Abstract: Electoral systems are modified by individuals who have incentives to bias the rules for their political advantage (i.e., gerrymandering). To prevent gerrymandering, legislative institutions can rely on mathematical tools to guarantee democratic fairness and territorial contiguity. These tools have been successfully used in the past; however, there is a need to accommodate additional meanings of the term fairness within the electoral systems of modern democracies. In this paper, we present an optimization framework that considers multiple criteria for drawing districts and assigning the number of representatives. Besides some typical districting criteria (malapportionment and contiguity), we introduce novel criteria for ensuring territorial equilibrium and incentives for candidates to deploy their representation efforts fairly during their campaign and period in office. We test the method, which we denote as Multi-criteria Pen, in a recent and a forthcoming reform of the Chilean electoral system. The results show the potential of our tool to improve the current territorial design and offers insights on the motivations, objectives, and deficiencies of both reform plans.

Abstract: Convexification based on convex envelopes is ubiquitous in the non-linear optimization literature. Thanks to considerable efforts of the optimization community for decades, we are able to compute the convex envelopes of a considerable number of functions that appear in practice, and thus obtain tight and tractable approximations to challenging problems. We contribute to this line of work by considering a family of functions that, to the best of our knowledge, has not been considered before in the literature. We call this family ray-concave functions. We show sufficient conditions that allow us to easily compute closed-form expressions for the convex envelope of ray-concave functions over arbitrary polytopes. With these tools, we are able to provide new perspectives to previously known convex envelopes and derive a previously unknown convex envelope for a function that arises in probability contexts.

Abstract: Constructed facilities should be robust with respect to the loss of load-bearing elements due to abnormal events. Yet, strengthening structures to withstand such damage has a significant impact on construction costs. Strengthening costs should be justified by the threat and should result in smaller expected costs of progressive collapse. In regular frame structures, beams and columns compete for the strengthening budget. In this paper, we present a risk-based formulation to address the optimal design of regular plane frames under element loss conditions. We address the threat probabilities for which strengthening has better cost-benefit than usual design, for different frame configurations, and study the impacts of strengthening extent and cost. The risk-based optimization reveals optimum points of compromise between competing failure modes: local bending of beams, local crushing of columns, and global pancake collapse, for frames of different aspect ratios. The conceptual study is based on a simple analytical model for progressive collapse, but it provides relevant insight for the design and strengthening of real structures.

Abstract: In this work, we consider a risk-averse ultimate pit problem where the grade of the mineral is uncertain. We derive conditions under which we can generate a set of nested pits by varying the risk level instead of using revenue factors. We propose two properties that we believe are desirable for the problem: risk nestedness, which means the pits generated for different risk aversion levels should be contained in one another, and additive consistency, which states that preferences in terms of order of extraction should not change if independent sectors of the mine are added as precedences. We show that only an entropic risk measure satisfies these properties and propose a two-stage stochastic programming formulation of the problem, including an efficient approximation scheme to solve it. We illustrate our approach in a small self-constructed example, and apply our approximation scheme to a real-world section of the Andina mine, in Chile.

Abstract: We study the structure of a simple dynamic optimization problem consisting of one state and one control variable, from a physicist's point of view. By using an analogy to a physical model, we study this system in the classical and quantum frameworks. Classically, the dynamic optimization problem is equivalent to a classical mechanics constrained system, so we must use the Dirac method to analyze it in a correct way. We find that there are two second-class constraints in the model: one fix the momenta associated with the control variables, and the other is a reminder of the optimal control law. The dynamic evolution of this constrained system is given by the Dirac's bracket of the canonical variables with the Hamiltonian. This dynamic results to be identical to the unconstrained one given by the Pontryagin equations, which are the correct classical equations of motion for our physical optimization problem. In the same Pontryagin scheme, by imposing a closed-loop lambda-strategy, the optimality condition for the action gives a consistency relation, which is associated to the Hamilton-Jacobi-Bellman equation of the dynamic programming method. A similar result is achieved by quantizing the classical model. By setting the wave function Psi (x, t) = e(is(x,t)) in the quantum Schrodinger equation, a non-linear partial equation is obtained for the S function. For the right-hand side quantization, this is the Hamilton-Jacobi-Bellman equation, when S(x, t) is identified with the optimal value function. Thus, the Hamilton-Jacobi-Bellman equation in Bellman's maximum principle, can be interpreted as the quantum approach of the optimization problem. (C) 2017 Elsevier B.V. All rights reserved.

Abstract: We consider physics-informed neural networks (PINNs) (Raissiet al., 2019) for forward physical problems. In order to find optimal PINNs configuration, we introduce a hyper-parameter optimization (HPO) procedure via Gaussian processes-based Bayesian optimization. We apply the HPO to Helmholtz equation for bounded domains and conduct a thorough study, focusing on: (i) performance, (ii) the collocation points density r and (iii) the frequency kappa, confirming the applicability and necessity of the method. Numerical experiments are performed in two and three dimensions, including comparison to finite element methods.

