arXiv subjects

Trivikram Dokka

Publications and source records attributed to Trivikram Dokka.

16 recordsLinked to original sources

arXiv2025

Algorithms for min-buying in networks

The paper is motivated by pricing decisions faced by forecourt fuel retailers across their outlets on a road network. Through our modelling approach we are able adapt the network structure to a bipartite graph with demand nodes representing volumes of fuel from customers using a specific route that connects to the seller's outlet nodes that intersect that route on the network. Customers may have their demand satisfied at the lowest priced competitor on their route. However, the seller can satisfy some or all of this demand by matching or beating this price via one of their outlets intersecting the route. We give a practical extension to min-pricing by considering a binary logit variant for buyers evaluating the choice between two sellers. We derive two MIP formulations for min-buying in the case of general demand. We also propose several constructive heuristics, based on insertion and selection operations, suitable for problem instances beyond the scope of the exact methods. The performance of models and algorithms are evaluated in a numerical study and develop insights from the results. Importantly, we are able to highlight the value of price-matching decisions under buyer demand sensitivity.

math.OC↗

arXiv2023

Parametric Distributionally Robust Optimisation Models for Budgeted Multi-period Newsvendor Problems

In this paper, we consider a static, multi-period newsvendor model under a budget constraint. In the case where the true demand distribution is known, we develop a heuristic algorithm to solve the problem. By comparing this algorithm with off-the-shelf solvers, we show that it generates near-optimal solutions in a short time. We then consider a scenario in which limited information on the demand distribution is available. It is assumed, however, that the true demand distribution lies within some given family of distributions and that samples can be obtained from it. We consider the cases of normal and Poisson demands. For each case, we show that using maximum likelihood estimates in place of the true parameters can lead to poor estimates of the true cost associated with an order quantity. Hence, we make use of likelihood inference to develop confidence sets for the true parameters. These are used as ambiguity sets in a distributionally robust model, where we enforce that the worst-case distribution lies in the same family as the true distribution. We solve these models by discretising the ambiguity set and reformulating them as piecewise linear models. We show that these models quickly become large as the ambiguity set grows, resulting in long computation times. To overcome this, we propose a heuristic cutting surface algorithm that exploits theoretical properties of the objective function to reduce the size of the ambiguity set. We illustrate that our cutting surface algorithm solves orders of magnitude faster than the piecewise linear model, while generating very near-optimal solutions.

math.OC↗

arXiv2022

Pricing and Electric Vehicle Charging Equilibria

We study equilibria in an Electric Vehicle (EV) charging game, a cost minimization game inherent to decentralized charging control strategy for EV power demand management. In our model, each user optimizes its total cost which is sum of direct power cost and the indirect dissatisfaction cost. We show that, taking player specific price independent dissatisfaction cost in to account, contrary to popular belief, herding only happens at lower EV uptake. Moreover, this is true for both linear and logistic dissatisfaction functions. We study the question of existence of price profiles to induce a desired equilibrium. We define two types of equilibria, distributed and non-distributed equilibria, and show that under logistic dissatisfaction, only non-distributed equilibria are possible by feasibly setting prices. In linear case, both type of equilibria are possible but price discrimination is necessary to induce distributed equilibria. Finally, we show that in the case of symmetric EV users, mediation cannot improve upon Nash equilibria.

