Inverse Optimal Approach Research Articles

Flexibility Characterization of Sustainable Power Systems in Demand Space: A Data-Driven Inverse Optimization Approach

Flexibility Characterization of Sustainable Power Systems in Demand Space: A Data-Driven Inverse Optimization Approach

Inverse Optimization for Routing Problems

We propose a method for learning decision makers’ behavior in routing problems using inverse optimization (IO). The IO framework falls into the supervised learning category and builds on the premise that the target behavior is an optimizer of an unknown cost function. This cost function is to be learned through historical data, and in the context of routing problems, can be interpreted as the routing preferences of the decision makers. In this view, the main contributions of this study are to propose an IO methodology with a hypothesis function, loss function, and stochastic first-order algorithm tailored to routing problems. We further test our IO approach in the Amazon Last Mile Routing Research Challenge, where the goal is to learn models that replicate the routing preferences of human drivers, using thousands of real-world routing examples. Our final IO-learned routing model achieves a score that ranks second compared with the 48 models that qualified for the final round of the challenge. Our examples and results showcase the flexibility and real-world potential of the proposed IO methodology to learn from decision-makers’ decisions in routing problems. History: This paper has been accepted for the Transportation Science Special Issue on TSL Conference 2023. Funding: This work was supported by the European Research Council [TRUST-949796].

An assessment of numerical and geometrical quality of bases on surface fitting on Powell–Sabin triangulations

It is well known that the problem of fitting a dataset by using a spline surface minimizing an energy functional can be carried out by solving a linear system. Such a linear system strongly depends on the underlying functional space and, particularly, on the basis considered. Some papers in the literature study the numerical behavior and processing of the above-mentioned linear systems in specific cases. The bases that have local support and constitute a partition of unity have been shown to be interesting in the frame of geometric problems. In this work, we investigate the numerical effects of considering these bases in the quadratic Powell–Sabin spline space. Specifically, we present a direct approach to explore different preconditioning strategies and assess whether the already known ‘good’ bases also possess favorable numerical properties. Additionally, we introduce an inverse optimization approach based on a nonlinear optimization model to identify new bases that exhibit both good geometric and numerical properties.

Stabilizing Grand Cooperation via Cost Adjustment: An Inverse Optimization Approach

For an unbalanced cooperative game, its grand coalition can be stabilized by some instruments, such as subsidization and penalization, that impose new cost terms to certain coalitions. In this paper, we study an alternative instrument, referred to as cost adjustment, that does not need to impose any new coalition-specific cost terms. Specifically, our approach is to adjust existing cost coefficients of the game under which (i) the game becomes balanced so that the grand coalition becomes stable, (ii) a desired way of cooperation is optimal for the grand coalition to adopt, and (iii) the total cost to be shared by the grand coalition is within a prescribed range. Focusing on a broad class of cooperative games, known as integer minimization games, we formulate the problem on how to optimize the cost adjustment as a constrained inverse optimization problem. We prove [Formula: see text]-hardness and derive easy-to-check feasibility conditions for the problem. Based on two linear programming reformulations, we develop two solution algorithms. One is a cutting-plane algorithm, which runs in polynomial time when the corresponding separation problem is polynomial time solvable. The other needs to explicitly derive all the inequalities of a linear program, which runs in polynomial time when the linear program contains only a polynomial number of inequalities. We apply our models and solution algorithms to two typical unbalanced games, including a weighted matching game and an uncapacitated facility location game, showing that their optimal cost adjustments can be obtained in polynomial time. History: Accepted by Area Editor Andrea Lodi for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72022018 and 72091210]; the Research Grants Council of the Hong Kong SAR, China [Grant 16210020]; Hong Kong Polytechnic University [Grant P0032007]; and the Youth Innovation Promotion Association, Chinese Academy of Sciences [Grant 2021454]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0268 .

FREEDOM: Validated Method for Rapid Assessment of Incipient Faults of Aerospace Systems

Model-based fault detection and isolation (FDI) methods allow to infer the health status of complex aerospace systems through a large quantity of data acquired in flight and evaluations of numerical models of the equipment. This results in an intensive computational procedure that can be addressed only grounding the aircraft. We introduce an original methodology to sensitively accelerate FDI by reducing the computational demand to identify the health status of the aircraft. Our scheme FREEDOM (Fast Reliability Estimate and Incipient Fault Detection of Multiphysics Aerospace Systems) proposes an original combination of a novel two-step compression strategy to compute offline a synthesized representation of the dynamical response of the system and uses an inverse Bayesian optimization approach to infer online the level of damage determined by multiple fault modes affecting the equipment. We demonstrate and validate FREEDOM against numerical and physical experiments for the case of an electromechanical actuator employed for secondary flight controls. Particular attention is dedicated to simultaneous incipient mechanical and electrical faults considering different experimental settings. The outcomes validate our FDI strategy, which permits to achieve the accurate identification of complex damages outperforming the computational time of state-of-the-art algorithms by two orders of magnitude.

