Mixed-Integer Linear Programming (MILP) Algorithms

Mixed-Integer Linear Programming Definition

A mixed-integer linear program (MILP) is a problem with

Linear objective function, f^Tx, where f is a column vector of constants, and x is the column vector of unknowns
Bounds and linear constraints, but no nonlinear constraints (for definitions, see Write Constraints)
Restrictions on some components of x to have integer values

In mathematical terms, given vectors f, lb, and ub, matrices A and Aeq, corresponding vectors b and beq, and a set of indices intcon, find a vector x to solve

$\min_{x} f^{T} x subject to {\begin{array}{l} x (intcon) are integers \\ A \cdot x \leq b \\ A e q \cdot x = b e q \\ l b \leq x \leq u b . \end{array}$

intlinprog Algorithm

Algorithm Overview
Linear Program Preprocessing
Linear Programming
Mixed-Integer Program Preprocessing
Cut Generation
Heuristics for Finding Feasible Solutions
Branch and Bound

Algorithm Overview

intlinprog uses this basic strategy to solve mixed-integer linear programs. intlinprog can solve the problem in any of the stages. If it solves the problem in a stage, intlinprog does not execute the later stages.

Reduce the problem size using Linear Program Preprocessing.
Solve an initial relaxed (noninteger) problem using Linear Programming.
Perform Mixed-Integer Program Preprocessing to tighten the LP relaxation of the mixed-integer problem.
Try Cut Generation to further tighten the LP relaxation of the mixed-integer problem.
Try to find integer-feasible solutions using heuristics.
Use a Branch and Bound algorithm to search systematically for the optimal solution. This algorithm solves LP relaxations with restricted ranges of possible values of the integer variables. It attempts to generate a sequence of updated bounds on the optimal objective function value.

Linear Program Preprocessing

According to the Mixed-Integer Linear Programming Definition, there are matrices A and Aeq and corresponding vectors b and beq that encode a set of linear inequalities and linear equalities

$\begin{matrix} A \cdot x \leq b \\ A e q \cdot x = b e q . \end{matrix}$

These linear constraints restrict the solution x.

Usually, it is possible to reduce the number of variables in the problem (the number of components of x), and reduce the number of linear constraints. While performing these reductions can take time for the solver, they usually lower the overall time to solution, and can make larger problems solvable. The algorithms can make solution more numerically stable. Furthermore, these algorithms can sometimes detect an infeasible problem.

Preprocessing steps aim to eliminate redundant variables and constraints, improve the scaling of the model and sparsity of the constraint matrix, strengthen the bounds on variables, and detect the primal and dual infeasibility of the model.

For details, see Andersen and Andersen [2] and Mészáros and Suhl [8].

Linear Programming

The initial relaxed problem is the linear programming problem with the same objective and constraints as Mixed-Integer Linear Programming Definition, but no integer constraints. Call x_LP the solution to the relaxed problem, and x the solution to the original problem with integer constraints. Clearly,

f^Tx_LP ≤ f^Tx,

because x_LP minimizes the same function but with fewer restrictions.

This initial relaxed LP (root node LP) and all generated LP relaxations during the branch-and-bound algorithm are solved using linear programming solution techniques.

Mixed-Integer Program Preprocessing

During mixed-integer program preprocessing, intlinprog analyzes the linear inequalities A*x ≤ b along with integrality restrictions to determine whether:

The problem is infeasible.
Some bounds can be tightened.
Some inequalities are redundant, so can be ignored or removed.
Some inequalities can be strengthened.
Some integer variables can be fixed.

The IntegerPreprocess option lets you choose whether intlinprog takes several steps, takes all of them, or takes almost none of them. If you include an x0 argument, intlinprog uses that value in preprocessing.

The main goal of mixed-integer program preprocessing is to simplify ensuing branch-and-bound calculations. Preprocessing involves quickly preexamining and eliminating some of the futile subproblem candidates that branch-and-bound would otherwise analyze.

For details about integer preprocessing, see Savelsbergh [10].