econ.TH↗

arXiv2022

Robust Markov decision processes under parametric transition distributions

This paper considers robust Markov decision processes under parametric transition distributions. We assume that the true transition distribution is uniquely specified by some parametric distribution, and explicitly enforce that the worst-case distribution from the model is uniquely specified by a distribution in the same parametric family. After formulating the parametric robust model, we focus on developing algorithms for carrying out the robust Bellman updates required to complete robust value iteration. We first formulate the update as a linear program by discretising the ambiguity set. Since this model scales poorly with problem size and requires large amounts of pre-computation, we develop two additional algorithms for solving the robust Bellman update. Firstly, we present a cutting surface algorithm for solving this linear program in a shorter time. This algorithm requires the same pre-computation, but only ever solves the linear program over small subsets of the ambiguity set. Secondly, we present a novel projection-based bisection search algorithm that completely eliminates the need for discretisation and does not require any pre-computation. We test our algorithms extensively on a dynamic multi-period newsvendor problem under binomial and Poisson demands. In addition, we compare our methods with the non-parametric phi-divergence based methods from the literature. We show that our projection-based algorithm completes robust value iteration significantly faster than our other two parametric algorithms, and also faster than its non-parametric equivalent.

math.OC↗

arXiv2022

Distributionally Robust Resource Planning Under Binomial Demand Intakes

In this paper, we consider a distributionally robust resource planning model inspired by a real-world service industry problem. In this problem, there is a mixture of known demand and uncertain future demand. Prior to having full knowledge of the demand, we must decide upon how many jobs we plan to complete on each day in the planning horizon. Any jobs that are not completed by the end of their due date incur a cost and become due the following day. We present two distributionally robust optimisation (DRO) models for this problem. The first is a non-parametric model with a phi-divergence based ambiguity set. The second is a parametric model, where we treat the number of uncertain jobs due on each day as a binomial random variable with an unknown success probability. We reformulate the parametric model as a mixed integer program and find that it scales poorly with the sizes of the ambiguity and uncertainty sets. Hence, we make use of theoretical properties of the binomial distribution to derive fast heuristics based on dimension reduction. One is based on cutting surface algorithms commonly seen in the DRO literature. The other operates on a small subset of the uncertainty set for the future demand. We perform extensive computational experiments to establish the performance of our algorithms. We compare decisions from the parametric and non-parametric models, to assess the benefit of including the binomial information.

math.OC↗

arXiv2021

Data-driven Heuristics for DC optimal transmission switching problem

The goal of Optimal Transmission Switching (OTS) problem for power systems is to identify a topology of the power grid that minimizes the cost of the system operation while satisfying the operational and physical constraints. Among the most popular methods to solve OTS is to construct approximation via integer linear programming formulations, which often come with big-M inequalities. These big-M inequalities increase, considerably, the difficulty of solving the resulting formulations. Moreover, choosing big-M values optimally is as hard as solving OTS itself. In this paper, we devise two data-driven big-M bound strengthening methods which take network structure, power demands and generation costs into account. We illustrate the robustness of our methods to load changes and impressive runtime improvements of mixed-integer solvers achieved by our methods with extensive experiments on benchmark instances. The speedup by one of the proposed methods is almost 13 times with respect to the exact method.

math.OC↗

arXiv2021

Understanding controlled EV charging impacts using scenario-based forecasting models

Electrification of transport is a key strategy in reducing carbon emissions. Many countries have adopted policies of complete but gradual transformation to electric vehicles (EVs). However, mass EV adoption also means a spike in load (kW), which in turn can disrupt existing electricity infrastructure. Smart or controlled charging is widely seen as a potential solution to alleviate this stress on existing networks. Learning from the recent EV trials in the UK and elsewhere we take into account two key aspects which are largely ignored in current research: EVs actually charging at any given time and wide range of EV types, especially battery capacity-wise. Taking a minimalistic scenario-based approach, we study forecasting models for mean number of active chargers and mean EV consumption for distinct scenarios. Focusing on residential charging the models we consider range from simple regression models to more advanced machine and deep learning models such as XGBoost and LSTMs. We then use these models to evaluate the impacts of different levels of future EV penetration on a specimen distribution transformer that captures typical real-world scenarios. In doing so, we also initiate the study of different types of controlled charging when fully controlled charging is not possible. This aligns with the outcomes from recent trials which show that a sizeable proportion of EV owners may not prefer fully controlled centralized charging. We study two possible control regimes and show that one is more beneficial from load-on-transformer point of view, while the other may be preferred for other objectives. We show that a minimum of 60% control is required to ensure that transformers are not overloaded during peak hours.