A small-gain approach to inverse optimal adaptive control of nonlinear systems with unmodeled dynamics

This work formulates and solves the problem of inverse optimal adaptive control for nonlinear systems with parametric and dynamic uncertainties. A concept and sufficient conditions for solving this problem are first introduced and derived based on adaptive control Lyapunov function method. One difficulty obstructing current works to solve this problem is that inverse optimality and stability of the closed-loop system are not necessarily achieved simultaneously in the presence of both parametric uncertainties and unmodeled dynamics. To this end, a new auxiliary system and a cost functional that puts integral penalty on the state, control, parameter estimate and unmodeled dynamics are proposed to develop a new inverse optimal adaptive feedback approach. Then a small-gain approach is proposed to establish the links between inverse optimality and stability of the closed-loop system. It is proved that if inverse optimality is achieved and the small-gain condition holds, the closed-loop system is uniformly bounded and the system state converges to an arbitrarily small neighborhood of the origin. Finally, an example is given to illustrate the obtained results.

Inverse optimization of integer programming games for parameter estimation arising from competitive retail location selection

When determining store locations, competing retailers must take customers’ store choice into consideration. Customers predominantly select which store to visit based on price, accessibility, and convenience. Incumbent retailers can estimate the weight of these factors (customer attraction parameters) using granular historical data. Their location decision under full information and simultaneous competition translates into an integer programming game. Unlike incumbents, new entrants lack this detailed information; however, they can observe the resulting location structure of incumbents. Assuming the observed location structure is (near-)optimal for all incumbent retailers, a new entrant can use these observations to estimate customer attraction parameters. To facilitate this estimation, we propose an “inverse optimization approach” for integer programming games (IPGs), enabling a new entrant to identify parameters that lead to the observed equilibrium solutions. We solve this “inverse IPG” via decomposition by solving a master problem and a subproblem. The master problem identifies parameter combinations for which the observations represent (approximate) Nash equilibria compared with optimal solutions enumerated in the subproblem. This row-generation approach extends prior methods for inverse integer optimization to competitive settings with (approximate) equilibria.We compare the decision-making of new entrants selecting locations based on scenarios, or information about the underlying distribution of customer attraction parameters (expected values), with new entrants using inversely estimated parameters for their location decisions. New entrants who rely on inversely optimized parameters can improve their profits. On average over a large set of synthetic numerical experiments, we observe improvements of 4–11%. This benefit can be realized with as little as one or two observations, yet additional observations help to increase prediction reliability significantly.

Inverse data envelopment analysis optimization approaches with flexible measures

PurposeThe purpose of this study is to estimate inputs (outputs) and flexible measures when outputs (inputs) are changed provided that the relative efficiency values remain without change.Design/methodology/approachA novel inverse data envelopment analysis (DEA) approach with flexible measures is proposed in this research to assess inputs (outputs) and flexible measures when outputs (inputs) are perturbed on condition that the relative efficiency scores remain unchanged. Furthermore, flexible inverse DEA approaches proposed in this study are used for a numerical example from the literature and an application of Iranian banking industry to clarify and validate them.FindingsThe findings show that including flexible measures into the investigation effects on the changes of performance measures estimated and leads to more reasonable achievements.Originality/valueThe traditional inverse DEA models usually investigate the changes of some determinate input-output factors for the changes of other given input-output indicators assuming that the efficiency values are preserved. However, there are situations that the changes of performance measures should be tackled while some measures, called flexible measures, can play either input or output roles. Accordingly, inverse DEA optimization models with flexible measures are rendered in this paper to address these issues.

Novel procedure to determine the stress-strain relation of Lithium-ion Batteries through indentation test

This study proposes a novel approach to convert the load–depth curve measured in the experimental indentation test into stress–strain curves. Due to the lack of the tensile test results for the cylindrical lithium-ion battery (LIB) cell, a combination of the analytical analysis and the inverse optimization approach is first conducted as a reference point to determine the elastoplastic constitutive equation, which can predict the response of cylindrical LIB cell cells under the cylindrical indenter. The obtained load-depth curves from the robust optimization algorithm are in good agreement with the experimental loading–unloading curves. This novel approach includes a new definition of the contact area between the cylindrical indenter and the cylindrical LIB cell. Finite element models validate the measured contact area values from the analytical approach. The analytical approach in this study can accurately capture the stress–strain curve validated with optimization results. Afterward, different single indentation tests are conducted to estimate the reduction of Young’s modulus through the increasing indentation depth. Noteworthy, a definition of the indentation strain can physically interpret the indentation strain in the case of two crossed cylinders. The cell voltage is measured during the indentation test to detect the internal short circuit in the LIB cell. In homogenized modeling, the damage of the LIB cell is considered, which is an essential indicator of the internal short circuit of a LIB cell and possible thermal runaway.