Abstract: This paper investigates how the choice of stochastic approaches and distribution assumptions impacts strategic investment decisions in energy planning problems. We formulate a two-stage stochastic programming model assuming different distributions for the input parameters and show that there is significant discrepancy among the associated stochastic solutions and other robust solutions published in the literature. To remedy this sensitivity issue, we propose a combined machine learning and distributionally robust optimization (DRO) approach which produces more robust and stable strategic investment decisions with respect to uncertainty assumptions. DRO is applied to deal with ambiguous probability distributions and Machine Learning is used to restrict the DRO model to a subset of important uncertain parameters ensuring computational tractability. Finally, we perform an out-of-sample simulation process to evaluate solutions performances. The Swiss energy system is used as a case study all along the paper to validate the approach.

Abstract: The appointment scheduling problem (ASP) studies how to manage patient arrivals to a healthcare system to improve system performance. An important challenge occurs when some patients may not show up for an appointment. Although the ASP is well studied in the literature, the vast majority of the existing work does not consider the well-observed phenomenon that patient no-show is influenced by the appointment time, the usual decision variable in the ASP. This paper studies the ASP with random service time (exogenous uncertainty) with known distribution and patient decision-dependent no-show behavior (endogenous uncertainty). This problem belongs to the class of stochastic optimization with decision-dependent uncertainties. Such problems are notoriously difficult as they are typically nonconvex. We propose a stochastic projected gradient path (SPGP) method to solve the problem, which requires the development of a gradient estimator of the objective function-a nontrivial task, as the literature on gradient-based optimization algorithms for problems with decision-dependent uncertainty is very scarce and unsuitable for our model. Our method can solve the ASP problem under arbitrarily smooth show-up probability functions. We present solutions under different patterns of no-show behavior and demonstrate that breaking the assumption of constant show-up probability substantially changes the scheduling solutions. We conduct numerical experiments in a variety of settings to compare our results with those obtained with a distributionally robust optimization method developed in the literature. The cost reduction obtained with our method, which we call the value of distribution information, can be interpreted as how much the system performance can be improved by knowing the distribution of the service times, compared to not knowing it. We observe that the value of distribution information is up to 31% of the baseline cost, and that such value is higher when the system is crowded or/and the waiting time cost is relatively high.
Summary of Contribution: This paper studies an appointment scheduling problem under time-dependent patient no-show behavior, a situation commonly observed in practice. The problem belongs to the class of stochastic optimization problems with decision-dependent uncertainties in the operations research literature. Such problems are notoriously difficult to solve as a result of the lack of convexity, and the present case requires different techniques because of the presence of continuous distributions for the service times. A stochastic projected gradient path method, which includes the development of specialized techniques to estimate the gradient of the objective function, is proposed to solve the problem. For problems with a similar structure, the algorithm can be applied once the gradient estimator of the objective function is obtained. Extensive numerical studies are presented to demonstrate the quality of the solutions, the importance of modeling time-dependent no-shows in appointment scheduling, and the value of using distribution information about the service times.

Abstract: This paper develops a novel procedure to conduct a Freight Origin-Destination Synthesis (FODS) that jointly estimates the trip distribution, mode choice, and the empty trips by truck and rail that provide the best match to the observed freight traffic counts. Four models are integrated: (1) a gravity model for trip distribution, (2) a binary logit model for mode choice, (3) a Noortman and Van Es' model for truck, and (4) a Noortman and Van Es' model for rail empty trips. The estimation process entails an iterative minimization of a nonconvex objective function, the summation of squared errors of the estimated truck and rail traffic counts with respect to the five model parameters. Of the two methods tested to address the nonconvexity, an interior point method with a set of random starting points (Multi-Start algorithm) outperformed the Ordinary Least Squared (OLS) inference technique. The potential of this methodology is examined using a hypothetical example of developing a nationwide freight demand model for Bangladesh. This research improves the existing FODS techniques that use readily available secondary data such as traffic counts and link costs, allowing transportation planners to evaluate policy outcomes without needing expensive freight data collection. This paper presents the results, model validation, limitations, and future scope for improvements.

Abstract: Mine planning optimization aims at maximizing the profit obtained from extracting valuable ore. Beyond its theoretical complexity-the open-pit mining problem with capacity constraints reduces to a knapsack problem with precedence constraints, which is NP-hard-practical instances of the problem usually involve a large to very large number of decision variables, typically of the order of millions for large mines. Additionally, any comprehensive approach to mine planning ought to consider the underlying geostatistical uncertainty as only limited information obtained from drill hole samples of the mineral is initially available. In this regard, as blocks are extracted sequentially, information about the ore grades of blocks yet to be extracted changes based on the blocks that have already been mined. Thus, the problem lies in the class of multi-period large scale stochastic optimization problems with decision-dependent information uncertainty. Such problems are exceedingly hard to solve, so approximations are required. This paper presents an adaptive optimization scheme for multi-period production scheduling in open-pit mining under geological uncertainty that allows us to solve practical instances of the problem. Our approach is based on a rolling-horizon adaptive optimization framework that learns from new information that becomes available as blocks are mined. By considering the evolution of geostatistical uncertainty, the proposed optimization framework produces an operational policy that reduces the risk of the production schedule. Our numerical tests with mines of moderate sizes show that our rolling horizon adaptive policy gives consistently better results than a non-adaptive stochastic optimization formulation, for a range of realistic problem instances.