Data-driven Inverse Optimization with Imperfect Information

In data-driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal. Thus, the observer seeks the agent's objective function that best explains a historical sequence of signals and corresponding optimal actions. We focus here on situations where the observer has imperfect information, that is, where the agent's true objective function is not contained in the search space of candidate objectives, where the agent suffers from bounded rationality or implementation errors, or where the observed signal-response pairs are corrupted by measurement noise. We formalize this inverse optimization problem as a distributionally robust program minimizing the worst-case risk that the {\em predicted} decision ({\em i.e.}, the decision implied by a particular candidate objective) differs from the agent's {\em actual} response to a random signal. We show that our framework offers rigorous out-of-sample guarantees for different loss functions used to measure prediction errors and that the emerging inverse optimization problems can be exactly reformulated as (or safely approximated by) tractable convex programs when a new suboptimality loss function is used. We show through extensive numerical tests that the proposed distributionally robust approach to inverse optimization attains often better out-of-sample performance than the state-of-the-art approaches

Constrained Optimization of Rank-One Functions with Indicator Variables

Optimization problems involving minimization of a rank-one convex function over constraints modeling restrictions on the support of the decision variables emerge in various machine learning applications. These problems are often modeled with indicator variables for identifying the support of the continuous variables. In this paper we investigate compact extended formulations for such problems through perspective reformulation techniques. In contrast to the majority of previous work that relies on support function arguments and disjunctive programming techniques to provide convex hull results, we propose a constructive approach that exploits a hidden conic structure induced by perspective functions. To this end, we first establish a convex hull result for a general conic mixed-binary set in which each conic constraint involves a linear function of independent continuous variables and a set of binary variables. We then demonstrate that extended representations of sets associated with epigraphs of rank-one convex functions over constraints modeling indicator relations naturally admit such a conic representation. This enables us to systematically give perspective formulations for the convex hull descriptions of these sets with nonlinear separable or non-separable objective functions, sign constraints on continuous variables, and combinatorial constraints on indicator variables. We illustrate the efficacy of our results on sparse nonnegative logistic regression problems

Developing a DEMATEL method to prioritize distribution centers in supply chain

During the past two decades, there have been significant numbers of studies focusing on supply chain management for evaluating important factors on the success of a supply chain program. In this paper, we present a method to prioritize the locations of distribution centers in a supply chain. The proposed model of this paper uses balanced scorecard (BSC) to categorize the most important attributes affecting the location of distribution centers and the attributes are ranked based on decision making trial and evaluation laboratory (DEMATEL) method. The implementation of the proposed model of this paper is also applied for a real-world case study of oil company and the results are analyzed under different scenarios

Reliability-based Bayesian Updating using Machine Learning

Bayesian updating is a powerful tool for model calibration and uncertainty quantification when new observations are available. By reformulating Bayesian updating into a structural reliability problem and introducing an auxiliary random variable, the state-of-the-art BUS algorithm has showcased large potential to achieve higher accuracy and efficiency compared with the conventional Markov-Chain-Monte-Carlo approach. However, BUS faces a number of limitations. The transformed reliability problem often investigates a very rare event problem especially when the number of measurements increases. Moreover, conventional reliability analysis techniques are not efficient and in some cases not capable of accurately estimating the probability of rare events. To overcome the aforementioned limitations, we propose integrating BUS algorithm with adaptive Kriging-based reliability analysis method. This approach improves the accuracy of the Bayesian updating and requires considerably smaller number of evaluations of the time-consuming likelihood function, compared to BUS.This research has been partly funded by the U.S. National Science Foundation (NSF) through awards CMMI-1462183, 1563372, and 1635569. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation

A Parallel Learning Strategy for Adaptive Kriging-based Reliability Analysis Methods

Adaptive Kriging-based reliability analysis methods have shown great advantages over conventional methods for their computational efficiency and accuracy. However, the widely accepted learning strategies such as Expected Feasibility function and U function can select one training point for each iteration, and therefore are not suitable for parallel processing. To address this limitation, the uncertainty of the failure probability is estimated through Adaptive Kriging with probabilistic classification-based Monte Carlo simulation based on the fact that the total number of failure points follows a Poisson Binomial distribution. By maximally reducing the uncertainty of the estimated failure probability, the theoretically optimal learning strategy is derived in this paper. Due to the computational difficulty in implementing the optimal learning strategy, a pseudo optimal parallel learning strategy is proposed to closely reach the optimal solution. The efficiency of the proposed parallel learning strategy is investigated here by implementing two benchmark reliability problems. Results indicate that the total number of evaluations to the performance function through the proposed parallel learning strategy can be even close to the approach based on single training point enriching.This research has been partly funded by the U.S. National Science Foundation (NSF) through awards CMMI-1462183, 1563372, and 1635569. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation