Wasserstein Distributionally Robust Inverse Multiobjective Optimization

Inverse multiobjective optimization provides a general framework for the unsupervised learning task of inferring parameters of a multiobjective decision making problem (DMP), based on a set of observed decisions from the human expert. However, the performance of this framework relies critically on the availability of an accurate DMP, sufficient decisions of high quality, and a parameter space that contains enough information about the DMP. To hedge against the uncertainties in the hypothetical DMP, the data, and the parameter space, we investigate in this paper the distributionally robust approach for inverse multiobjective optimization. Specifically, we leverage the Wasserstein metric to construct a ball centered at the empirical distribution of these decisions. We then formulate a Wasserstein distributionally robust inverse multiobjective optimization problem (WRO-IMOP) that minimizes a worst-case expected loss function, where the worst case is taken over all distributions in the Wasserstein ball. We show that the excess risk of the WRO-IMOP estimator has a sub-linear convergence rate. Furthermore, we propose the semi-infinite reformulations of the WRO-IMOP and develop a cutting-plane algorithm that converges to an approximate solution in finite iterations. Finally, we demonstrate the effectiveness of our method on both a synthetic multiobjective quadratic program and a real world portfolio optimization problem.Comment: 19 page

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

Inverse Optimization and Inverse Multiobjective Optimization

Inverse optimization is a powerful paradigm for learning preferences and restrictions that explain the behavior of a decision maker, based on a set of external signal and the corresponding decision pairs. However, most inverse optimization algorithms are designed specifically in a batch setting, where all data is available in advance. As a consequence, there has been rare use of these methods in an online setting that is more suitable for real-time applications. To change such a situation, we propose a general framework for inverse optimization through online learning. Specifically, we develop an online learning algorithm that uses an implicit update rule which can handle noisy data. We also note that the majority of existing studies assumes that the decision making problem is with a single objective function, and attributes data divergence to noises, errors or bounded rationality, which, however, could lead to a corrupted inference when decisions are tradeoffs among multiple criteria. We take a data-driven approach and design a more sophisticated inverse optimization formulation to explicitly infer parameters of a multiobjective decision making problem from noisy observations. This framework, together with our mathematical analyses and advanced algorithm developments, demonstrates a strong capacity in estimating critical parameters, decoupling interpretable components from noises or errors, deriving the denoised optimal decisions, and ensuring statistical significance. In particular, for the whole decision maker population, if suitable conditions hold, we will be able to understand the overall diversity and the distribution of their preferences over multiple criteria. Additionally, we propose a distributionally robust approach to inverse multiobjective optimization. Specifically, we study the problem of learning the objective functions or constraints of a multiobjective decision making model, based on a set of observed decisions. In particular, these decisions might not be exact and possibly carry measurement noises or are generated with the bounded rationality of decision makers. We use the Wasserstein metric to construct the uncertainty set centered at the empirical distribution of these decisions. We show that this framework has statistical performance guarantees. We also develop an algorithm to solve the resulting minmax problem and prove its finite convergence

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Distributionally Robust Optimization: A Review

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization

Embedding quasi-static time series within a genetic algorithm for stochastic optimization: the case of reactive power compensation on distribution systems

This paper presents a methodology for the optimal placement and sizing of reactive power compensation devices in a distribution system (DS) with distributed generation. Quasi-static time series is embedded in an optimization method based on a genetic algorithm to adequately represent the uncertainty introduced by solar photovoltaic generation and electricity demand and its effect on DS operation. From the analysis of a typical DS, the reactive power compensation rating power results in an increment of 24.9% when compared to the classical genetic algorithm model. However, the incorporation of quasi-static time series analysis entails an increase of 26.8% on the computational time required

Situation Awareness for Smart Distribution Systems

In recent years, the global climate has become variable due to intensification of the greenhouse effect, and natural disasters are frequently occurring, which poses challenges to the situation awareness of intelligent distribution networks. Aside from the continuous grid connection of distributed generation, energy storage and new energy generation not only reduces the power supply pressure of distribution network to a certain extent but also brings new consumption pressure and load impact. Situation awareness is a technology based on the overall dynamic insight of environment and covering perception, understanding, and prediction. Such means have been widely used in security, intelligence, justice, intelligent transportation, and other fields and gradually become the research direction of digitization and informatization in the future. We hope this Special Issue represents a useful contribution. We present 10 interesting papers that cover a wide range of topics all focused on problems and solutions related to situation awareness for smart distribution systems. We sincerely hope the papers included in this Special Issue will inspire more researchers to further develop situation awareness for smart distribution systems. We strongly believe that there is a need for more work to be carried out, and we hope this issue provides a useful open-access platform for the dissemination of new ideas