This work deals with the problem of semantic optimization of the central class of conjunctive queries (CQs). Since CQ evaluation is NP-complete, a long line of research has focussed on identifying fragments of CQs that can be efficiently evaluated. One of the most general such restrictions corresponds to generalized hypetreewidth bounded by a fixed constant k ≥ 1; the associated fragment is denoted GHWk. A CQ is semantically in GHWk if it is equivalent to a CQ in GHWk. The problem of checking whether a CQ is semantically in GHWk has been studied in the constraint-free case, and it has been shown to be NP-complete. However, in case the database is subject to constraints such as tuple-generating dependencies (TGDs) that can express, e.g., inclusion dependencies, or equality-generating dependencies (EGDs) that capture, e.g., key dependencies, a CQ may turn out to be semantically in GHWk under the constraints, while not being semantically in GHWk without the constraints. This opens avenues to new query optimization techniques. In this paper, we initiate and develop the theory of semantic optimization of CQs under constraints. More precisely, we study the following natural problem: Given a CQ and a set of constraints, is the query semantically in GHWk , for a fixed k ≥ 1, under the constraints, or, in other words, is the query equivalent to one that belongs to GHWk over all those databases that satisfy the constraints? We show that, contrary to what one might expect, decidability of CQ containment is a necessary but not a sufficient condition for the decidability of the problem in question. In particular, we show that checking whether a CQ is semantically in GHW1 is undecidable in the presence of full TGDs (i.e., Datalog rules) or EGDs. In view of the above negative results, we focus on the main classes of TGDs for which CQ containment is decidable, and that do not capture the class of full TGDs, i.e., guarded, non-recursive and sticky sets of TGDs, and show that the problem in question is decidable, while its complexity coincides with the complexity of CQ containment. We also consider key dependencies over unary and binary relations, and show that the problem in question is decidable in elementary time. Furthermore, we investigate whether being semantically in GHWk alleviates the cost of query evaluation. Finally, in case a CQ is not semantically in GHWk , we discuss how it can be approximated via a CQ that falls in GHWk in an optimal way. Such approximations might help finding "quick" answers to the input query when exact evaluation is intractable.