MixedInteger Linear Programming (MILP) Algorithms
MixedInteger Linear Programming Definition
A mixedinteger linear program (MILP) 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 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
$$\underset{x}{\mathrm{min}}{f}^{T}x\text{subjectto}\{\begin{array}{l}x(\text{intcon})\text{areintegers}\hfill \\ A\cdot x\le b\hfill \\ Aeq\cdot x=beq\hfill \\ lb\le x\le ub.\hfill \end{array}$$
Legacy intlinprog
Algorithm
Algorithm Overview
The 'legacy'
intlinprog
algorithm uses this basic strategy to solve
mixedinteger 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 MixedInteger Program Preprocessing to tighten the LP relaxation of the mixedinteger problem.
Try Cut Generation to further tighten the LP relaxation of the mixedinteger problem.
Try to find integerfeasible 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 MixedInteger 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 [3] and Mészáros and Suhl [10].
Linear Programming
The initial relaxed problem is the linear programming problem with the same objective and constraints as MixedInteger 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 branchandbound algorithm are solved using linear programming solution techniques.
MixedInteger Program Preprocessing
During mixedinteger 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, and therefore some variables can be fixed.
Some inequalities are redundant, so can be ignored or removed.
Some inequalities can be strengthened.
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 mixedinteger program preprocessing is to simplify ensuing branchandbound calculations. Preprocessing involves quickly preexamining and eliminating some of the futile subproblem candidates that branchandbound would otherwise analyze.
For details about integer preprocessing, see Achterberg et al. [1].
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:
Mixedinteger rounding cuts
Gomory cuts
Clique cuts
Cover cuts
Flow cover cuts
Furthermore, if the problem is purely integer (all variables are
integervalued), then intlinprog
also uses the following
cuts:
Strong ChvatalGomory cuts
Zerohalf cuts
'intermediate'
cuts include all 'basic'
cuts, plus:
Simple liftandproject cuts
Simple pivotandreduce cuts
Reduceandsplit cuts
'advanced'
cuts include all
'intermediate'
cuts except reduceandsplit cuts,
plus:
Strong ChvatalGomory cuts
Zerohalf cuts
For purely integer problems, 'intermediate'
uses the most
cut types, because it uses reduceandsplit 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 [6] and, for clique cuts, Atamtürk, Nemhauser, and Savelsbergh [4].
Heuristics for Finding Feasible Solutions
To get an upper bound on the objective function, the branchandbound
procedure must find feasible points. A solution to an LP relaxation during
branchandbound can be integer feasible, which can provide an improved upper
bound to the original MILP. Certain techniques find feasible points faster
before or during branchandbound. intlinprog
uses these
techniques at the root node and during some branchandbound 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 integerfeasible solution. Or heuristics can be
improvement heuristics, which start at an
integerfeasible point and attempt to find a better integerfeasible point,
meaning one with lower objective function value. The
intlinprog
improvement heuristics are
'rins'
, 'rss'
, 1opt, 2opt, 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 
'intermediate'  The solver runs rounding heuristics twice with
different parameters, then runs diving heuristics twice with
different parameters. If there is an integerfeasible
solution, the solver then runs 
'advanced'  The solver runs rounding heuristics twice with
different parameters, then runs diving heuristics twice with
different parameters. If there is an integerfeasible
solution, the solver then runs 
'rins' or the equivalent
'rinsdiving' 

'rss' or the equivalent
'rssdiving' 

'round' 

'rounddiving'  The solver works in a similar way to

'diving' 
Diving heuristics generally select one variable that should be integervalued, for which the current solution is fractional. The heuristics then introduce a bound that forces the variable to be integervalued, 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 [5], Section 3.1. 
'none' 