eess.SP↗

arXiv2020

Automatic Generation of Algorithms for Black-Box Robust Optimisation Problems

We develop algorithms capable of tackling robust black-box optimisation problems, where the number of model runs is limited. When a desired solution cannot be implemented exactly the aim is to find a robust one, where the worst case in an uncertainty neighbourhood around a solution still performs well. This requires a local maximisation within a global minimisation. To investigate improved optimisation methods for robust problems, and remove the need to manually determine an effective heuristic and parameter settings, we employ an automatic generation of algorithms approach: Grammar-Guided Genetic Programming. We develop algorithmic building blocks to be implemented in a Particle Swarm Optimisation framework, define the rules for constructing heuristics from these components, and evolve populations of search algorithms. Our algorithmic building blocks combine elements of existing techniques and new features, resulting in the investigation of a novel heuristic solution space. As a result of this evolutionary process we obtain algorithms which improve upon the current state of the art. We also analyse the component level breakdowns of the populations of algorithms developed against their performance, to identify high-performing heuristic components for robust problems.

math.OC↗

arXiv2020

An Efficient Approach to Distributionally Robust Network Capacity Planning

In this paper, we consider a network capacity expansion problem in the context of telecommunication networks, where there is uncertainty associated with the expected traffic demand. We employ a distributionally robust stochastic optimization (DRSO) framework where the ambiguity set of the uncertain demand distribution is constructed using the moments information, the mean and variance. The resulting DRSO problem is formulated as a bilevel optimization problem. We develop an efficient solution algorithm for this problem by characterizing the resulting worst-case two-point distribution, which allows us to reformulate the original problem as a convex optimization problem. In computational experiments the performance of this approach is compared to that of the robust optimization approach with a discrete uncertainty set. The results show that solutions from the DRSO model outperform the robust optimization approach on highly risk-averse performance metrics, whereas the robust solution is better on the less risk-averse metric.

math.OC↗

arXiv2020

Particle Swarm Metaheuristics for Robust Optimisation with Implementation Uncertainty

We consider global non-convex optimisation problems under uncertainty. In this setting, it is not possible to implement a desired solution exactly. Instead, any other solution within some distance to the intended solution may be implemented. The aim is to find a robust solution, i.e., one where the worst possible solution nearby still performs as well as possible. Problems of this type exhibit another maximisation layer to find the worst case solution within the minimisation level of finding a robust solution, which makes them harder to solve than classic global optimisation problems. So far, only few methods have been provided that can be applied to black-box problems with implementation uncertainty. We improve upon existing techniques by introducing a novel particle swarm based framework which adapts elements of previous approaches, combining them with new features in order to generate more effective techniques. In computational experiments, we find that our new method outperforms state of the art comparator heuristics in almost 80% of cases.

math.OC↗

arXiv2019

The Multi-level Bottleneck Assignment Problem: Complexity and Solution Methods

We study the multi-level bottleneck assignment problem (MBA), which has important applications in scheduling and quantitative finance. Given a weight matrix, the task is to rearrange entries in each column such that the maximum sum of values in each row is as small as possible. We analyze the complexity of this problem in a generalized setting, where there are restrictions in how values in columns can be permuted. We present a lower bound on its approximability by giving a non-trivial gap reduction from three-dimensional matching to MBA. To solve MBA, a greedy method has been used in the literature. We present new solution methods based on an extension of the greedy method, an integer programming formulation, and a column generation heuristic. In computational experiments we show that it is possible to outperform the standard greedy approach by around 10% on random instances.

cs.CC↗

arXiv2019

Approximate policy iteration using neural networks for storage problems

We consider the stochastic single node energy storage problem (SNES) and revisit Approximate Policy Iteration (API) to solve SNES. We show that the performance of API can be boosted by using neural networks as an approximation architecture at the policy evaluation stage. To achieve this, we use a model different to that in literature with aggregate variables reducing the dimensionality of the decision vector, which in turn makes it viable to use neural network predictions in the policy improvement stage. We show that performance improvement by neural networks is even more significant in the case when charging efficiency of storage systems is low.

