# reformulating inverse problem as multi-objective optimization

I'm working on an inverse problem for my Ph.D. research, for which I'll write the objective functional as

$J(\theta) = E(G(\theta) - u^o)$,

where $\theta$ are the parameters, $G$ is the forward map from parameters to observations, $u^o$ are the observations, and $E$ is some metric of the error. The forward map $G$ comes from solving an elliptic PDE in which $\theta$ is a coefficient, which for definiteness' sake we'll say is the Poisson equation:

$-\nabla\cdot ke^\theta\nabla u = f$.

In general I'll say that the forward map from $\theta$ to $u$ amounts to solving the nonlinear system $F(u, \theta) = 0$. I've been solving this using the adjoint method so far. This can also be written as a constrained optimization problem

$J(\theta; u, \lambda) = E(u - u^o) + \langle F(u, \theta), \lambda\rangle$

but computationally all the procedures are the same.

What both formulations do not exploit is the fact that the forward map is, itself, another optimization problem, i.e. the solution $u$ of the Poisson equation is the minimizer of the Dirichlet energy

$D(u, \theta) = \int_\Omega\left(\frac{1}{2}ke^\theta|\nabla u|^2 - fu\right)dx$.

Can this inverse problem be reformulated as a multi-objective optimization problem for the pair of objectives $\{E(u - u^o)$, $D(u, \theta)\}$? If so, is this useful, either for computational purposes or for proving things?

I don't know a lot about multi-objective optimization, so references to any useful literature would be helpful. All I could turn up on a google search for multi-objective optimization and inverse problems were a bunch of papers about genetic algorithms.

• Note that given $\theta$, $D$ has a unique minimizer $u$; and any $u$ that is not a minimizer of $D$ is not a solution to your forward mapping. So there's no room for trade-offs. At best, you could think of this as a bilevel optimization problem -- but, again, since the minimizer is unique and $D$ is convex, you can equivalently replace the lower-level problem with the necessary optimality condition (the PDE). Multi-objective optimization only makes sense if you have two independent objectives; one example would be a fitting and a regularization term... – Christian Clason Nov 28 '16 at 22:38
• ... but regularization theory usually (but not always) is the better way of picking the correct balance between fitting and regularization. – Christian Clason Nov 28 '16 at 22:39
• @ChristianClason thanks for the reply, if you rewrite it as an answer I'll accept it. I figured that, if this idea did work, you could eliminate the need for the adjoint state $\lambda$, but I guess it's a dead end. Oh well! – Daniel Shapero Nov 30 '16 at 1:07

Multi-objective optimization is concerned with the simultaneous optimization of two (or more) competing objectives where you do not wish to decide on the trade-off between the two beforehand. However, this is not the situation you have: Only the exact minimizer of the Dirichlet energy $D$ is a solution to your forward mapping; the actual value of the functional is irrelevant (in particular, any $u$ that does not attain the minimal value is completely worthless). So the only "trade-off" that makes sense is to ensure that $D$ is minimized, and use the remaining freedom (in choosing $\theta$) to minimize $E$. This would lead to a bilevel optimization problem ("minimize the minimizer"), but since $D$ is strictly convex and thus admits a unique minimizer, you can equivalently replace the optimization problem for $D$ with the first-order necessary condition $F(u)=0$. This is exactly what you are doing with your Lagrangian $J$ -- by taking the supremum over all $\lambda$, you are enforcing minimality of $D$.
You mention that you would like to eliminate the adjoint state. In principle, you need some way of "backpropagating" the difference between state and data back to the parameter space. This is usually provided by the adjoint state $\lambda$; however, if the equation is sufficiently linear in parameter and/or state separately, you might be able to replace the adjoint state with a suitable surrogate. In this context, you might find the following paper useful: http://dx.doi.org/10.1088/0266-5611/19/6/010.