Learning Automata (LA) is a popular decision-making mechanism to "determine the optimal action out of a set of allowable actions" [1]. The distinguishing characteristic of automata-based learning is that the search for an optimal parameter (or decision) is conducted in the space of probability distributions defined over the parameter space, rather than in the parameter space itself [2]. In this paper, we propose a novel LA paradigm that can solve a large class of deterministic optimization problems. Although many LA algorithms have been devised in the literature, those LA schemes are not able to solve deterministic optimization problems as they suppose that the environment is stochastic. In this paper, our proposed scheme can be seen as the counterpart of the family of pursuit LA developed for stochastic environments [3]. While classical pursuit LAs can pursue the action with the highest reward estimate, our pursuit LA rather pursues the collection of actions that yield the highest performance by invoking a team of LA. The theoretical analysis of the pursuit scheme does not follow classical LA proofs, and can pave the way towards more schemes where LA can be applied to solve deterministic optimization problems. Furthermore, we analyze the scheme under both a constant learning parameter and a time-decaying learning parameter. We provide some experimental results that show how our Pursuit-LA scheme can be used to solve the Maximum Satisfiability (Max-SAT) problem. To avoid premature convergence and better explore the search space, we enhance our scheme with the concept of artificial barriers recently introduced in [4]. Interestingly, although our scheme is simple by design, we observe that it performs well compared to sophisticated state-of-the-art approaches.