math.OC↗

arXiv2019

Mixed Uncertainty Sets for Robust Combinatorial Optimization

In robust optimization, the uncertainty set is used to model all possible outcomes of uncertain parameters. In the classic setting, one assumes that this set is provided by the decision maker based on the data available to her. Only recently it has been recognized that the process of building useful uncertainty sets is in itself a challenging task that requires mathematical support. In this paper, we propose an approach to go beyond the classic setting, by assuming multiple uncertainty sets to be prepared, each with a weight showing the degree of belief that the set is a "true" model of uncertainty. We consider theoretical aspects of this approach and show that it is as easy to model as the classic setting. In an extensive computational study using a shortest path problem based on real-world data, we auto-tune uncertainty sets to the available data, and show that with regard to out-sample performance, the combination of multiple sets can give better results than each set on its own.

math.OC↗

arXiv2018

Algorithms and Uncertainty Sets for Data-Driven Robust Shortest Path Problems

We consider robust shortest path problems, where the aim is to find a path that optimizes the worst-case performance over an uncertainty set containing all relevant scenarios for arc costs. The usual approach for such problems is to assume this uncertainty set given by an expert who can advise on the shape and size of the set. Following the idea of data-driven robust optimization, we instead construct a range of uncertainty sets from the current literature based on real-world traffic measurements provided by the City of Chicago. We then compare the performance of the resulting robust paths within and outside the sample, which allows us to draw conclusions what the most suited uncertainty set is. Based on our experiments, we then focus on ellipsoidal uncertainty sets, and develop a new solution algorithm that significantly outperforms a state-of-the-art solver.

math.OC↗

arXiv2017

Robust toll pricing: A novel approach

We study a robust toll pricing problem where toll setters and users have different level of information when taking their decisions. Toll setters do not have full information on the costs of the network and rely on historical information when determining toll rates, whereas users decide on the path to use from origin to destination knowing toll rates and having, in addition, more accurate traffic data. Toll setters often also face constraints on price experimentation which means less opportunity for price revision. Motivated by this we propose a novel robust pricing methodology for fixing prices where we take non-adversarial view of nature different from the existing robust approaches. We show that our non-adversarial robustness results in less conservative pricing decisions compared to traditional adversarial nature setting. We start by first considering a single origin-destination parallel network in this new robust setting and formulate the robust toll pricing problem as a distributionally robust optimization problem, for which we develop an exact algorithm based on a mixed-integer programming formulation and a heuristic based on two-point support distribution. We further extend our formulations to more general networks and show how our algorithms can be adapted for the general networks. Finally, we illustrate the usefulness of our approach by means of numerical experiments both on randomly generated networks and on the data recorded on the road network of the city of Chicago.

math.OC↗

arXiv2017

An Experimental Comparison of Uncertainty Sets for Robust Shortest Path Problems

Through the development of efficient algorithms, data structures and preprocessing techniques, real-world shortest path problems in street networks are now very fast to solve. But in reality, the exact travel times along each arc in the network may not be known. This lead to the development of robust shortest path problems, where all possible arc travel times are contained in a so-called uncertainty set of possible outcomes. Research in robust shortest path problems typically assumes this set to be given, and provides complexity results as well as algorithms depending on its shape. However, what can actually be observed in real-world problems are only discrete raw data points. The shape of the uncertainty is already a modelling assumption. In this paper we test several of the most widely used assumptions on the uncertainty set using real-world traffic measurements provided by the City of Chicago. We calculate the resulting different robust solutions, and evaluate which uncertainty approach is actually reasonable for our data. This anchors theoretical research in a real-world application and allows us to point out which robust models should be the future focus of algorithmic development.

math.OC↗