Inverse Optimization with Noisy Data

Inverse optimization refers to the inference of unknown parameters of an optimization problem based on knowledge of its optimal solutions. This paper considers inverse optimization in the setting where measurements of the optimal solutions of a convex optimization problem are corrupted by noise. We first provide a formulation for inverse optimization and prove it to be NP-hard. In contrast to existing methods, we show that the parameter estimates produced by our formulation are statistically consistent. Our approach involves combining a new duality-based reformulation for bilevel programs with a regularization scheme that smooths discontinuities in the formulation. Using epi-convergence theory, we show the regularization parameter can be adjusted to approximate the original inverse optimization problem to arbitrary accuracy, which we use to prove our consistency results. Next, we propose two solution algorithms based on our duality-based formulation. The first is an enumeration algorithm that is applicable to settings where the dimensionality of the parameter space is modest, and the second is a semiparametric approach that combines nonparametric statistics with a modified version of our formulation. These numerical algorithms are shown to maintain the statistical consistency of the underlying formulation. Lastly, using both synthetic and real data, we demonstrate that our approach performs competitively when compared with existing heuristics

Similarity of Scenic Bilevel Images

This paper has been submitted to IEEE Transaction on Image Processing in May 2015.This paper presents a study of bilevel image similarity, including new objective metrics intended to quantify similarity consistent with human perception, and a subjective experiment to obtain ground truth for judging the performance of the objective similarity metrics. The focus is on scenic bilevel images, which are complex, natural or hand-drawn images, such as landscapes or portraits. The ground truth was obtained from ratings by 77 subjects of 44 distorted versions of seven scenic images, using a modified version of the SDSCE testing methodology. Based on hypotheses about human perception of bilevel images, several new metrics are proposed that outperform existing ones in the sense of attaining significantly higher Pearson and Spearman-rank correlation coefficients with respect to the ground truth from the subjective experiment. The new metrics include Adjusted Percentage Error, Bilevel Gradient Histogram and Connected Components Comparison. Combinations of these metrics are also proposed, which exploit their complementarity to attain even better performance. These metrics and the ground truth are then used to assess the relative severity of various kinds of distortion and the performance of several lossy bilevel compression methods.http://deepblue.lib.umich.edu/bitstream/2027.42/111737/2/Similarity of Scenic Bilevel Images.pdfDescription of Similarity of Scenic Bilevel Images.pdf : Main article ("Similarity of Scenic Bilevel Images"

Prescribing net demand for two-stage electricity generation scheduling

We consider a two-stage generation scheduling problem comprising a forward dispatch and a real-time re-dispatch. The former must be conducted facing an uncertain net demand that includes non-dispatchable electricity consumption and renewable power generation. The latter copes with the plausible deviations with respect to the forward schedule by making use of balancing power during the actual operation of the system. Standard industry practice deals with the uncertain net demand in the forward stage by replacing it with a good estimate of its conditional expectation (usually referred to as a point forecast), so as to minimize the need for balancing power in real time. However, it is well known that the cost structure of a power system is highly asymmetric and dependent on its operating point, with the result that minimizing the amount of power imbalances is not necessarily aligned with minimizing operating costs. In this paper, we propose a bilevel program to construct, from the available historical data, a prescription of the net demand that does account for the power system’s cost asymmetry. Furthermore, to accommodate the strong dependence of this cost on the power system’s operating point, we use clustering to tailor the proposed prescription to the foreseen net-demand regime. By way of an illustrative example and a more realistic case study based on the European power system, we show that our approach leads to substantial cost savings compared to the customary way of doing.European Research Council (ERC) under the EU Horizon 2020 research and innovation program (grant agreement No. 755705); Spanish Ministry of Science and Innovation (AEI/10.13039/501100011033) through project PID2020-115460GB-I00 and through the State Training Subprogram 2018 of the State Program for the Promotion of Talent and its Employability in R&D&I, within the framework of the State Plan for Scientific and Technical Research and Innovation 2017–2020 (with the support of the European Social Fund), reference PRE2018-083722 and ENE2017-83775-P); Junta de Andalucía (JA) and the European Regional Development Fund (FEDER) through the research project P20_00153; Partial funding for open access charge: Universidad de Málaga / CBUA