitivities. Our motivation is to compute Greeks in cases where direct computation is expensive, such as in local volatility models, or can only ever be done approximately. We provide a detailed analysis of numerous aspects of GP surrogates, including choice of kernel family, simulation design, choice of trend function and impact of noise. We further discuss the application to Delta hedging, including a new Lemma that relates quality of the Delta approximation to discrete-time hedging loss. Results are illustrated with two extensive case studies that consider estimation of Delta, Theta and Gamma and benchmark approximation quality and uncertainty quantification using a variety of statistical metrics. Among our key take-aways are the recommendation to use Matern kernels, the benefit of including virtual training points to capture boundary conditions, and the significant loss of fidelity when training on stock-path-based datasets.
Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, with even small approximations of components of a problem translating into large errors in the solutions. We specify conditions under which approximations are well behaved in the sense of minimizers, stationary points, and level-sets and this leads to a framework of consistent approximations. The framework is developed for a broad class of composite problems, which are neither convex nor smooth. We demonstrate the framework using examples from stochastic optimization, neural-network based machine learning, distributionally robust optimization, penalty and augmented Lagrangian methods, interior-point methods, homotopy methods, smoothing methods, extended nonlinear programming, difference-of-convex programming, and multi-objective optimization. An enhanced proximal method illustrates the algorithmic possibilities. A quantitative analysis supplements the development by furnishing rates of convergence.
Multi-fidelity modeling and calibration are data fusion tasks that ubiquitously arise in engineering design. In this paper, we introduce a novel approach based on latent-map Gaussian processes (LMGPs) that enables efficient and accurate data fusion. In our approach, we convert data fusion into a latent space learning problem where the relations among different data sources are automatically learned. This conversion endows our approach with attractive advantages such as increased accuracy, reduced costs, flexibility to jointly fuse any number of data sources, and ability to visualize correlations between data sources. This visualization allows the user to detect model form errors or determine the optimum strategy for high-fidelity emulation by fitting LMGP only to the subset of the data sources that are well-correlated. We also develop a new kernel function that enables LMGPs to not only build a probabilistic multi-fidelity surrogate but also estimate calibration parameters with high accuracy and consistency. The implementation and use of our approach are considerably simpler and less prone to numerical issues compared to existing technologies. We demonstrate the benefits of LMGP-based data fusion by comparing its performance against competing methods on a wide range of examples.
We study timed systems in which some timing features are unknown parameters. Parametric timed automata (PTAs) are a classical formalism for such systems but for which most interesting problems are undecidable. Notably, the parametric reachability emptiness problem, i.e., the emptiness of the parameter valuations set allowing to reach some given discrete state, is undecidable. Lower-bound/upper-bound parametric timed automata (L/U-PTAs) achieve decidability for reachability properties by enforcing a separation of parameters used as upper bounds in the automaton constraints, and those used as lower bounds. In this paper, we first study reachability. We exhibit a subclass of PTAs (namely integer-points PTAs) with bounded rational-valued parameters for which the parametric reachability emptiness problem is decidable. Using this class, we present further results improving the boundary between decidability and undecidability for PTAs and their subclasses such as L/U-PTAs. We then study liveness. We prove that: (1) the existence of at least one parameter valuation for which there exists an infinite run in an L/U-PTA is PSPACE-complete; (2) the existence of a parameter valuation such that the system has a deadlock is however undecidable; (3) the problem of the existence of a valuation for which a run remains in a given set of locations exhibits a very thin border between decidability and undecidability.
We address the problem of production planning and distribution in multi-echelon supply chains. We consider uncertain demands and lead times which makes the problem stochastic and non-linear. A Markov Decision Process formulation and a Non-linear Programming model are presented. As a sequential decision-making problem, Deep Reinforcement Learning (RL) is a possible solution approach. This type of technique has gained a lot of attention from Artificial Intelligence and Optimization communities in recent years. Considering the good results obtained with Deep RL approaches in different areas there is a growing interest in applying them in problems from the Operations Research field. We have used a Deep RL technique, namely Proximal Policy Optimization (PPO2), to solve the problem considering uncertain, regular and seasonal demands and constant or stochastic lead times. Experiments are carried out in different scenarios to better assess the suitability of the algorithm. An agent based on a linearized model is used as a baseline. Experimental results indicate that PPO2 is a competitive and adequate tool for this type of problem. PPO2 agent is better than baseline in all scenarios with stochastic lead times (7.3-11.2%), regardless of whether demands are seasonal or not. In scenarios with constant lead times, the PPO2 agent is better when uncertain demands are non-seasonal (2.2-4.7%). The results show that the greater the uncertainty of the scenario, the greater the viability of this type of approach.
This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.
The difficulty in specifying rewards for many real-world problems has led to an increased focus on learning rewards from human feedback, such as demonstrations. However, there are often many different reward functions that explain the human feedback, leaving agents with uncertainty over what the true reward function is. While most policy optimization approaches handle this uncertainty by optimizing for expected performance, many applications demand risk-averse behavior. We derive a novel policy gradient-style robust optimization approach, PG-BROIL, that optimizes a soft-robust objective that balances expected performance and risk. To the best of our knowledge, PG-BROIL is the first policy optimization algorithm robust to a distribution of reward hypotheses which can scale to continuous MDPs. Results suggest that PG-BROIL can produce a family of behaviors ranging from risk-neutral to risk-averse and outperforms state-of-the-art imitation learning algorithms when learning from ambiguous demonstrations by hedging against uncertainty, rather than seeking to uniquely identify the demonstrator's reward function.
Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.
Simultaneous Localization And Mapping (SLAM) is a fundamental problem in mobile robotics. While point-based SLAM methods provide accurate camera localization, the generated maps lack semantic information. On the other hand, state of the art object detection methods provide rich information about entities present in the scene from a single image. This work marries the two and proposes a method for representing generic objects as quadrics which allows object detections to be seamlessly integrated in a SLAM framework. For scene coverage, additional dominant planar structures are modeled as infinite planes. Experiments show that the proposed points-planes-quadrics representation can easily incorporate Manhattan and object affordance constraints, greatly improving camera localization and leading to semantically meaningful maps. The performance of our SLAM system is demonstrated in //youtu.be/dR-rB9keF8M .
Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space, such as the simplex, the time-discretisation error can dominate when we are near the boundary of the space. We demonstrate that while current SGMCMC methods for the simplex perform well in certain cases, they struggle with sparse simplex spaces; when many of the components are close to zero. However, most popular large-scale applications of Bayesian inference on simplex spaces, such as network or topic models, are sparse. We argue that this poor performance is due to the biases of SGMCMC caused by the discretization error. To get around this, we propose the stochastic CIR process, which removes all discretization error and we prove that samples from the stochastic CIR process are asymptotically unbiased. Use of the stochastic CIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.
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