Inductive queries are queries to an inductive database that generate a set of patterns in a data mining context. Inductive querying poses new challenges to database and data mining technology. We study conjunctive inductive queries, which are queries that can be written as a conjunction of a monotonic and an anti-monotonic subquery. We introduce the conjunctive inductive query optimization problem, which is concerned with minimizing the cost of computing the answer set to a conjunctive query. In the optimization problem, it is assumed that there are costs ca and cm associated to evaluating a pattern w.r.t. a monotonic and an anti-monotonic subquery respectively. The aim is then to minimize the total cost needed to compute all solutions to the query. Secondly, we present an algorithm that aims at optimizing conjunctive inductive queries in the context of the pattern domain of strings and evaluate it on a challenging data set in computational biology.