If you are working on optimization with partial differential equations as constraints, you may be interested in the website

“OPTPDE – A Collection of Problems in PDE-Constrained Optimization”, http://www.optpde.net.

If you have developed an algorithm which can handle a certain class of optimization problems you need to do evaluations and tests on how well the method performs. To do so, you need well constructed test problems. This could be either problems where the optimal solution is known analytically our problems where the solution is known with a rigorous error bound obtained with a bullet-proof solver. Both things are not always easy to obtain and OPTPDE shall serve as a resource for such problems. It has been designed by Roland Herzog, Arnd RĂ¶sch, Stefan Ulbrich and Winnifried Wollner.

The generation of test instance for optimization problems seems quite important to me and indeed, several things can go wrong if this is not done right. Frequently, one sees tests for optimization routines on problems where the optimal solution is not known. Since there are usually different ways to express optimality conditions it is not always clear how to check for optimality; even more so, if you only check for “approximate optimality”, e.g. up to machine precision. A frequently observed effect is a kind of “trusted method bias”. By this I mean that an optimal solution is calculated by some trusted method and comparing the outcome of the tested routine with this solution. However, the trusted method uses some stopping criterion usually based on some specific set of formulations of optimality conditions and these can be different from what the new method has been tuned to. And most often, the stopping criteria do not give a rigorous error bound for the solution or the optimal objective value.

For sparse reconstruction problems, I dealt with this issue in “Constructing test instance for Basis Pursuit Denoising” (preprint available here) but I think this methodology could be used for other settings as well.

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