The inverse kinematics problem can be stated as a nonlinear constrained least-squares optimization, which is in its simplest form as follows:
$$
\mathbf{q}^*=\arg\min_{\mathbf{q} \in \mathbb{R}^n} \left\| \mathbf{x}_d-K \left( \mathbf{q} \right) \right\|^2 \\
\text{s.t.} \quad \mathbf{q}_l<\mathbf{q}<\mathbf{q}_u
$$
where $\mathbf{q}$ is the vector of the $n$ independent joint angles, $\mathbf{x}_d$ is the desired Cartesian pose comprising target position and orientation of the end-effector, and $K\left(\cdot\right)$ is the forward kinematic map. The minimization must be carried out without violating the lower and upper joints bounds, $\mathbf{q}_l$ and $\mathbf{q}_u$ respectively.
Therefore, you are right - in a sense - saying that there exists only one measurement you have to account for, if compared with the traditional least-squares setting.
Concerning how we can solve the problem, there's a large number of traditional techniques that do not resort to sophisticated optimizations: Jacobian transposed, Jacobian pseudoinverse, a synergy of the two termed Damped Least-Squares, to cite the most known. Thus, it's not necessary to employ Ceres, for example, as back-end to implement your solver.
Nonetheless, lately there's been an increasing attention in the community to the use of Sequential Quadratic Programming and more generally nonlinear methods: see for example [1] for a extensive survey. The reason for that is the significant amount of benefits you'll get with a nonlinear optimizer:
- robustness against singularities and out-of-reach targets
- better handling of joints bounds
- possibility to deal with higher number of linear/nonlinear constraints
- scalability with the number of DoFs
- and more...
Personally, I developed in the past a nonlinear inverse kinematics solver using Ipopt for a humanoid robot ([2]) and I found that I managed to easily outperform other standard techniques, as those provided by KDL.
Thereby, my warm suggestion is to go with Ceres, Ipopt or other equivalent optimizers.
References
[1] TRAC-IK: An open-source library for improved solving of generic inverse kinematics.
[2] An experimental evaluation of a novel minimum-jerk cartesian controller for humanoid robots.