  Linear programming has played a crucial role in shaping decision-making,
resource allocation, and cost reduction in various domains. In this paper, we
investigate the application of overparametrized neural networks and their
implicit bias in solving linear programming problems. Specifically, our
findings reveal that training diagonal linear networks with gradient descent,
while optimizing the squared $L_2$-norm of the slack variable, leads to
solutions for entropically regularized linear programming problems. Remarkably,
the strength of this regularization depends on the initialization used in the
gradient descent process. We analyze the convergence of both discrete-time and
continuous-time dynamics and demonstrate that both exhibit a linear rate of
convergence, requiring only mild assumptions on the constraint matrix. For the
first time, we introduce a comprehensive framework for solving linear
programming problems using diagonal neural networks. We underscore the
significance of our discoveries by applying them to address challenges in basis
pursuit and optimal transport problems.