The main difference between 'intermediate'
and
'advanced'
is that 'advanced'
runs
heuristics more frequently during branchandbound 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 [9].
1opt — This heuristic runs whenever an algorithm finds a new integerfeasible 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.
2opt — This heuristic runs whenever an algorithm finds a new integerfeasible solution. 2opt finds all pairs of integer variables that affect the same constraint, meaning they have nonzero entries in the same row of an
A
orAeq
constraint matrix. For each pair, 2opt takes an integerfeasible solution and moves the values of the variable pairs up or down using all four possible moves (upup, updown, downup, and downdown), 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 integerfeasible 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
integerfeasible point. So when you provide x0
, you can obtain good results
by setting the 'Heuristics'
option to 'rinsdiving'
or
another setting that uses 'rins'
.
Branch and Bound
The branchandbound 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 branchandbound 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 branchandbound 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.'strongpscost'
— Similar to'maxpscost'
, but instead of the pseudocost being initialized to1
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 [2]).Order all potential branching variables (those that are currently fractional but should be integer) by their current pseudocostbased 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 pseudocostbased score does not change for
k
consecutive variables, wherek
is an internally chosen value, usually between 5 and 10.Branch on the variable with the highest pseudocostbased 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 branchandbound 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 tok2
times for each variable, wherek2
is a small integer such as 4 or 8. Therefore,'reliability'
has even slower branching, but potentially fewer branchandbound iterations, compared to'strongpscost'
.'mostfractional'
— Choose the variable with fractional part closest to1/2
.'maxfun'
— Choose the variable with maximal corresponding absolute value in the objective vectorf
.
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 integerinfeasible component x(i) in the node, add up the smaller of p_{i}^{–} and p_{i}^{+}, wherep_{i}^{–} = x(i) – ⌊x(i)⌋
p_{i}^{+} = 1 – p_{i}^{–}.'simplebestproj'
— Choose the node with the best projection.
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 branchandbound 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
orRelativeGapTolerance
tolerances.The number of explored nodes exceeds the
MaxNodes
option.The number of integer feasible points exceeds the
MaxFeasiblePoints
option.
For details about the branchandbound procedure, see Nemhauser and Wolsey [11] and Wolsey [13].
HiGHS MILP Algorithm
Overview of HiGHS
The intlinprog
"highs"
algorithm is based on the HiGHS opensource software.
intlinprog
converts MATLAB^{®}formatted inputs and options into equivalent HiGHS arguments, and
converts the returned solution into standard MATLAB format as well.
Algorithm Outline
The "highs"
algorithm performs these steps.
Get a branch/bound node (if none then stop)
Repeat until a stopping condition:
Search until a stopping condition:
Perform Plunge/Diving
Propagate domain
Prune nodes, update bounds, exit if infeasibility detected or
ObjectiveCutOff
option value is reachedIf restart conditions are met, return to step 2
Install the next node:
Choose and evaluate a node
If evaluation prunes this node, return to step 5
Generate cuts for the node
If domain is infeasible, cut off the node, and bring open nodes into the node queue
Update the basis
Go to step 4
Presolve
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.
Presolve 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 background, see Andersen and Andersen [3] and Mészáros and Suhl [10].
During mixedinteger 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.
For background about integer preprocessing, see Achterberg et al. [1].
Evaluate Root Node
To evaluate the root node, the algorithm performs the following steps.
Detect symmetry and simplify problem.
Evaluate root LP.
Generate and add LP cuts (see Cut Generation).
Apply randomized rounding.
Generate and add more LP cuts.
Perform cut generation and heuristics in a loop.
Check for restart conditions, restart at step 2 if warranted.
After the root node evaluation completes, the algorithm proceeds with BranchandBound.
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. For background about cut generation algorithms
(also called cutting plane methods), see Cornuéjols [6] and Atamtürk, Nemhauser, and Savelsbergh [4].
Plunge/Diving
To find integerfeasible points, intlinprog
uses
heuristics that are similar to branchandbound 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.”
Diving heuristics generally select one variable that should be integervalued, for which the current solution is fractional. The heuristics then introduce a bound that forces the variable to be integervalued, 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 [5], Section 3.1.
