The AIMMS Presolver

The need for a presolver

Of all nonlinear solvers in AIMMS only a couple use (limited) preprocessing techniques. Therefore, AIMMS itself has implemented a presolve algorithm with the goal to reduce the size of the problem and to tighten the variable bounds, which may help the solver to solve nonlinear problems faster. Besides the BARON global solver, all nonlinear solvers in AIMMS are local solvers, i.e. the solution found by the solver is a local solution and cannot be guaranteed to be a global solution. The presolve algorithm may help the solver in finding a better solution. A local solver might sometimes fail to find a solution and then it is often not clear whether that is caused by the problem being infeasible or by the solver failing to find a solution for a feasible problem. The presolve algorithm may reveal inconsistent constraints and/or variable bounds and hence identify a problem as infeasible.

Presolve techniques

Consider the following constrained nonlinear optimization problem:

\[\begin{split}\begin{align} & \text{minimize} & & f(x) \\ & \text{subject to} & & g(x) \leq d & & \\ &&& Ax \leq b & & \\ &&& l\leq x \leq u & & \\ \end{align}\end{split}\]

The objective function \(f(x)\) can either be linear or nonlinear, while \(g(x)\) is a nonlinear function. The AIMMS presolve algorithm will (amongst others)

  • remove singleton rows by moving the bounds to the variables,

  • reduce variable bounds from linear and nonlinear constraints that contain bounded variables,

  • delete fixed variables,

  • remove one variable of a doubleton, and

  • delete redundant constraints.

A detailed description of each of these techniques can be found in [FG94].

Singleton rows

A singleton row is a linear constraint that contains only one variable. An equality singleton row fixes the variable to the right-hand-side value of the row, and unless this value conflicts with the current bounds of the variable in which case the problem is infeasible, AIMMS can remove both the row and variable from the problem. An inequality singleton row introduces a new bound on the variable which can be redundant, tighter than an existing bound in which case AIMMS will update the bound, or infeasible. The AIMMS presolve algorithm will remove all singleton rows.

Deleting fixed variables

If a variable is fixed then sometimes another row becomes a singleton row, and if that row is an equality row AIMMS can fix the remaining variable and remove it from the problem. By repeating this process AIMMS can solve any triangular system of linear equations that is part of the problem.

Bound reductions using linear constraints

The following analysis applies to a linear “less than or equal to” constraint. A similar analysis applies to other constraint types. Assume we have a linear constraint \(i\) that originally has the form

(1)\[ \sum_j a_{ij}x_j \leq b_i\]

Assuming that all variables in this constraint have finite bounds, we can determine the following lower and upper limits on constraint \(i\)

(2)\[ \underline{b_i} = \sum_{j\in P_i} a_{ij}l_j + \sum_{j\in N_i} a_{ij}u_j\]

and

(3)\[ \overline{b_i} = \sum_{j\in P_i} a_{ij}u_j + \sum_{j\in N_i} a_{ij}l_j\]

where \(P_i = \{j \mid a_{ij} > 0\}\) and \(N_i = \{j \mid a_{ij} < 0\}\) define the sets of variables with a positive coefficient and negative coefficient in constraint \(i\) respectively.

Bound analysis

By comparing the lower and upper limits of a constraint with the right-hand-side value we obtain one of the following situations:

  • \(\underline{b_i} > b_i\): constraint (1) cannot be satisfied and is infeasible.

  • \(\underline{b_i} = b_i\): constraint (1) can only be satisfied if all variables in the constraint are fixed on their lower bound if they have a positive coefficient, or fixed on their upper bound if they have a negative coefficient. The constraint and all its variables can be removed from the problem.

  • \(\overline{b_i} \leq b_i\): constraint (1) is redundant, i.e. will always be satisfied, and can be removed from the problem.

  • \(\underline{b_i} < b_i < \overline{b_i}\): constraint (1) cannot be eliminated but can often be used to improve the bounds of one or more variables as we will see below.

If \(\underline{b_i} < b_i < \overline{b_i}\), then combining (1) with (2) gives the following bounds on a variable \(k\) in constraint \(i\):

(4)\[ x_k \leq l_k + (b_i - \underline{b_i})/a_{ik}\qquad\mbox{if $a_{ik} > 0$}\]

and

(5)\[ x_k \geq u_k + (b_i - \underline{b_i})/a_{ik}\qquad\mbox{if $a_{ik} < 0$}\]

If the upper bound given by (4) is smaller than the current lower bound of variable \(k\) then the problem is infeasible. If it is smaller then the current upper bound of variable \(k\), AIMMS will update the upper bound for variable \(k\). Something similar holds for the lower bound as given by (5). Note that bounds (4) and (5) can only be derived if all bounds \(l_j\) and \(u_j\) in (2) are finite. But also if exactly one of the bounds in (2) is an infinite bound, AIMMS can still find an implied bound for the corresponding variable.

