A mixed-integer linear program is a problem with

Linear objective function,

*f*^{T}*x*, where*f*is a column vector of constants, and*x*is the column vector of unknownsBounds 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

$$\underset{x}{\mathrm{min}}{f}^{T}x\text{subjectto}\{\begin{array}{l}x(\text{intcon})\text{areintegers}\hfill \\ A\cdot x\le b\hfill \\ Aeq\cdot x=beq\hfill \\ lb\le x\le ub.\hfill \end{array}$$

`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.

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

$$\begin{array}{c}A\text{\hspace{0.05em}}\xb7\text{\hspace{0.05em}}x\le b\\ Aeq\text{\hspace{0.05em}}\xb7\text{\hspace{0.05em}}x=beq.\end{array}$$

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 [1] and Mészáros and Suhl [5].

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*^{T}*x*_{LP} ≤ *f*^{T}*x*,

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.

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.

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 [7].

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

Cliques 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 [2].

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. There are techniques
for finding feasible points faster before or during branch-and-bound.
Currently, `intlinprog`

uses these techniques only
at the root node, not during the branch-and-bound iterations. These
techniques are heuristic, meaning they are algorithms that can succeed
but can also fail.

Set the `intlinprog`

heuristics in the `'Heuristics'`

option.
The options are:

`'rss'`

(default) —`intlinprog`

applies a hybrid procedure combining ideas from`'rins'`

and local branching to search for integer-feasible solutions.`'none'`

—`intlinprog`

does not search for a feasible point. It simply takes any feasible point it encounters in its branch-and-bound search.`'rins'`

—`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 [3].`'round'`

—`intlinprog`

takes the LP solution to the relaxed problem at a node. It rounds the integer components in a way that attempts to maintain feasibility.`'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, hence the name "diving." Currently,`intlinprog`

uses four diving heuristics sequentially, and halts when it obtains an integer-feasible point:Vector length diving

Coefficient diving

Fractional diving

Guided diving (applies when

`intlinprog`

already found at least one integer-feasible point)

`intlinprog`

uses diving heuristics only when it has found no integer-feasible point with a relative gap less than`5e-2`

.Diving heuristics generally select one variable that is supposed to be integer-valued, for which the current solution is fractional. They then introduce a bound that forces that 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 Grötschel [4], Section 3.1.

`'rss-diving'`

,`'rins-diving'`

, or`'round-diving'`

—`intlinprog`

tries the first named heuristic method (`'rins'`

,`'rss'`

, or`'round'`

). If this heuristic fails to find an integer-feasible point,`intlinprog`

then tries`'diving'`

.

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*^{T}*x*.
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*^{T}*x*_{feas} ≥ *f*^{T}*x*,

because *f*^{T}*x* 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*^{T}*x*_{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*.`'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*and_{i}^{–}*p*, where_{i}^{+}*p*=_{i}^{–}*x*(*i*) – ⌊*x*(*i*)⌋*p*= 1 –_{i}^{+}*p*._{i}^{–}`'simplebestproj'`

— Choose the node with the*best projection*.

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 [6] and Wolsey [8].

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

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

[3] 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.

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

.

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

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

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

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

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