Query optimizers of future database management systems are likely to face large access plan spaces in their task. Exhaustively searching such access plan spaces is unacceptable. We propose a query optimization algorithm based on simulated annealing, which is a probabilistic hill climbing algorithm. We show the specific formulation of the algorithm for the case of optimizing complex non-recursive queries that arise in the study of linear recursion. The query answer is explicitly represented and manipulated within the closed semiring of linear relational operators. The optimization algorithm is applied to a state space that is constructed from the equivalent algebraic forms of the query answer. A prototype of the simulated annealing algorithm has been built and few experiments have been performed for a limited class of relational operators. Our initial experience is that, in general, the algorithm converges to processing strategies that are very close to the optimal. Moreover, the traditional processing strategies (e.g., the semi-naive evaluation) have been found to be, in general, suboptimal.