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1. On Cm solutions to systems of linear inequalities

   O'Neill, Kevin; Luli, Garving K (April 2023, Advances in mathematics)

   Recent work of C. Fefferman and the first author [8] has demonstrated that the linear system of equations has a solution if and only if satisfy a certain finite collection of partial differential equations. Here, the are fixed semialgebraic functions. In this paper, we consider the analogous problem for systems of linear inequalities: Our main result is a negative one, demonstrated by counterexample: the existence of a solution F may not, in general, be determined via an analogous finite set of partial differential inequalities in . 

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2. An adaptive and stability-promoting layerwise training approach for sparse deep neural network architecture

   https://doi.org/10.1016/j.cma.2025.117938  

   Krishnanunni, CG; Bui-Thanh, Tan (June 2025, Computer Methods in Applied Mechanics and Engineering)

   This work presents a two-stage adaptive framework for progressively developing deep neural network (DNN) architectures that generalize well for a given training dataset. In the first stage, a layerwise training approach is adopted where a new layer is added each time and trained independently by freezing parameters in the previous layers. We impose desirable structures on the DNN by employing manifold regularization, sparsity regularization, and physics-informed terms. We introduce a $$\ epsilon-\delta$$ stability-promoting concept as a desirable property for a learning algorithm and show that employing manifold regularization yields a $$\epsilon-\delta$$ stability-promoting algorithm. Further, we also derive the necessary conditions for the trainability of a newly added layer and investigate the training saturation problem. In the second stage of the algorithm (post-processing), a sequence of shallow networks is employed to extract information from the residual produced in the first stage, thereby improving the prediction accuracy. Numerical investigations on prototype regression and classification problems demonstrate that the proposed approach can outperform fully connected DNNs of the same size. Moreover, by equipping the physics-informed neural network (PINN) with the proposed adaptive architecture strategy to solve partial differential equations, we numerically show that adaptive PINNs not only are superior to standard PINNs but also produce interpretable hidden layers with provable stability. We also apply our architecture design strategy to solve inverse problems governed by elliptic partial differential equations. 

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3. Solving Composite Fixed Point Problems with Block Updates

   https://doi.org/10.1515/anona-2020-0173  

   Combettes, Patrick L.; Glaudin, Lilian E. (January 2021, Advances in Nonlinear Analysis)

   null (Ed.)

   Abstract Various strategies are available to construct iteratively a common fixed point of nonexpansive operators by activating only a block of operators at each iteration. In the more challenging class of composite fixed point problems involving operators that do not share common fixed points, current methods require the activation of all the operators at each iteration, and the question of maintaining convergence while updating only blocks of operators is open. We propose a method that achieves this goal and analyze its asymptotic behavior. Weak, strong, and linear convergence results are established by exploiting a connection with the theory of concentrating arrays. Applications to several nonlinear and nonsmooth analysis problems are presented, ranging from monotone inclusions and inconsistent feasibility problems, to variational inequalities and minimization problems arising in data science. 

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4. On the impact of running intersection inequalities for globally solving polynomial optimization problems

   https://doi.org/10.1007/s12532-019-00169-z  

   Del Pia, Alberto; Khajavirad, Aida; Sahinidis, Nikolaos V. (August 2019, Mathematical Programming Computation)

   We consider global optimization of nonconvex problems whose factorable reformulations contain a collection of multilinear equations. Important special cases include multilinear and polynomial optimization problems. The multilinear polytope is the convex hull of the set of binary points z satisfying the system of multilinear equations given above. Recently Del Pia and Khajavirad introduced running intersection inequalities, a family of facet-defining inequalities for the multilinear polytope. In this paper we address the separation problem for this class of inequalities. We first prove that separating flower inequalities, a subclass of running intersection inequalities, is NP-hard. Subsequently, for multilinear polytopes of fixed degree, we devise an efficient polynomial-time algorithm for separating running intersection inequalities and embed the proposed cutting-plane generation scheme at every node of the branch-and-reduce global solver BARON. To evaluate the effectiveness of the proposed method we consider two test sets: randomly generated multilinear and polynomial optimization problems of degree three and four, and computer vision instances from an image restoration problem Results show that running intersection cuts significantly improve the performance of BARON and lead to an average CPU time reduction of 50% for the random test set and of 63% for the image restoration test set. 

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5. On the asymptotic behavior of the solutions to parabolic variational inequalities

   https://doi.org/https://doi.org/10.1515/crelle-2019-0041  

   Spolaor, Luca; Colombo, Maria; Velichkov, Bozhidar (January 2020, Journal für die reine und angewandte Mathematik)

   We consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Łojasiewicz inequality. The difficulty lies in the fact that, since the constraint is non-analytic, the pioneering method of L. Simon ([]) does not apply and we have to exploit a better understanding on the constraint itself. We then apply this inequality to two associated problems. First we combine it with an abstract result on parabolic variational inequalities, to prove the convergence at infinity of the strong global solutions to the parabolic obstacle and thin-obstacle problems to a unique stationary solution with a rate. Secondly, we give an abstract proof, based on a parabolic approach, of the epiperimetric inequality, which we then apply to the singular points of the obstacle and thin-obstacle problems. 

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