Bound reductions using nonlinear constraints

We can rewrite a nonlinear constraint \(g_i(x)\leq d_i\) as

(6)\[ \sum_j a_{ij}x_i + h_i(y) \leq d_i\]

separating the linear variables \(x\) in this constraint from the nonlinear variables \(y\). As before, we can find lower and upper limits on the linear part of the constraint, and again we denote them by \(\underline{b_i}\) and \(\overline{b_i}\) respectively. For this constraint we can derive the following upper bound on the nonlinear term in (6):

(7)\[ h_i(y) \leq d_i - \underline{b_i}\]

Note that if there are no linear terms in constraint (6) then \(\underline{b_i} = 0\).

Nonlinear analysis using expression trees

Nonlinear expressions in AIMMS are stored in an expression tree. By going through the expression tree from the top node to the leafs we can sometimes derive bounds on some of the variables in the expression. For example, assume we have the constraint

\[\sqrt{\ln x} \leq 2\]

with \(x\) unbounded. It follows that the \(\ln x\) sub-expression should be in the range \([0,4]\) since \(\sqrt{y}\) is not defined for \(y<0\), which in turn implies that \(x\) should be in the range \((1,e^4]\).

Types of nonlinear analysis

AIMMS can analyze nonlinear expressions for various types of reductions, and uses various types of techniques, such as:

  • operator domain analysis: reduce bounds on operator arguments by the implicit domains of operators such as \(\sqrt{x}\) or \(\ln x\),

  • operator range analysis: compute the bounds of a nonlinear expression on the basis of known bounds on the argument(s) and use those bounds for further reductions, and

  • for invertible functions, compute bounds on operator arguments on the basis of bounds on a known operator range.

Supported operators

The presolve algorithm can handle nonlinear expressions build up by the operators listed in this table. If a nonlinear constraint contains an operator that is not in this table then it will be ignored by the presolve algorithm.

Table 34 Operators used by the presolve algorithm

Operators

\(\log_{10} x\), \(\ln x\), \(\exp x\), \(e^x\)

\(x^a\), \(a^x\) (\(a \neq 0\)), \(x^y\)

\(\sin x\), \(\cos x\), \(\tan x\), \(\arcsin x\), \(\arccos x\), \(\arctan x\)

\(x+y\), \(x-y\), \(x \cdot y\), \(x/y\)

Doubletons

If a problem contains a constraint of the form \(x = ay\), \(a \neq 0\), then the variables \(x\) and \(y\) define a doubleton. If the presolve algorithm detects a doubleton then it will replace the variable \(x\) by the term \(ay\) in every constraint in which \(x\) appears, and remove the variable \(x\) from the problem. For some problems good initial values are given to the variables. In case the initial value given to \(x\) does not match the initial value of \(y\) according to the relationship \(x = a y\), it is unclear which initial value we should assign to \(y\). Preliminary test results showed that in such a case it is better not to remove the doubleton, and pass both variables to the solver with their own initial value. This has become the default behavior of our presolve algorithm regarding doubletons.

The presolve algorithm

The AIMMS Presolver iteratively applies all reduction techniques discussed above until no further reductions are available anymore, or an iteration limit has been reached. Various options are available in the Solvers general - AIMMS presolver section of the option tree to steer the presolve algorithm. For instance a user can choose to only use linear constraints for reducing bounds, or to not remove doubletons.

Mixed integer programming problems

If the optimization problem contains binary variables then the AIMMS Presolver can apply probing which is a technique that looks at the logical implications of fixing a binary variable to 0 or 1. Probing can be used to reduce more variables bounds, reformulate constraints or improve coefficients. In some cases quadratic constraints containing binary variables can be reformulated as linear constraints. Coefficient improvement is a process of improving the coefficients of the binary variables such that the relaxation becomes more tight. A detailed description of probing and coefficient improvement can be found in [Sav94].

Successes may vary

The benefits of using the AIMMS Presolver may vary from model to model. The solution of presolved NLPs may become better or worse compared to the original NLP. Presolving may change infeasible NLPs to feasible problems for a given starting point, or vice versa. Also, presolving may make the model more degenerate and harder to solve. Finaly, for eliminated constraints and variables dual information is lost, and AIMMS makes no effort yet to recover the lost dual information, as this may be very hard in the presence of nonlinear reductions.