This study addresses the optimal design and operation of produced water and urban water networks, problems that are inherently difficult due to their nonconvex optimization formulations. A key source of this difficulty is the nonlinear representation of frictional pressure changes, commonly modeled through the Hazen-Williams equation. For produced water networks, we analytically prove that a standard piecewise linear formulation can yield near-optimal solutions for general network configurations. Computational experiments on real-world cases confirm the efficiency and effectiveness of this approach, providing high-quality solutions with significant runtime savings. In contrast, urban water network design problems pose challenges where direct piecewise linearization is ineffective and inefficient. To overcome these, we propose a general-purpose primal heuristic for mixed-integer nonlinear programs (MINLPs) with continuous nonlinearities. Our method iteratively solves linearized approximations with progressively reduced variable domains, ultimately recovering high quality feasible solutions. Applied to benchmark instances from the literature, the heuristic discovers superior solutions for three of nine long-standing test cases.
In this paper we define the first class of algebraically equivalent transformation (AET) functions for predictor-corrector (PC) interior-point algorithms (IPAs) for sufficient linear complementarity problems (LCPs) and we provide the unified complexity analysis of the introduced PC IPA. This new class is called the class of AET functions with bounded proportional growth rate for PC IPAs. The main difference between our PC IPA and those from the literature is that the AET function is an input data of our algorithm. The new class of AET functions is defined by several inequalities that are directly used in the analysis of the new PC IPA. We show that the PC IPA using any function of the new class of AET functions has polynomial iteration complexity in the size of the problem, the starting point’s duality gap, the accuracy parameter and the handicap of the problem’s matrix. We also provide promising numerical results on a newly generated test set of problems that contain matrices with large handicap, as well.
Hierarchical Nash game models such as Multi-Leader-Follower games have recently attracted the interest of research as they are an important modeling tool in various applications to study a strategic decision process of individuals, where the individuals are split into a hierarchy of different groups. In this paper, we address these game models using a parametric optimization approach in order to reduce the number of levels to a single-level game. In particular, we further consider the potential followers game to reduce the complexity of the problem. On the one hand, we study various mathematical structures and properties of such nonlinear hierarchical games. On the other hand, we provide suitable numerical solution approaches for the resulting single-level, but nonconvex, non-smooth or disjunctive optimization problems based on the specific theoretical properties. Finally, we finish the paper with computational results for some numerical experiments that support our approaches.
The security-constrained unit commitment problem seeks the minimum cost schedule of electricity generation units that meets demand over a 24h horizon and remains robust to failures of the transmission branches of the network linking generators and loads. Due to the size of realistic problems, which involve millions of variables and constraints, state-of-the-art solution techniques rely on linearizing the physical laws governing energy flows through branches and applying decomposition methods. We formulate this problem using high fidelity second order conic expressions for energy flows, yielding a large-scale mixed integer second order conic problem. We solve it using a decomposition procedure that separates the problem into a master problem and a large number of subproblems, and that combines Benders and column and constraint generation ideas, along with an accurate outer linearization for the master problem. We conduct computational experiments on realistic case studies to demonstrate the efficiency of the proposed methodology. For a number of realistic instances, we compare our results with those obtained using state-of-the-art commercial solvers.
Counterfactual explanations (CEs) are powerful tools within the thriving field of explainable machine learning, consisting of finding a minimal perturbation to a given instance so that its output in a certain predictive model satisfies some desirable property. Since counterfactuals are computed from an empirical model trained on a dataset, it is not evident that the resulting counterfactual explanations satisfy the requirements of the ground-truth model, which may differ from the empirical one. In this paper, we address this issue and build robust counterfactuals, in the sense that they fulfill the desired property on the ground-truth model with a given probability. We propose a statistical test to obtain performance guarantees for our robust counterfactual explanations, not only under the empirical model, but also under the ground-truth model. We particularize to generalized linear models (GLMs), a broad family of models that includes both classification, such as the logistic regression model, and regression, such as the linear regression model. We show that, for GLMs, there is a strong connection between CEs with ground-truth model guarantees and parameter-robust CEs. Finding a robust counterfactual explanation in the GLM framework amounts to solving a second-order cone program, readily solvable by off-the-shelf solvers such as Gurobi.
