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1. Pareto Optimality and Compromise for Environmental Water Management

   https://doi.org/10.1029/2020WR028296  

   Null, Sarah E.; Olivares, Marcelo A.; Cordera, Felipe; Lund, Jay R. (October 2021, Water Resources Research)

   Abstract Water management usually considers economic and ecological objectives, and involves tradeoffs, conflicts, compromise, and cooperation among objectives. Pareto optimality often is championed in water management, but its relationships with the mathematical representation of objectives, and implications of tradeoffs for Pareto optimal decisions, are rarely examined. We evaluate the mathematical properties of optimized tradeoffs to identify promising regions for compromise, suggest strategies for reducing conflicts, and better understand whether decision‐makers are more or less likely to cooperate on environmental water allocations. Cooperation and compromise among objectives can be easier when tradeoff curves are concave and more adversarial when tradeoff curves are convex. “Knees,” or areas with maximum curvature, bulges, or breakpoints in concave Pareto frontiers, suggest more promising areas for compromise. Evaluating the shape of Pareto curves based on each objective's performance function can screen for the existence of knees amenable to compromise. We explore water management and restorations actions that improve and shift the location and prominence of knees in concave Pareto frontiers. Connecting river habitats and other non‐flow management actions may add knees on locally concave regions of Pareto frontiers. Managing multiple streams regionally, rather than individually, can sometimes turn convex local tradeoffs into concave regional tradeoffs more amenable to compromise. Overall, this analysis provides a deep investigation of how the shape of tradeoffs influences the range and promise of decisions to improve performance, and illustrates that management actions may encourage cooperation and reduce conflict. 

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2. Extended McCormick relaxation rules for handling empty arguments representing infeasibility

   https://doi.org/10.1007/s10898-023-01315-7  

   Ye, Jason; Scott, Joseph K. (July 2023, Journal of Global Optimization)

   McCormick’s relaxation technique is one of the most versatile and commonly used methods for computing the convex relaxations necessary for deterministic global optimization. The core of the method is a set of rules for propagating relaxations through basic arithmetic operations. Computationally, each rule operates on four-tuples describing each input argument in terms of a lower bound value, an upper bound value, a convex relaxation value, and a concave relaxation value. We call such tuples McCormick objects. This paper extends McCormick’s rules to accommodate input objects that are empty (i.e., the convex relaxation value lies above the concave, or both relaxation values lie outside the bounds). Empty McCormick objects provide a natural way to represent infeasibility and are readily generated by McCormick-based domain reduction techniques. The standard McCormick rules are strictly undefined for empty inputs and applying them anyway can yield relaxations that are non-convex/concave on infeasible parts of their domains. In contrast, our extended rules always produce relaxations that are well-defined and convex/concave on their entire domain. This capability has important applications in reduced-space global optimization, global dynamic optimization, and domain reduction. 

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3. Video: Dynamic fully immersive virtual reality of supersonic flows

   https://doi.org/10.1103/APS.DFD.2022.GFM.V0026  

   Paeres, David; Lagares, Christian; Araya, Guillermo (November 2022, 75th APS-DFD November 2022)

   In this video, we show high-fidelity numerical results of supersonic spatially-developing turbulent boundary layers (SDTBL) under strong concave and concave curvatures and Mach = 2.86. The selected numerical tool is Direct Numerical Simulation (DNS) with high spatial/temporal resolution. The prescribed concave geometry is based on the experimental study by Donovan et al. (J. Fluid Mech., 259, 1-24, 1994). Turbulent inflow conditions are based on extracted data from a previous DNS over a flat plate (i.e., turbulence precursors). The comprehensive DNS information sheds important light on the transport phenomena inside turbulent boundary layers subject to strong deceleration or Adverse Pressure Gradient (APG) caused by concave walls as well as to strong acceleration or Favorable Pressure Gradient (FPG) caused by convex walls at different wall thermal conditions (i.e., cold, adiabatic and hot walls). In this opportunity, the selected scientific visualization tool is Virtual Reality (VR) by extracting vortex core iso-surfaces via the Q-criterion to convert them to a file format readable by the HTC Vive VR toolkit. The reader is invited to visit our Virtual Wind Tunnel (VWT) under a fully immersive environment for further details. The video is available at: https://gfm.aps.org/meetings/dfd-2022/6313a60c199e4c2da9a946bc 

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4. Log-Concave and Multivariate Canonical Noise Distributions for Differential Privacy

   Awan, Jordan; Dong, Jinshuo (January 2022, Advances in neural information processing systems)

   Koyejo, S.; Mohamed, S.; Agarwal, A.; Belgrave, D.; Cho, K.; Oh, A. (Ed.)

   A canonical noise distribution (CND) is an additive mechanism designed to satisfy f-differential privacy (f-DP), without any wasted privacy budget. f-DP is a hypothesis testing-based formulation of privacy phrased in terms of tradeoff functions, which captures the difficulty of a hypothesis test. In this paper, we consider the existence and construction of both log-concave CNDs and multivariate CNDs. Log-concave distributions are important to ensure that higher outputs of the mechanism correspond to higher input values, whereas multivariate noise distributions are important to ensure that a joint release of multiple outputs has a tight privacy characterization. We show that the existence and construction of CNDs for both types of problems is related to whether the tradeoff function can be decomposed by functional composition (related to group privacy) or mechanism composition. In particular, we show that pure epsilon-DP cannot be decomposed in either way and that there is neither a log-concave CND nor any multivariate CND for epsilon-DP. On the other hand, we show that Gaussian-DP, (0,delta)-DP, and Laplace-DP each have both log-concave and multivariate CNDs. 

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5. Tight Analysis of Extra-gradient and Optimistic Gradient Methods For Nonconvex Minimax Problems

   Mahdavinia, P; Deng, Y; Li, H; Mahdavi, M (October 2022, Neural Information Processing Systems (NeurIPS))

   Despite the established convergence theory of Optimistic Gradient Descent Ascent (OGDA) and Extragradient (EG) methods for the convex-concave minimax problems, little is known about the theoretical guarantees of these methods in nonconvex settings. To bridge this gap, for the first time, this paper establishes the convergence of OGDA and EG methods under the nonconvex-strongly-concave (NC-SC) and nonconvex-concave (NC-C) settings by providing a unified analysis through the lens of single-call extra-gradient methods. We further establish lower bounds on the convergence of GDA/OGDA/EG, shedding light on the tightness of our analysis. We also conduct experiments supporting our theoretical results. We believe our results will advance the theoretical understanding of OGDA and EG methods for solving complicated nonconvex minimax real-world problems, e.g., Generative Adversarial Networks (GANs) or robust neural networks training. 

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