An Inverse Optimal Value Approach for Synchronously Optimizing Activity Durations and Worker Assignments with a Project Ideal Cost

Most companies survive the pain of cost and schedule overruns because of inaccurate project activity time settings. In order to deliver a project with a target cost and on schedule, this research proposes an inverse optimal value approach to optimize activity durations and the corresponding worker assignments synchronously to make the optimal project cost infinitely close to an ideal cost. The leader model reflects cost orientation and adjustability of activity durations, the follower model reflects the complexity of activity sequence, critical path completion time, cost pressure, skill matching, energy consumption, and other costs. Through upper-level and lower-level feedback and interaction of activity durations and worker assignments it is possible to deliver a project with an ideal cost. With considerations of the mathematical model characteristics of bi-level programming, nonlinearity, NP hard, and MAX functions, an improved genetic algorithm combining adaptive artificial fish swarms is designed. From the comparison results of random examples and an actual example, the error rate of the optimal value of the improved algorithm is acceptable. Numerical experiments show that the inverse optimal approach can deliver a project without delay and with an ideal cost. The inverse optimization method is more in line with the idea of target management, and can help managers achieve the purpose of cost control.

An inverse optimal approach for distributed preview consensus tracking

This paper develops a distributed consensus control protocol that guarantees consensus tracking and global optimality with respect to a positive quadratic performance index for identical general linear multi-agent systems on an undirected graph. By using the previewable future reference signal, the consensus tracking control problem is converted into a preview control based optimal problem. The inverse optimality theory is introduced to overcome the difficulty of designing a distributed optimal controller, which is caused by the fact that the communication topology always interplays with the system dynamics and subsequently the global information is usually required in a conventional optimal controller design. First, a distributed optimal controller for tracking a reference signal with stable mode is designed. Then, an internal model based distributed optimal controller is further designed to track a time-varying reference signal. The structured quadratic performance indexes are derived for both cases, respectively, and based on which, two distributed optimal tracking control protocols are implemented to guarantee the asymptotic stability of the augmented systems only using local information. Simulation results show that an appropriate preview length can effectively improve the tracking performance.

H∞ Control of Scalar Nonlinear Systems

The £2 disturbance attenuation control or the nonlinear H∞ control of a nonlinear system requires solving the Hamilton-Jacobi-Isaacs (HJI) partial differential inequality, which is generally a difficult undertaking. Though there are inverse optimal approaches, they have limited utilization from a practical point of view. In this work, a novel approach of state feedback H∞ control is proposed for scalar nonlinear systems using piecewise affine bounds of scalar-valued nonlinear functions. An LMI based systematic procedure is proposed. To illustrate the proposed method, an example of a scalar nonlinear system possessing significant nonlinearity has been considered. To demonstrate the effectiveness of the proposed scheme, it is compared with Linear Parameter Varying (LPV) based state feedback H∞ control, using quasi-LPV models based on Taylor series expansion.

Objective Selection for Cancer Treatment: An Inverse Optimization Approach

The quality of radiation therapy treatment plans and the efficiency of the planning process are heavily affected by the choice of planning objectives. Although simple objectives enable efficient treatment planning, the resulting treatment quality might not be clinically acceptable; complex objectives can generate high-quality treatment, yet the planning process becomes computationally prohibitive. In “Objective Selection for Cancer Treatment: An Inverse Optimization Approach,” by integrating inverse optimization and feature selection techniques, Ajayi, Lee, and Schaefer propose a novel objective selection method that uses historical radiation therapy treatment data to infer a set of planning objectives that are tractable and parsimonious yet clinically effective. Although the objective selection problem is a large-scale bilevel mixed-integer program, the authors propose various solution approaches inspired by feature selection greedy algorithms and patient-specific anatomical characteristics.

Optimality-based clustering: An inverse optimization approach

We propose a new clustering approach, called optimality-based clustering, that clusters data points based on their latent decision-making preferences. We assume that each data point is a decision generated by a decision-maker who (approximately) solves an optimization problem and cluster the data points by identifying a common objective function of the optimization problems for each cluster such that the worst-case optimality error is minimized. We propose three different clustering models and test them in the diet recommendation application.

