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1. On local quantum Gibbs states

   https://doi.org/10.1063/5.0058574  

   Duboscq, Romain; Pinaud, Olivier (October 2022, Journal of Mathematical Physics)

   We address in this work the problem of minimizing quantum entropies under local constraints. We suppose that macroscopic quantities, such as the particle density, current, and kinetic energy, are fixed at each point of Rd and look for a density operator over L2(Rd), minimizing an entropy functional. Such minimizers are referred to as local Gibbs states. This setting is in contrast with the classical problem of prescribing global constraints, where the total number of particles, total current, and total energy in the system are fixed. The question arises, for instance, in the derivation of fluid models from quantum dynamics. We prove, under fairly general conditions, that the entropy admits a unique constrained minimizer. Due to a lack of compactness, the main difficulty in the proof is to show that limits of minimizing sequences satisfy the local energy constraint. We tackle this issue by introducing a simpler auxiliary minimization problem and by using a monotonicity argument involving the entropy. 

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2. Throughput bound minimization for random access channel assignment

   https://doi.org/10.23919/WiOpt52861.2021.9589241  

   Dutta, Abhinanda; Weber, Steven (October 2021, 2021 19th International Symposium on Modeling and Optimization in Mobile, Ad hoc, and Wireless Networks (WiOpt))

   Throughput extremization is an important facet of performance modeling for low-power wide-area network (LP-WAN) wireless networks (e.g., LoRaWAN) as it provides insight into the best and worst case behavior of the network. Our previous work on throughput extremization established lower and upper bounds on throughput for random access channel assignment over a collision erasure channel in which the lower bound is expressed in terms of the number of radios and sum load on each channel. In this paper the lower bound is further characterized by identifying two local minimizers (a load balanced assignment and an imbalanced assignment) where the decision variables are the number of radios assigned to each channel and the total load on each channel. A primary focus is to characterize how macro-parameters of the optimization, i.e., the total number of radios, their total load, and the minimum load per radio, determine the regions under which each of the local minimizers is in fact the global minimizer. 

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3. Escaping Saddle-Point Faster under Interpolation-like Conditions

   Roy, A.; Balasubramanian, K.; Ghadimi, S.; Mohapatra, P. (December 2020, 34th Conference on Neural Information Processing Systems (NeurIPS 2020))

   In this paper, we show that under over-parametrization several standard stochastic optimization algorithms escape saddle-points and converge to local-minimizers much faster. One of the fundamental aspects of over-parametrized models is that they are capable of interpolating the training data. We show that, under interpolation-like assumptions satisfied by the stochastic gradients in an overparametrization setting, the first-order oracle complexity of Perturbed Stochastic Gradient Descent (PSGD) algorithm to reach an \epsilon-local-minimizer, matches the corresponding deterministic rate of ˜O(1/\epsilon^2). We next analyze Stochastic Cubic-Regularized Newton (SCRN) algorithm under interpolation-like conditions, and show that the oracle complexity to reach an \epsilon-local-minimizer under interpolation-like conditions, is ˜O(1/\epsilon^2.5). While this obtained complexity is better than the corresponding complexity of either PSGD, or SCRN without interpolation-like assumptions, it does not match the rate of ˜O(1/\epsilon^1.5) corresponding to deterministic Cubic-Regularized Newton method. It seems further Hessian-based interpolation-like assumptions are necessary to bridge this gap. We also discuss the corresponding improved complexities in the zeroth-order settings. 

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4. Inexact Fixed-Point Proximity Algorithm for the $$\ell _0$$ Sparse Regularization Problem

   https://doi.org/10.1007/s10915-024-02600-7  

   Fang, Ronglong; Xu, Yuesheng; Yan, Mingsong (July 2024, Journal of Scientific Computing)

   Abstract We studyinexactfixed-point proximity algorithms for solving a class of sparse regularization problems involving the$$\ell _0$$ ${\ell }_0$norm. Specifically, the$$\ell _0$$ ${\ell }_0$model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the$$\ell _0$$ ${\ell }_0$norm regularization term. Such an$$\ell _0$$ ${\ell }_0$model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the$$\ell _0$$ ${\ell }_0$models found by the proposed inexact algorithm outperform global minimizers of the corresponding$$\ell _1$$ ${\ell }_1$models, in terms of approximation accuracy and sparsity of the solutions. 

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5. Well-posedness and regularity for a polyconvex energy

   https://doi.org/10.1051/cocv/2023041  

   Gangbo, Wilfrid; Jacobs, Matt; Kim, Inwon (January 2023, ESAIM: Control, Optimisation and Calculus of Variations)

   We prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional in two and three dimensions, which corresponds to theH¹-projection of measure-preserving maps. Our result introduces a new criteria on the uniqueness of the minimizer, based on the smallness of the lagrange multiplier. No estimate on the second derivatives of the pressure is needed to get a unique global minimizer. As an application, we construct a minimizing movement scheme to constructL^r-solutions of the Navier–Stokes equation (NSE) for a short time interval. 

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