Cut Generation

Cuts are additional linear inequality constraints that intlinprog adds to the problem. These inequalities attempt to restrict the feasible region of the LP relaxations so that their solutions are closer to integers. You control the type of cuts that intlinprog uses with the CutGeneration option.

'basic' cuts include:

Mixed-integer rounding cuts
Gomory cuts
Clique cuts
Cover cuts
Flow cover cuts

Furthermore, if the problem is purely integer (all variables are integer-valued), then intlinprog also uses the following cuts:

Strong Chvatal-Gomory cuts
Zero-half cuts

'intermediate' cuts include all 'basic' cuts, plus:

Simple lift-and-project cuts
Simple pivot-and-reduce cuts
Reduce-and-split cuts

'advanced' cuts include all 'intermediate' cuts except reduce-and-split cuts, plus:

Strong Chvatal-Gomory cuts
Zero-half cuts

For purely integer problems, 'intermediate' uses the most cut types, because it uses reduce-and-split cuts, while 'advanced' does not.

Another option, CutMaxIterations, specifies an upper bound on the number of times intlinprog iterates to generate cuts.

For details about cut generation algorithms (also called cutting plane methods), see Cornuéjols [5] and, for clique cuts, Atamtürk, Nemhauser, and Savelsbergh [3].

Heuristics for Finding Feasible Solutions

To get an upper bound on the objective function, the branch-and-bound procedure must find feasible points. A solution to an LP relaxation during branch-and-bound can be integer feasible, which can provide an improved upper bound to the original MILP. Certain techniques find feasible points faster before or during branch-and-bound. intlinprog uses these techniques at the root node and during some branch-and-bound iterations. These techniques are heuristic, meaning they are algorithms that can succeed but can also fail.

Heuristics can be start heuristics, which help the solver find an initial or new integer-feasible solution. Or heuristics can be improvement heuristics, which start at an integer-feasible point and attempt to find a better integer-feasible point, meaning one with lower objective function value. The intlinprog improvement heuristics are 'rins', 'rss', 1-opt, 2-opt, and guided diving.

Set the intlinprog heuristics using the 'Heuristics' option. The options are:

Option	Description
`'basic'` (default)	The solver runs rounding heuristics twice with different parameters, runs diving heuristics twice with different parameters, then runs `'rss'`. The solver does not run later heuristics when earlier heuristics lead to a sufficiently good integer-feasible solution.
`'intermediate'`	The solver runs rounding heuristics twice with different parameters, then runs diving heuristics twice with different parameters. If there is an integer-feasible solution, the solver then runs `'rins'` followed by `'rss'`. If `'rss'` finds a new solution, the solver runs `'rins'` again. The solver does not run later heuristics when earlier heuristics lead to a sufficiently good integer-feasible solution.
`'advanced'`	The solver runs rounding heuristics twice with different parameters, then runs diving heuristics twice with different parameters. If there is an integer-feasible solution, the solver then runs `'rins'` followed by `'rss'`. If `'rss'` finds a new solution, the solver runs `'rins'` again. The solver does not run later heuristics when earlier heuristics lead to a sufficiently good integer-feasible solution.
`'rins'` or the equivalent `'rins-diving'`	`intlinprog` searches the neighborhood of the current, best integer-feasible solution point (if available) to find a new and better solution. See Danna, Rothberg, and Le Pape [6]. When you select `'rins'`, the solver runs rounding heuristics twice with different parameters, runs diving heuristics twice with different parameters, then runs `'rins'`.
`'rss'` or the equivalent `'rss-diving'`	`intlinprog` applies a hybrid procedure combining ideas from `'rins'` and local branching to search for integer-feasible solutions. When you select `'rss'`, the solver runs rounding heuristics twice with different parameters, runs diving heuristics twice with different parameters, then runs `'rss'`. The solver does not run later heuristics when earlier heuristics lead to a sufficiently good integer-feasible solution. These settings perform the same heuristics as `'basic'`.
`'round'`	`intlinprog` takes the LP solution to the relaxed problem at a node, and rounds the integer components in a way that attempts to maintain feasibility. When you select `'round'`, the solver, at the root node, runs rounding heuristics twice with different parameters, then runs diving heuristics twice with different parameters. Thereafter, the solver runs only rounding heuristics at some branch-and-bound nodes.
`'round-diving'`	The solver works in a similar way to `'round'`, but also runs diving heuristics (in addition to rounding heuristics) at some branch-and-bound nodes, not just the root node.
`'diving'`	`intlinprog` uses heuristics that are similar to branch-and-bound steps, but follow just one branch of the tree down, without creating the other branches. This single branch leads to a fast “dive” down the tree fragment, thus the name “diving.” Currently, `intlinprog` uses six diving heuristics in this order: Vector length diving Coefficient diving Fractional diving Pseudo cost diving Line search diving Guided diving (applies when the solver already found at least one integer-feasible point) Diving heuristics generally select one variable that should be integer-valued, for which the current solution is fractional. The heuristics then introduce a bound that forces the variable to be integer-valued, and solve the associated relaxed LP again. The method of choosing the variable to bound is the main difference between the diving heuristics. See Berthold [4], Section 3.1.
`'none'`	`intlinprog` does not search for a feasible point. The solver simply takes any feasible point it encounters in its branch-and-bound search.

