{'fraction_non_alphanumeric': 0.08295625942684766, 'fraction_numerical': 0.00904977375565611, 'mean_word_length': 4.579831932773109, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Asymptotic Convergence of the Steepest Descent Method for the Exponential Penalty in Linear Programm

u̇(t) = −∇xf(u(t), r(t)), u(t0) = u0 where f(x, r) is the exponential penalty function associated with the linear program min{c′x : Ax ≤ b}, and r(t) decreases to 0 as t goes to ∞. We show that for each initial condition (t0, u0) the solution u(t) is defined on the whole interval [t0,∞) and, under suitable hypothesis on the rate of decrease of r(t), we establish the convergence of u(t) towards an optimal solution u∞ of the linear program. In particular we find sufficient conditions for u∞ to coincide with the limit of the unique minimizer x(r) of f(·, r).