## Friday, April 17, 2020

### Objective Constraints (Again)

Long ago, I did a couple of posts [1, 2] about constraints designed to bound objective functions. We are referring here to constraints that explicitly bound the value of the objective function in an integer or mixed-integer linear program. The typical application is when the user is minimizing $f(x)$ subject to $x\in X$ and specifies a bound $f(x) \le d$. (If maximizing the inequality becomes $f(x)\ge d$.) The reason for doing so is to help the solver prune inferior nodes (nodes where $f(x) > d$ when minimizing) faster.

One way to accomplish the goal is to set a feasible starting solution $x^0 \in X$ for which $f(x)\le d$. This of course requires you to know such a solution. Also, setting a starting solution, even a good one, will likely steer the solver in a different direction than what it would have taken without the starting solution (meaning it will build a different tree), and this can wind up either faster or slower than not having the start, depending on where you sit on Santa's naughty/nice list and assorted random factors. (Asserting the bound by any of the other methods listed below can also have unintended consequences. Pretty much anything you do with a MIP can have unintended consequences.)

Assuming you have a bound in mind but not a starting solution, you have a few options. The main takeaways from those two posts were basically the following.
1. If your solver has the capability, your best bet is probably to specify the bound via a parameter. (CPLEX has the "upper cutoff" parameter for min problems and the "lower cutoff" parameter for max problems to do just this.)
2. Failing that, you can introduce a variable $z$ to represent your objective function, add a defining constraint $z = f(x)$, minimize $z$ and then specify $d$ as an upper bound for $z$. This may slow the solver some (for reasons explained in the prior posts) but is likely not as bad as the last option.
3. The last option, which is the most obvious (and thus one users gravitate to), is to add the constraint $f(x) \le d$ to the model. This can slow the solver down noticeably.
The short version of why the last option is undesirable is that if the last constraint is not binding  (which will happen if $d$ is not the optimal value and the solver has found an optimal or near optimal solution), it is just excess baggage. If it is binding, it can cause dual degeneracy.

Someone recently asked about this, and I waved my hand and invoked "dual degeneracy", but I'm not sure how clear I was. So I thought I would augment the two previous posts with a small example.

Suppose that we are solving a MIP model, and at some node we are staring at the following LP relaxation:\begin{alignat*}{5} \min & {}-{}5x_{1} & {}+{}40x_{2} & {}-{}5x_{3} & {}+{}5x_{4}\\ \textrm{s.t.} & \phantom{\{\}-}x_{1} & {}-{}\phantom{4}6x_{2} & & {}-{}3x_{4} & {}+{}s_{1} & & & =-3\\ & \phantom{\{\}-}x_{1} & {}-{}\phantom{4}2x_{2} & {}+\phantom{5}{}x_{3} & {}+{}\phantom{4}x_{4} & & {}+{}s_{2} & & =\phantom{-}0\\ & {}-{}5x_{1} & {}+{}40x_{2} & {}-{}5x_{3} & {}+{}5x_{4} & & & {}+{}s_{3} & =-6 &\quad (*)\end{alignat*}where the variables are nonnegative, the $s$ variables are slacks, and the constraint (*) is our way of imposing an upper bound of -6 on the objective function. In matrix terms the problem is\begin{align} \min\quad & \bar{c}'\bar{x}\\ \textrm{s.t.}\quad & \bar{A}\bar{x}=\bar{b}\\ & \bar{x}\ge0 \end{align} with $\bar{x}=(x_1,\dots,x_4,s_1,\dots,s_3)'$, $\bar{c}=(-5,40,-5,5,0,0,0)'$, $\bar{b}=(-3,0,-6)'$ and $$\bar{A}=\left[\begin{array}{rrrrrrr} 1 & -6 & 0 & -3 & 1 & 0 & 0\\ 1 & -2 & 1 & 1 & 0 & 1 & 0\\ -5 & 40 & -5 & 5 & 0 & 0 & 1 \end{array}\right].$$The initial basis would be the slack variables, giving us an infeasible solution $x=0$, $s=(-3,0,-6)$ with reduced costs $r = \bar{c}$. The negative values of $s_1$ and $s_3$ cause the infeasibility.

MIP solvers commonly use the dual simplex method to eliminate infeasibility in a node LP. Dual simplex would pivot in the row $i$ with the most negative right-hand side value $\bar{b}_i$, and in the column $j$ for which the ratio $r_j/\bar{a}_{ij}$ is minimal among those where $\bar{a}_{ij}\lt 0$. Here $i=3$ and $j$ is either 1 or 3 (the ratio in both column 1 and column 3 being $-5/-5=1$). Suppose that the solver chooses column 1, making the new basis (in row order) $(s_1, s_2, x_1).$ After the pivot, the reduced cost vector becomes $\hat{r}=c(0,0,0,0,0,0,-1)$, the new right-hand side vector is $\hat{b}=(-4.2, -1.2, 1.2)'$, and the new constraint matrix is $$\hat{A} = \left[\begin{array}{rrrrrrr} 0 & 2 & -1 & -2 & 1 & 0 & 0.2\\ 0 & 6 & 0 & 2 & 0 & 1 & 0.2\\ 1 & -8 & 1 & -1 & 0 & 0 & -0.2 \end{array}\right].$$The solution is still infeasible, and dual simplex will look to pivot in row 1 (where $\hat{b}$ is most negative. There are two possible pivot columns, columns 3 and 4, but the ratio used to distinguish them is zero in both cases because the reduced cost vector is all zeros (except for $s_3$, the slack in the objective constraint).

The same thing happens if we pivot in column 3 rather than column 1, and in fact it is possible to show that the reduced cost vector will be all zeros other than the slack in the constraint limit as long as the slack in the constraint limit is nonbasic. Since that slack variable will typically be nonbasic so long as the constraint is binding, and the constraint is useful only when binding, we can expect to see a lot of LPs where this occurs.  The tie is survivable (we've already seen one tie for pivot column), but picture this occurring where there are many dual pivots required, with perhaps many eligible columns (negative coefficients) for each pivot, and they all have ratio 0. The solver will be flying somewhat blind when it picks pivot columns, which could plausibly slow things down.

### References

[1] "Objective Functions Make Poor Constraints"
[2] "Objective Constraints: The Sequel"