An exact dynamic programming approach to segmented isotonic regression

• Published in: Omega (Oxford) Vol. 105; p. 102516
• Main Authors: Bucarey, Víctor, Labbé, Martine, Morales, Juan M., Pineda, Salvador
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.12.2021
• Subjects: Cardinality-constrained shortest path problem; Consumers’ price-response; Data clustering; Inverse optimization; Isotonic regression; Segmented regression; Consumers’ price-response; Segmented regression; Inverse optimization; Isotonic regression; Cardinality-constrained shortest path problem; Data clustering
• ISSN: 0305-0483; 1873-5274
• DOI: 10.1016/j.omega.2021.102516

•We propose a polynomial-time algorithm for segmented isotonic regression.•The regression is formulated as a cardinality-constrained shortest path problem.•Dynamic programming is used to find the path providing the optimal curve fit.•Lower and upper bounds are exploited to accelerate the proposed solution algorithm.•Useful tool for a pool of small flexible consumers to partake in electricity markets. This paper proposes a polynomial-time algorithm to construct the monotone stepwise curve that minimizes the sum of squared errors with respect to a given cloud of data points. The fitted curve is also constrained on the maximum number of steps it can be composed of and on the minimum step length. Our algorithm relies on dynamic programming and is built on the basis that said curve-fitting task can be tackled as a shortest-path type of problem. Numerical results on synthetic and realistic data sets reveal that our algorithm is able to provide the globally optimal monotone stepwise curve fit for samples with thousands of data points in less than a few hours. Furthermore, the algorithm gives a certificate on the optimality gap of any incumbent solution it generates. From a practical standpoint, this piece of research is motivated by the roll-out of smart grids and the increasing role played by the small flexible consumption of electricity in the large-scale integration of renewable energy sources into current power systems. Within this context, our algorithm constitutes an useful tool to generate bidding curves for a pool of small flexible consumers to partake in wholesale electricity markets.