## Wednesday, June 5, 2024

### The dangers of large bounds

Let's say we have a well scaled optimization problem with a very good and robust primal-dual optimal solution, which solves without any numerical difficulties. It could be a linear problem in standard form $$$$\begin{array}{ll}\textrm{minimize}&c^Tx\\ \textrm{subject to}& Ax\leq b,\end{array}$$$$ where the primal solution $x$ and dual solution $y$ are of reasonable magnitude and satisfy $Ax\leq b$, $y\geq 0$, $A^Ty+c=0$, $b^Ty=c^Tx$ with the best achievable precision. In other words, everything is as nice as it could be.

Obviously (right?) nothing should change if we extend the problem with huge bounds on $x$, so large that they are completely redundant for the given problem: $$$$\begin{array}{ll}\textrm{minimize}&c^Tx\\ \textrm{subject to}& Ax\leq b,\\ & x\leq M.\end{array}$$$$

To see what might go wrong let us first understand what the solver might think about problems whose feasible set lies very far out. The simplest example might be $$$$\begin{array}{ll}\textrm{minimize}&x\\ \textrm{subject to}& x=10^{12}.\end{array}$$$$ with the optimal solution $x=10^{12}$. But is that problem feasible from a practical perspective? If we work in the realm of practical applications with solutions of reasonable magnitude, then one might argue that this simple problem is for all intents and purposes infeasible, since the first feasible point lives further than we care to look. This can be made precise using Farkas lemma: the infeasibility certificate for this problem would be a value $y$ such that $y\leq 0$ and $10^{12}y=1$. A very good practical example of such a $y$ is $y=10^{-12}$ - it violates the conditions by only $10^{-12}$, failing to satisfy $y\leq 0$ by this narrow margin. This was just a toy example, but it easy to imagine that for a nontrivial optimization problem the solver might converge either to a solution or to such an infeasibility certificate, both of great quality! That makes problems with large solutions potentially hard to numerically distinguish from infeasible.

The same logic can now be applied to the problem (2) with redundant large bounds. The infeasibility certificate for it will be a pair of vectors $y,\tilde{y}$ such that $$A^Ty+\tilde{y}=0,\quad b^Ty+M\cdot \mathbb{1}^T\tilde{y}=-1,\quad y\geq 0,\quad \tilde{y}\geq 0.$$ If we let $y$ be the dual solution to the original problem (1) then by choosing a very small negative $\tilde{y}$ (in the order of $M^{-1}$) we can easily satisfy the second condition in (4), while the first and fourth one will only be violated by a tiny amount (of order $\tilde{y}$). As counterintuitive as it might be the math works out and we get certificate of primal infeasibility perfectly acceptable in finite precision arithmetic, with only tiny violations.

The rule of thumb one can remember connecting examples (2) and (3) is the following: the bound $x\leq M$ can be written in equality form as $x+\tilde{x}=M,\ \tilde{x}\geq 0$, so even with reasonable values of $x$ we get that the value of $\tilde{x}$ in any solution must be huge, putting us in position of example (3).

Similar considerations apply to distinguishing primal and dual infeasibility from feasibility in the case of large coefficients in the objective and other scaling problems; this applies for instance to big-M models in integer optimization, large penalties etc.

Will this come up in practice? Yes. Perhaps not for simple linear problems and with the extensive presolve MOSEK has, but it is not hard to set up a relatively straightforward conic problem, solve it, add huge bounds, turn presolve off and see MOSEK return a high quality infeasibility certificate. We leave these experiments to the reader.