Static Pricing Problems under Mixed Multinomial Logit Demand

Price differentiation is a common strategy for many transport operators. In this paper, we study a static multiproduct price optimization problem with demand given by a continuous mixed multinomial logit model. To solve this new problem, we design an efficient iterative optimization algorithm that asymptotically converges to the optimal solution. To this end, a linear optimization (LO) problem is formulated, based on the trust-region approach, to find a "good" feasible solution and approximate the problem from below. Another LO problem is designed using piecewise linear relaxations to approximate the optimization problem from above. Then, we develop a new branching method to tighten the optimality gap. Numerical experiments show the effectiveness of our method on a published, non-trivial, parking choice model

Aspects of quadratic optimization - nonconvexity, uncertainty, and applications

Quadratic Optimization (QO) has been studied extensively in the literature due to its application in real-life problems. This thesis deals with two complicated aspects of QO problems, namely nonconvexity and uncertainty. A nonconvex QO problem is intractable in general. The first part of this thesis presents methods to approximate a nonconvex QP problem. Another important aspect of a QO problem is taking into account uncertainties in the parameters since they are mostly approximated/estimated from data. The second part of the thesis contains analyses of two methods that deal with uncertainties in a convex QO problem, namely Static and Adjustable Robust Optimization problems. To test the methods proposed in this thesis, the following three real-life applications have been considered: pooling problem, portfolio problem, and norm approximation problem

Intervention Grouping Strategy for Multi-component Interconnected Systems:A Scalable Optimization Approach

The well-being of modern societies depends on the functioning of their infrastructure networks. During their service lives, infrastructure networks are subject to different stresses (e.g., deterioration, hazards, etc.). Interventions are performed to ensure the continuous fulfillment of the infrastructure's functional goals. To guarantee a high level of infrastructure availability and serviceability with minimal intervention costs, preventive intervention planning is essential.Finding the optimal grouping strategy of intervention activities is an NP-hard problem that is well studied in the literature and for which various economic models and optimization approaches are proposed. This research focuses on a new efficient optimization model to cope with the intervention grouping problem of interconnected multi-component systems. We propose a scalable two-step intervention grouping model based on a clustering technique. The clustering technique is formulated using Integer Linear Programing, which guarantees the convergence to global optimal solutions of the considered problem. The proposed optimization model can account for the interactions between multiple infrastructure networks and the impact on multiple stakeholders (e.g., society and infrastructure operators). The model can also accommodate different types of intervention, such as maintenance, removal, and upgrading.We show the performance of the proposed model using a demonstrative example. Results reveal a substantial reduction in net costs. In addition, the optimal intervention plan obtained in the analysis shows repetitive patterns, which indicates that a rolling horizon strategy could be adopted so that the analysis is only performed for a short time horizon

Robust Optimization using a new Volume-Based Clustering approach

We propose a new data-driven technique for constructing uncertainty sets for robust optimization problems. The technique captures the underlying structure of sparse data through volume-based clustering, resulting in less conservative solutions than most commonly used robust optimization approaches. This can aid management in making informed decisions under uncertainty, allowing a better understanding of the potential outcomes and risks associated with possible decisions. The paper demonstrates how clustering can be performed using any desired geometry and provides a mathematical optimization formulation for generating clusters and constructing the uncertainty set. In order to find an efficient solution to the problem, we explore different approaches since the method may be computationally expensive. This contribution to the field provides a novel data-driven approach to uncertainty set construction for robust optimization that can be applied to real-world scenarios

Efficient Sensitivity Analysis for Parametric Robust Markov Chains

We provide a novel method for sensitivity analysis of parametric robust Markov chains. These models incorporate parameters and sets of probability distributions to alleviate the often unrealistic assumption that precise probabilities are available. We measure sensitivity in terms of partial derivatives with respect to the uncertain transition probabilities regarding measures such as the expected reward. As our main contribution, we present an efficient method to compute these partial derivatives. To scale our approach to models with thousands of parameters, we present an extension of this method that selects the subset of $k$ parameters with the highest partial derivative. Our methods are based on linear programming and differentiating these programs around a given value for the parameters. The experiments show the applicability of our approach on models with over a million states and thousands of parameters. Moreover, we embed the results within an iterative learning scheme that profits from having access to a dedicated sensitivity analysis

Efficient Sensitivity Analysis for Parametric Robust Markov Chains

We provide a novel method for sensitivity analysis of parametric robust Markov chains. These models incorporate parameters and sets of probability distributions to alleviate the often unrealistic assumption that precise probabilities are available. We measure sensitivity in terms of partial derivatives with respect to the uncertain transition probabilities regarding measures such as the expected reward. As our main contribution, we present an efficient method to compute these partial derivatives. To scale our approach to models with thousands of parameters, we present an extension of this method that selects the subset of $k$ parameters with the highest partial derivative. Our methods are based on linear programming and differentiating these programs around a given value for the parameters. The experiments show the applicability of our approach on models with over a million states and thousands of parameters. Moreover, we embed the results within an iterative learning scheme that profits from having access to a dedicated sensitivity analysis.Comment: To be presented at CAV 202

LiBiT algorithm

The algorithm designed to solve static pricing problems under mixed multinomial logit deman

Static Pricing Problems under Mixed Multinomial Logit Demand

Price differentiation is a common strategy for many transport operators. In this paper, we study a static multiproduct price optimization problem with demand given by a continuous mixed multinomial logit model. To solve this new problem, we design an efficient iterative optimization algorithm that asymptotically converges to the optimal solution. To this end, a linear optimization (LO) problem is formulated, based on the trust-region approach, to find a "good" feasible solution and approximate the problem from below. Another LO problem is designed using piecewise linear relaxations to approximate the optimization problem from above. Then, we develop a new branching method to tighten the optimality gap. Numerical experiments show the effectiveness of our method on a published, non-trivial, parking choice model

When are static and adjustable robust optimization problems with constraint-wise uncertainty equivalent?

Adjustable Robust Optimization (ARO) yields, in general, better worst-case solutions than static Robust Optimization (RO). However, ARO is computationally more difficult than RO. In this paper, we derive conditions under which the worst-case objective values of ARO and RO problems are equal. We prove that if the uncertainty is constraint-wise and the adjustable variables lie in a compact set, then under one of the following sets of conditions robust solutions are optimal for the corresponding (ARO) problem: (i) the problem is fixed recourse and the uncertainty set is compact, (ii) the problem is convex with respect to the adjustable variables and concave with respect to the parameters defining constraint-wise uncertainty. Furthermore, if we have both constraint-wise and nonconstraint-wise uncertainty, under similar sets of assumptions we prove that there is an optimal decision rule for the Adjustable Robust Optimization problem that does not depend on the parameters defining constraint-wise uncertainty. Also, we show that for a class of problems, using affine decision rules that depend on both types of uncertain parameters yields the same optimal value as ones depending solely on the nonconstraint-wise uncertain parameter. Additionally, we provide several examples not only to illustrate our results, but also to show that the assumptions are crucial and omitting one of them can make the optimal worst-case objective values different