http://nathanyeung.com Scrawny to Tennis Fit Thu, 14 Apr 2011 21:23:48 +0000 hourly 1 http://wordpress.org/?v=3.3.2 http://nathanyeung.com/2011/04/monte-carlo/comment-page-1/#comment-417 Enoch Yeung Thu, 14 Apr 2011 21:23:48 +0000 http://nathanyeung.com/?p=776#comment-417 Some randomized algorithms do have guarantees of optimality or convergence towards the desired solution - however, because of the underlying stochastic formulation, the strength of such guarantees are usually limited to almost sure convergence - which is a convergence that occurs as the number of iterations approaches infinity. Thus, the randomized algorithm is guaranteed to solve an NP-hard problem, but only after an infinite amount of time. For example, in my markov chains class, I learned about simulated annealing - one of the most basic algorithm to finding an optimal solution over a given energy landscape/profile. The guarantee for this algorithm is almost sure convergence to the optimal solution, provided an appropriate annealing schedule. Hence, there are plenty of open problems! Some randomized algorithms do have guarantees of optimality or convergence towards the desired solution – however, because of the underlying stochastic formulation, the strength of such guarantees are usually limited to almost sure convergence – which is a convergence that occurs as the number of iterations approaches infinity. Thus, the randomized algorithm is guaranteed to solve an NP-hard problem, but only after an infinite amount of time. For example, in my markov chains class, I learned about simulated annealing – one of the most basic algorithm to finding an optimal solution over a given energy landscape/profile. The guarantee for this algorithm is almost sure convergence to the optimal solution, provided an appropriate annealing schedule. Hence, there are plenty of open problems!

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