An Inverse Optimization Approach of Vehicle Routing Problem

An Inverse Optimization Approach of Vehicle Routing Problem

Inverse optimal missile guidance law under constraints based on prescribed-time explicit reference governor

In this paper, the utilization of inverse-optimality-based prescribed-time explicit reference governor is investigated for missile intercepting against unknown maneuvering targets under performance and control input constraints. With an arctangent-based disturbance observer equipped for disturbance elimination, incorporating the inverse optimality approach into the missile interception guarantees the minimization of a performance index. In the framework of prescribed-time explicit reference governor, the control constraint is translated into a restriction of time-varying invariant set, and it follows the prescribed-time regulation of the applied reference along with the restriction satisfaction. The combined prescribed-time explicit reference governor approach could be transformed into a linear matrix inequality optimization problem, and its online solution over a receding horizon gives a Lyapunov function value for reference regulation and then control decisions in the continuous time. Simulation studies are performed to illustrate the performance of the proposed guidance control law.

Optimization-Based Network Identification for Thermal Transient Measurements

Network identification by deconvolution is a proven method for determining the thermal structure function of a given device. The method allows to derive the thermal capacitances as well as the resistances of a one-dimensional thermal path from the thermal step response of the device. However, the results of this method are significantly affected by noise in the measured data, which is unavoidable to a certain extent. In this paper, a post-processing procedure for network identification from thermal transient measurements is presented. This so-called optimization-based network identification provides a much more accurate and robust result compared to approaches using Fourier or Bayesian deconvolution in combination with Foster-to-Cauer transformation. The thermal structure function obtained from network identification by deconvolution is improved by repeatedly solving the inverse problem in a multi-dimensional optimization process. The result is a non-diverging thermal structure function, which agrees well with the measured thermal impedance. In addition, the associated time constant spectrum can be calculated very accurately. This work shows the potential of inverse optimization approaches for network identification.

Optimized Bézier-curve-based command generation and robust inverse optimal control for attitude tracking of spacecraft

In this paper we address the optimized attitude command generation and robust inverse optimal control design for the spacecraft attitude tracking with large angle slews. Firstly, with the reference attitude command described in a Bézier curve through the spherical linear interpolation, an optimized reference command is generated to meet the principles of small energy consumption and short turning time. The applied optimization combines the particle swarm optimization algorithm and genetic algorithm based on information entropy to guarantee the diversity of the population and improve the global search capabilities. Next, a robust inverse optimal control is developed for the attitude tracking of the obtained optimized reference attitude command, consisting of an inverse optimal Lyapunov approach for performance guarantee and an arctangent-based integral sliding mode control for the disturbance attenuation. A stronger generality of the control Lyapunov function is obtained for the inverse optimality. Furthermore, the developed control efforts are of smaller amplitudes due to introducing the auxiliary angular velocity for the smooth transition to the reference attitude command. Finally, simulation results are included to show the performance of the proposed scheme.

An Inverse Optimization Approach to Measuring Clinical Pathway Concordance

Clinical pathways outline standardized processes in the delivery of care for a specific disease. Patient journeys through the healthcare system, however, can deviate substantially from these pathways. Given the positive benefits of clinical pathways, it is important to measure the concordance of patient pathways so that variations in health system performance or bottlenecks in the delivery of care can be detected, monitored, and acted upon. This paper proposes the first data-driven inverse optimization approach to measuring pathway concordance in any problem context. Our specific application considers clinical pathway concordance for stage III colon cancer. We develop a novel concordance metric and demonstrate using real patient data from Ontario, Canada that it has a statistically significant association with survival. Our methodological approach considers a patient’s journey as a walk in a directed graph, where the costs on the arcs are derived by solving an inverse shortest path problem. The inverse optimization model uses two sources of information to find the arc costs: reference pathways developed by a provincial cancer agency (primary) and data from real-world patient-related activity from patients with both positive and negative clinical outcomes (secondary). Thus, our inverse optimization framework extends existing models by including data points of both varying “primacy” and “alignment.” Data primacy is addressed through a two-stage approach to imputing the cost vector, whereas data alignment is addressed by a hybrid objective function that aims to minimize and maximize suboptimality error for different subsets of input data.This paper was accepted by Chung Piaw Teo, Management Science Special Section on Data-Driven Prescriptive Analytics.

A critical evaluation of the index of patch quality

The inverse optimality approach can allow us to learn about an animal's environment by assuming their behaviour is optimal. This approach has been applied to animals diving underwater for food to produce the index of patch quality (IPQ), which aims to provide a proxy for prey abundance or quality in a foraging patch based on the animal's diving behaviour. The IPQ has been used in several empirical studies but has never been evaluated theoretically. Here, we discuss the strengths and weaknesses of the IPQ approach from a theoretical angle and review the empirical evidence supporting its use. We highlight several potential issues, in particular with the gain function—the function describing the energetic gain of an animal during a dive—used to calculate the IPQ. We investigate an alternative gain function which is appropriate in some cases, provide a new model based on this function, and discuss differences between the IPQ model and ours. We also find that there is little supporting empirical evidence justifying the general use of the IPQ and suggest future empirical validation methods which could help strengthen the case for the IPQ. Our findings have implications for the field of diving ecology and habitat assessment.