Pooling Problems with Single-Flow Constraints

The pooling problem is a frequently studied extension of the traditional minimum cost flow problem, in which the composition of the flow is subject to restrictions. In a network consisting of three layers of nodes, the composition is given at the source layer. In the intermediate nodes, referred to as pools, the composition is a weighted average of the compositions in entering flow streams. The same is true at the sink layer, where upper bounds on the concentration of each component apply. Motivated by practical applications, and needs for heuristic methods for the standard pooling problem, the current work focuses on pooling problems where the flow graph is restricted to satisfy certain sparsity conditions. We consider in particular the requirements that each pool receives flow from at most one neighboring source, or sends flow to at most one neighboring sink. We prove that the pooling problem remains NP-hard after this and other similar extensions. It is also demonstrated how the single-flow constrained extensions can be modeled by means of mixed integer linear programming (MILP), without introducing bilinear terms. We also show that such MILP-models are useful for computing good feasible solutions to the original problem.acceptedVersio

A Review of Formal Methods applied to Machine Learning

We review state-of-the-art formal methods applied to the emerging field of the verification of machine learning systems. Formal methods can provide rigorous correctness guarantees on hardware and software systems. Thanks to the availability of mature tools, their use is well established in the industry, and in particular to check safety-critical applications as they undergo a stringent certification process. As machine learning is becoming more popular, machine-learned components are now considered for inclusion in critical systems. This raises the question of their safety and their verification. Yet, established formal methods are limited to classic, i.e. non machine-learned software. Applying formal methods to verify systems that include machine learning has only been considered recently and poses novel challenges in soundness, precision, and scalability. We first recall established formal methods and their current use in an exemplar safety-critical field, avionic software, with a focus on abstract interpretation based techniques as they provide a high level of scalability. This provides a golden standard and sets high expectations for machine learning verification. We then provide a comprehensive and detailed review of the formal methods developed so far for machine learning, highlighting their strengths and limitations. The large majority of them verify trained neural networks and employ either SMT, optimization, or abstract interpretation techniques. We also discuss methods for support vector machines and decision tree ensembles, as well as methods targeting training and data preparation, which are critical but often neglected aspects of machine learning. Finally, we offer perspectives for future research directions towards the formal verification of machine learning systems

Relaxations and discretizations for the pooling problem

The pooling problem is a folklore NP-hard global optimization problem that finds applications in industries such as petrochemical refining, wastewater treatment and mining. This paper assimilates the vast literature on this problem that is dispersed over different areas and gives new insights on prevalent techniques. We also present new ideas for computing dual bounds on the global optimum by solving high-dimensional linear programs. Finally, we propose discretization methods for inner approximating the feasible region and obtaining good primal bounds. Valid inequalities are derived for the discretized models, which are formulated as mixed integer linear programs. The strength of our relaxations and usefulness of our discretizations is empirically validated on random test instances. We report best known primal bounds on some of the large-scale instances

Uncertain demand prediction for guaranteed automated vehicle fleet performance

Mobility-on-demand (MoD) services offer a convenient and efficient transportation option, using technology to replace traditional modes. However, the flexibility of MoD services also presents challenges in controlling the system. One of the major issues is supply-demand imbalance, caused by uneven stochastic travel demand. To address this, it is crucial to predict the network behavior and proactively adapt to future travel demand.In this thesis, we present a stochastic model predictive controller (SMPC) that accounts for uncertainties in travel demand predictions. Our method make use of Gaussian Process Regression (GPR) to estimate passenger travel demand and predict time patterns with uncertainty bounds. The SMPC integrates these demand predictions into a receding horizon MoD optimization and uses a probabilistic constraining method with a user-defined confidence interval to guarantee constraint satisfaction. This result in a Chance Constrained Model Predictive Control (CCMPC) solution. Our approach has two benefits: incorporating travel demand uncertainty into the MoD optimization and the ability to relax the solution into a simpler Mixed-Integer Linear Program (MILP). Our simulation results demonstrate that this method reduces median customer wait time by 4% compared to using only the mean prediction from GPR. By adjusting the confidence bound, near-optimal performance can be achieved