An optimization based 3D-1D coupling strategy is proposed for the simulation of fluid and chemical exchanges between a growing capillary network and the surrounding tissue, in the context of tumor-induced angiogenesis. Under proper assumptions on the regularity of the solution, a well posed mathematical model is worked out, based on the coupling between a three-dimensional and a one-dimensional equation. The novelty of the proposed approach lies in the solving strategy, resorting to a PDE-constrained optimization formulation of the problem: two auxiliary variables are introduced to approximate the value of the unknowns on the capillary wall and a properly designed cost functional is minimized constrained by the 3D-1D system of PDEs. The resulting approach appears to be highly robust and flexible in handling complex evolving networks. First of all, no conformity between the 3D computational mesh and the capillaries is required, allowing to avoid remeshing as the network grows. Furthermore, a proper choice of the interface variables ends up in a well conditioned discrete problem, regardless of the mutual sizes of the 3D and 1D meshes. Finally, interface variables are directly available, without the need of post processing. The optimization based strategy is adopted to model fluid pressure and oxygen concentration in the tissue and in the growing network. The dispersion of the chemical factor being responsible for the vessel growth is also accounted for and modeled by a 3D equation with a singular sink term. The vessel growth is modeled by a discrete tip-tracking strategy which provides, together with rules for branching and anastomosis, a realistic representation of the capillary network.