The main difference between 'intermediate' and 'advanced' is that 'advanced' runs heuristics more frequently during branch-and-bound iterations.

In addition to the previous table, the following heuristics run when the Heuristics option is 'basic', 'intermediate', or 'advanced'.

ZI round — This heuristic runs whenever an algorithm solves a relaxed LP. The heuristic goes through each fractional integer variable to attempt to shift it to a neighboring integer without affecting the feasibility with respect to other constraints. For details, see Hendel [7].
1-opt — This heuristic runs whenever an algorithm finds a new integer-feasible solution. The heuristic goes through each integer variable to attempt to shift it to a neighboring integer without affecting the feasibility with respect to other constraints, while lowering the objective function value.
2-opt — This heuristic runs whenever an algorithm finds a new integer-feasible solution. 2-opt finds all pairs of integer variables that affect the same constraint, meaning they have nonzero entries in the same row of an A or Aeq constraint matrix. For each pair, 2-opt takes an integer-feasible solution and moves the values of the variable pairs up or down using all four possible moves (up-up, up-down, down-up, and down-down), looking for a feasible neighboring solution that has a better objective function value. The algorithm tests each integer variable pair by calculating the largest size (same magnitude) of shifts for each variable in the pair that satisfies the constraints and also improves the objective function value.

At the beginning of the heuristics phase, intlinprog runs the trivial heuristic unless Heuristics is 'none' or you provide an initial integer-feasible point in the x0 argument. The trivial heuristic checks the following points for feasibility:

All zeros
Upper bound
Lower bound (if nonzero)
"Lock" point

The "lock" point is defined only for problems with finite upper and lower bounds for all variables. The "lock" point for each variable is its upper or lower bound, chosen as follows. For each variable j, count the number of corresponding positive entries in the linear constraint matrix A(:,j) and subtract the number corresponding negative entries. If the result is positive, use the lower bound for that variable, lb(j). Otherwise, use the upper bound for that variable, ub(j). The "lock" point attempts to satisfy the largest number of linear inequality constraints for each variable, but is not necessarily feasible.

After each heuristic completes with a feasible solution, intlinprog calls output functions and plot functions. See intlinprog Output Function and Plot Function Syntax.

If you include an x0 argument, intlinprog uses that value in the 'rins' and guided diving heuristics until it finds a better integer-feasible point. So when you provide x0, you can obtain good results by setting the 'Heuristics' option to 'rins-diving' or another setting that uses 'rins'.

Branch and Bound

The branch-and-bound method constructs a sequence of subproblems that attempt to converge to a solution of the MILP. The subproblems give a sequence of upper and lower bounds on the solution f^Tx. The first upper bound is any feasible solution, and the first lower bound is the solution to the relaxed problem. For a discussion of the upper bound, see Heuristics for Finding Feasible Solutions.

