We propose a principled optimization-based interactive query relaxation framework for queries that return no answers. Given an initial query that returns an empty-answer set, our framework dynamically computes and suggests alternative queries with fewer conditions than those the user has initially requested, in order to help the user arrive at a query with a non-empty-answer, or at a query for which no matter how many additional conditions are ignored, the answer will still be empty. Our proposed approach for suggesting query relaxations is driven by a novel probabilistic framework based on optimizing a wide variety of application-dependent objective functions. We describe optimal and approximate solutions of different optimization problems using the framework. Moreover, we discuss two important extensions to the base framework: the specification of a minimum size on the number of results returned by a relaxed query and the possibility of proposing multiple conditions at the same time. We analyze the proposed solutions, experimentally verify their efficiency and effectiveness, and illustrate their advantages over the existing approaches.