Inexact Stabilized Benders' Decomposition Approaches, with Application to Chance-Constrained Problems with Finite Support

We explore modifications of the standard cutting-plane approach for minimizing a convex nondifferentiable function, given by an oracle, over a combinatorial set, which is the basis of the celebrated (generalized) Benders' decomposition approach. Specifically, we combine stabilization—in two ways: via a trust region in the L1 norm, or via a level constraint—and inexact function computation (solution of the subproblems). Managing both features simultaneously requires a nontrivial convergence analysis; we provide it under very weak assumptions on the handling of the two parameters (target and accuracy) controlling the informative on-demand inexact oracle corresponding to the subproblem, strengthening earlier know results. This yields new versions of Benders' decomposition, whose numerical performance are assessed on a class of hybrid robust and chance-constrained problems that involve a random variable with an underlying discrete distribution, are convex in the decision variable, but have neither separable nor linear probabilistic constraints. The numerical results show that the approach has potential, especially for instances that are difficult to solve with standard techniques

The L-shaped method for large-scale mixed-integer waste management decision making problems

It is without a doubt that deciding upon strategic issues requires us to somehow anticipate and consider possible variations of the future. Unfortunately, when it comes to the actual modelling, the sheer size of the problems that accurately describe the uncertainty is often extremely hard to work with. This paper aims to describe a possible way of dealing with the issue of large-scale mixed integer models (in term of the number of possible future scenarios it can handle) for the studied waste management decision making problem. The algorithm is based on the idea of decomposing the overall problem alongside the different scenarios and solving these smaller problems instead. The use of the algorithm is demonstrated on a strategic waste management problem of choosing the optimal sites to build new incineration plants, while minimizing the expected cost of waste transport and processing. The uncertainty was modelled by 5,000 scenarios and the problem was solved to high accuracy using relatively modest means (in terms of computational power and needed software)

Incremental bundle methods using upper models

We propose a family of proximal bundle methods for minimizing sum-structured convex nondifferentiable functions which require two slightly uncommon assumptions, that are satisfied in many relevant applications: Lipschitz continuity of the functions and oracles which also produce upper estimates on the function values. In exchange, the methods: i) use upper models of the functions that allow to estimate function values at points where the oracle has not been called; ii) provide the oracles with more information about when the function computation can be interrupted, possibly diminishing their cost; iii) allow to skip oracle calls entirely for some of the component functions, not only at "null steps" but also at "serious steps"; iv) provide explicit and reliable a-posteriori estimates of the quality of the obtained solutions; v) work with all possible combinations of different assumptions on how the oracles deal with not being able to compute the function with arbitrary accuracy. We also discuss the introduction of constraints (or, more generally, of easy components) and use of (partly) aggregated models

Standard Bundle Methods: Untrusted Models and Duality

We review the basic ideas underlying the vast family of algorithms for nonsmooth convex optimization known as "bundle methods|. In a nutshell, these approaches are based on constructing models of the function, but lack of continuity of first-order information implies that these models cannot be trusted, not even close to an optimum. Therefore, many different forms of stabilization have been proposed to try to avoid being led to areas where the model is so inaccurate as to result in almost useless steps. In the development of these methods, duality arguments are useful, if not outright necessary, to better analyze the behaviour of the algorithms. Also, in many relevant applications the function at hand is itself a dual one, so that duality allows to map back algorithmic concepts and results into a "primal space" where they can be exploited; in turn, structure in that space can be exploited to improve the algorithms' behaviour, e.g. by developing better models. We present an updated picture of the many developments around the basic idea along at least three different axes: form of the stabilization, form of the model, and approximate evaluation of the function

Intelligent Traffic Management: From Practical Stochastic Path Planning to Reinforcement Learning Based City-Wide Traffic Optimization

This research focuses on intelligent traffic management including stochastic path planning and city scale traffic optimization. Stochastic path planning focuses on finding paths when edge weights are not fixed and change depending on the time of day/week. Then we focus on minimizing the running time of the overall procedure at query time utilizing precomputation and approximation. The city graph is partitioned into smaller groups of nodes and represented by its exemplar. In query time, source and destination pairs are connected to their respective exemplars and the path between those exemplars is found. After this, we move toward minimizing the city wide traffic congestion by making structural changes include changing the number of lanes, using ramp metering, varying speed limit, and modifying signal timing is possible. We propose a multi agent reinforcement learning (RL) framework for improving traffic flow in city networks. Our framework utilizes two level learning: a) each single agent learns the initial policy and b) multiple agents (changing the environment at the same time) update their policy based on the interaction with the dynamic environment and in agreement with other agents. The goal of RL agents is to interact with the environment to learn the optimal modification for each road segment through maximizing the cumulative reward over the set of possible actions in state space