As explained in Linear Programming, any solution to the linear programming relaxed problem has a lower objective function value than the solution to the MILP. Also, any feasible point x_feas satisfies

f^Tx_feas ≥ f^Tx,

because f^Tx is the minimum among all feasible points.

In this context, a node is an LP with the same objective function, bounds, and linear constraints as the original problem, but without integer constraints, and with particular changes to the linear constraints or bounds. The root node is the original problem with no integer constraints and no changes to the linear constraints or bounds, meaning the root node is the initial relaxed LP.

From the starting bounds, the branch-and-bound method constructs new subproblems by branching from the root node. The branching step is taken heuristically, according to one of several rules. Each rule is based on the idea of splitting a problem by restricting one variable to be less than or equal to an integer J, or greater than or equal to J+1. These two subproblems arise when an entry in x_LP, corresponding to an integer specified in intcon, is not an integer. Here, x_LP is the solution to a relaxed problem. Take J as the floor of the variable (rounded down), and J+1 as the ceiling (rounded up). The resulting two problems have solutions that are larger than or equal to f^Tx_LP, because they have more restrictions. Therefore, this procedure potentially raises the lower bound.

The performance of the branch-and-bound method depends on the rule for choosing which variable to split (the branching rule). The algorithm uses these rules, which you can set in the BranchRule option:

'maxpscost' — Choose the fractional variable with maximal pseudocost.
Pseudocost
The pseudocost of a variable i is based on empirical estimates of the change in the lower bound when i has been chosen as the branching variable, combined with the fractional part of the i component of the current point x. The fractional part p is in two pieces, the lower part and the upper part:
p_i^– = x(i) – ⌊x(i)⌋
p_i⁺ = 1 – p_i^–.
Let x_i^– be the solution of the linear program restricted to have x(i) ≤ ⌊x(i)⌋, and let the change in objective function be denoted
Δ_i^– = f^Tx_i^– – f^Tx.
Similarly, Δ_i⁺ is the change in objective function when the problem is restricted to have x(i) ≥ ⌈x(i)⌉.
The objective gain per unit change in variable x_i is
$d_{i}^{-} = \frac{Δ_{i}^{-}}{p_{i}^{-}} or d_{i}^{+} = \frac{Δ_{i}^{+}}{p_{i}^{+}} .$
Let s_i^– and s_i⁺ be the empirical averages of d_i^– and d_i⁺ during the branch-and-bound algorithm up to this point. The empirical values are initialized to the absolute value of the objective coefficient f(i) for the terms before there are any observations. Then the 'maxpscost' rule is to branch on a node i that maximizes, for some positive weights w⁺ and w^–, the quantity
w^– * p_i^– * s_i^– + w⁺ * p_i⁺ * s_i⁺.
Roughly speaking, this rule chooses a coefficient that is likely to increase the lower bound maximally.
'strongpscost' — Similar to 'maxpscost', but instead of the pseudocost being initialized to 1 for each variable, the solver attempts to branch on a variable only after the pseudocost has a more reliable estimate. To obtain a more reliable estimate, the solver does the following (see Achterberg, Koch, and Martin [1]).
- Order all potential branching variables (those that are currently fractional but should be integer) by their current pseudocost-based scores.
- Run the two relaxed linear programs based on the current branching variable, starting from the variable with the highest score (if the variable has not yet been used for a branching calculation). The solver uses these two solutions to update the pseudocosts for the current branching variable. The solver can halt this process early to save time in choosing the branch.
- Continue choosing variables in the list until the current highest pseudocost-based score does not change for k consecutive variables, where k is an internally chosen value, usually between 5 and 10.
- Branch on the variable with the highest pseudocost-based score. The solver might have already computed the relaxed linear programs based on this variable during an earlier pseudocost estimation procedure.
Because of the extra linear program solutions, each iteration of 'strongpscost' branching takes longer than the default 'maxpscost'. However, the number of branch-and-bound iterations typically decreases, so the 'strongpscost' method can save time overall.
'reliability' — Similar to 'strongpscost', but instead of running the relaxed linear programs only for uninitialized pseudocost branches, 'reliability' runs the programs up to k2 times for each variable, where k2 is a small integer such as 4 or 8. Therefore, 'reliability' has even slower branching, but potentially fewer branch-and-bound iterations, compared to 'strongpscost'.
'mostfractional' — Choose the variable with fractional part closest to 1/2.
'maxfun' — Choose the variable with maximal corresponding absolute value in the objective vector f.

