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1. Spectrahedral Regression

   https://doi.org/10.1137/21M1455899  

   O’Reilly, Eliza; Chandrasekaran, Venkat (June 2023, SIAM Journal on Optimization)

   Convex regression is the problem of fitting a convex function to a data set consisting of input-output pairs. We present a new approach to this problem called spectrahedral regression, in which we fit a spectrahedral function to the data, i.e., a function that is the maximum eigenvalue of an affine matrix expression of the input. This method represents a significant generalization of polyhedral (also called max-affine) regression, in which a polyhedral function (a maximum of a fixed number of affine functions) is fit to the data. We prove bounds on how well spectrahedral functions can approximate arbitrary convex functions via statistical risk analysis. We also analyze an alternating minimization algorithm for the nonconvex optimization problem of fitting the best spectrahedral function to a given data set. We show that this algorithm converges geometrically with high probability to a small ball around the optimal parameter given a good initialization. Finally, we demonstrate the utility of our approach with experiments on synthetic data sets as well as real data arising in applications such as economics and engineering design. 

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2. Data driven verification of positive invariant sets for discrete, nonlinear systems

   Strong, Amy K; Bridgeman, Leila J (July 2024, PMLR)

   Invariant sets are essential for understanding the stability and safety of nonlinear systems. However, certifying the existence of a positive invariant set for a nonlinear model is difficult and often requires knowledge of the system’s dynamic model. This paper presents a data driven method to certify a positive invariant set for an unknown, discrete, nonlinear system. A triangulation of a subset of the state space is used to query data points. Then, linear programming is used to create a continuous piecewise affine function that fulfills the criteria of the Extended Invariant Set Principle by leveraging an inequality error bound that uses the Lipschitz constant of the unknown system. Numerical results demonstrate the program’s ability to certify positive invariant sets from sampled data. 

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3. PWA-CTM: An Extended Cell-Transmission Model based on Piecewise Affine Approximation of the Fundamental Diagram

   https://doi.org/10.1109/MED54222.2022.9837131  

   Alimardani, Fatemeh; Baras, John S. (June 2022, Proceedings of the 30th Mediterranean Conference on Control and Automation (MED 2022))

   Throughout the past decades, many different versions of the widely used first-order Cell-Transmission Model (CTM) have been proposed for optimal traffic control. Highway traffic management techniques such as Ramp Metering (RM) are typically designed based on an optimization problem with nonlinear constraints originating in the flow-density relation of the Fundamental Diagram (FD). Most of the extended CTM versions are based on the trapezoidal approximation of the flow-density relation of the Fundamental Diagram (FD) in an attempt to simplify the optimization problem. However, this relation is naturally nonlinear, and crude approximations can greatly impact the efficiency of the optimization solution. In this study, we propose a class of extended CTMs that are based on piecewise affine approximations of the flow-density relation such that (a) the integrated squared error with respect to the true relation is greatly reduced in comparison to the trapezoidal approximation, and (b) the optimization problem remains tractable for real-time application of ramp metering optimal controllers. A two-step identification method is used to approximate the FD with piecewise affine functions resulting in what we refer to as PWA-CTMs. The proposed models are evaluated by the performance of the optimal ramp metering controllers, e.g. using the widely used PI-ALINEA approach, in complex highway traffic networks. Simulation results show that the optimization problems based on the PWA-CTMs require less computation time compared to other CTM extensions while achieving higher accuracy of the flow and density evolution. Hence, the proposed PWA-CTMs constitute one of the best approximation approaches for first-order traffic flow models that can be used in more general and challenging modeling and control applications. 

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4. Adjustable robust optimization through multi-parametric programming

   https://doi.org/10.1007/s11590-019-01438-5  

   Avraamidou, Styliani; Pistikopoulos, Efstratios N. (July 2019, Optimization Letters)

   Adjustable robust optimization (ARO) involves recourse decisions (i.e. reactive actions after the realization of the uncertainty, ‘wait-and-see’) as functions of the uncertainty, typically posed in a two-stage stochastic setting. Solving the general ARO problems is challenging, therefore ways to reduce the computational effort have been proposed, with the most popular being the affine decision rules, where ‘wait-and-see’ decisions are approximated as affine adjustments of the uncertainty. In this work we propose a novel method for the derivation of generalized affine decision rules for linear mixed-integer ARO problems through multi-parametric programming, that lead to the exact and global solution of the ARO problem. The problem is treated as a multi-level programming problem and it is then solved using a novel algorithm for the exact and global solution of multi-level mixed-integer linear programming problems. The main idea behind the proposed approach is to solve the lower optimization level of the ARO problem parametrically, by considering ‘here-and-now’ variables and uncertainties as parameters. This will result in a set of affine decision rules for the ‘wait-and-see’ variables as a function of ‘here-and-now’ variables and uncertainties for their entire feasible space. A set of illustrative numerical examples are provided to demonstrate the potential of the proposed novel approach. 

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5. White-box Induction From SVM Models: Explainable AI with Logic Programming

   https://doi.org/10.1017/S1471068420000356  

   Shakerin, Farhad; Gupta, Gopal (December 2020, Theory and practice of logic programming)

   Ricca, Francesco; Russo, Alessandra (Ed.)

   We focus on the problem of inducing logic programs that explain models learned by the support vector machine (SVM) algorithm. The top-down sequential covering inductive logic programming (ILP) algorithms (e.g., FOIL) apply hill-climbing search using heuristics from information theory. A major issue with this class of algorithms is getting stuck in local optima. In our new approach, however, the data-dependent hill-climbing search is replaced with a model-dependent search where a globally optimal SVM model is trained first, then the algorithm looks into support vectors as the most influential data points in the model, and induces a clause that would cover the support vector and points that are most similar to that support vector. Instead of defining a fixed hypothesis search space, our algorithm makes use of SHAP, an example-specific interpreter in explainable AI, to determine a relevant set of features. This approach yields an algorithm that captures the SVM model’s underlying logic and outperforms other ILP algorithms in terms of the number of induced clauses and classification evaluation metrics. 

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