After the algorithm branches, there are two new nodes to explore. The algorithm chooses which node to explore among all that are available using one of these rules:

'minobj' — Choose the node that has the lowest objective function value.
'mininfeas' — Choose the node with the minimal sum of integer infeasibilities. This means for every integer-infeasible component x(i) in the node, add up the smaller of p_i^– and p_i⁺, where
p_i^– = x(i) – ⌊x(i)⌋
p_i⁺ = 1 – p_i^–.
'simplebestproj' — Choose the node with the best projection.
Best Projection
Let x_B denote the best integer-feasible point found so far, x_R demote the LP relaxed solution at the root node, and x denote the node we examine. Let in(x) denote the sum of integer infeasibilities at the node x (see 'mininfeas'). The best projection rule is to minimize
$f^{T} x + \frac{f^{T} x_{B} - f^{T} x_{R}}{i n (x_{R})} i n (x) .$
If there is no integer-feasible point found so far, set f^Tx_B = 0.

intlinprog skips the analysis of some subproblems by considering information from the original problem such as the objective function’s greatest common divisor (GCD).

The branch-and-bound procedure continues, systematically generating subproblems to analyze and discarding the ones that won’t improve an upper or lower bound on the objective, until one of these stopping criteria is met:

The algorithm exceeds the MaxTime option.
The difference between the lower and upper bounds on the objective function is less than the AbsoluteGapTolerance or RelativeGapTolerance tolerances.
The number of explored nodes exceeds the MaxNodes option.
The number of integer feasible points exceeds the MaxFeasiblePoints option.

For details about the branch-and-bound procedure, see Nemhauser and Wolsey [9] and Wolsey [11].

References

[1] Achterberg, T., T. Koch and A. Martin. Branching rules revisited. Operations Research Letters 33, 2005, pp. 42–54. Available at https://opus4.kobv.de/opus4-zib/files/788/ZR-04-13.pdf.

[2] Andersen, E. D., and Andersen, K. D. Presolving in linear programming. Mathematical Programming 71, pp. 221–245, 1995.

[3] Atamtürk, A., G. L. Nemhauser, M. W. P. Savelsbergh. Conflict graphs in solving integer programming problems. European Journal of Operational Research 121, 2000, pp. 40–55.

[4] Berthold, T. Primal Heuristics for Mixed Integer Programs. Technischen Universität Berlin, September 2006. Available at https://www.zib.de/groetschel/students/Diplom-Berthold.pdf.

[5] Cornuéjols, G. Valid inequalities for mixed integer linear programs. Mathematical Programming B, Vol. 112, pp. 3–44, 2008.

[6] Danna, E., Rothberg, E., Le Pape, C. Exploring relaxation induced neighborhoods to improve MIP solutions. Mathematical Programming, Vol. 102, issue 1, pp. 71–90, 2005.

[7] Hendel, G. New Rounding and Propagation Heuristics for Mixed Integer Programming. Bachelor's thesis at Technische Universität Berlin, 2011. PDF available at https://opus4.kobv.de/opus4-zib/files/1332/bachelor_thesis_main.pdf.

[8] Mészáros C., and Suhl, U. H. Advanced preprocessing techniques for linear and quadratic programming. OR Spectrum, 25(4), pp. 575–595, 2003.

[9] Nemhauser, G. L. and Wolsey, L. A. Integer and Combinatorial Optimization. Wiley-Interscience, New York, 1999.

[10] Savelsbergh, M. W. P. Preprocessing and Probing Techniques for Mixed Integer Programming Problems. ORSA J. Computing, Vol. 6, No. 4, pp. 445–454, 1994.

[11] Wolsey, L. A. Integer Programming. Wiley-Interscience, New York, 1998.