Randomized Rounding, RINS, and RENS
To find new integerfeasible points, intlinprog
searches
the neighborhood of the current, best integerfeasible solution point (if
available) to find a new and better solution. See Danna, Rothberg, and Le Pape
[8]. Similarly, to find new integerfeasible points,
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. By taking randomized rounding steps,
intlinprog
can sometimes find a new feasible point.
RENS, which stands for Relaxation Enforced Neighborhood Search, is another
search technique for finding integerfeasible points. See Berthold [5].
BranchandBound
The branchandbound 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. These bounds are called the primal and dual bounds. For a minimization problem, the first upper bound (primal) is any feasible solution, and the first lower bound (dual) is the solution to the relaxed problem. For a maximization problem, the primal bound is the lower bound and the dual bound is the upper bound. As explained in Linear Programming, for a minimization problem, 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,  (1) 
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 branchandbound 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, for a minimization problem this
procedure potentially raises the lower bound.
After the algorithm branches, there are two new nodes to explore.
intlinprog
skips the analysis of some subproblems by
comparing their objective function values with the current global bounds.
The branchandbound procedure continues, systematically generating subproblems to analyze and discarding the ones that will not improve an upper or lower bound on the objective, until one of these stopping criteria is met:
The problem is proved to be infeasible.
The objective function value reaches the
ObjectiveCutOff
limit.The algorithm exceeds the
MaxTime
option.The difference between the lower and upper bounds on the objective function is less than the
AbsoluteGapTolerance
orRelativeGapTolerance
tolerances.The number of explored nodes exceeds the
MaxNodes
option.
For background about the branchandbound procedure, see Nemhauser and Wolsey [11] and Wolsey [13].
Iterative Display
When you select iterative display by setting the Display
option to the default "iter"
, the solver displays some of its
steps. HiGHS iterative display is more extensive and complicated than the
iterative display of other solvers. Furthermore, the HiGHS algorithm can restart
its branchandbound search, leading to an iterative display that also
restarts.
To select iterative display:
options = optimoptions("intlinprog",Display="iter"); [x,fval,exitflag,output] = intlinprog(f,intcon,A,b,Aeq,beq,... lb,ub,options)
Preamble. The iterative display begins by displaying the results of "presolve." The presolve algorithm reduces the complexity of the original problem by identifying and removing redundant rows and columns in the linear constraint matrices and performing related simplifications of the problem. For example,
Presolving model 18018 rows, 26027 cols, 248579 nonzeros 15092 rows, 24343 cols, 217277 nonzeros Objective function is integral with scale 1
Root Node Evaluation. The root node is the linear programming solution of the problem, not
considering any integer constraints. For a minimization problem, the
objective function value of the root node is a lower bound on the objective
function value of the solution to the problem including integer constraints.
For a minimization problem, the upper bound (if any) comes from a feasible
point with respect to all constraints. If there is no feasible point yet
found, the upper bound is Inf
.
The iterative display shows the size of the problem after presolve.
Solving MIP model with: 15092 rows 24343 cols (24343 binary, 0 integer, 0 implied int., 0 continuous) 217277 nonzeros
binary
is the number of binary variables.integer
is the number of integer variables.implied int
is the number of variables that are implied to be integer. For example, ifx(1)
is integer andx(1) + x(2) = 5
, thenx(2)
is implied to be integer.continuous
is the number of continuous variables.
The total number of variables is the number of columns in the model.
Dynamic Constraint Creation. The software starts by creating "dynamic constraints," which have three headers in the iterative display:
Cuts — Number of active cuts
inLp — Number of nonmodel rows in the LP matrix
Confl. — Number of conflicts
The software can further extend the constraints by restarting the dynamic constraint creation, which can create more constraints by starting the creation process from a new state.
In the last step before beginning the branchandbound process, the software reports the results of symmetry detection and the number of generators and orbitopes found. For example, this is the portion of the iterative display that appears in the preamble and dynamic constraint creation phase:
Nodes  B&B Tree  Objective Bounds  Dynamic Constraints  Work Proc. InQueue  Leaves Expl.  BestBound BestSol Gap  Cuts InLp Confl.  LpIters Time 0 0 0 0.00% 0 inf inf 0 0 0 0 1.5s 0 0 0 0.00% 0 inf inf 0 0 5 4934 3.6s … 0 0 0 0.00% 0 inf inf 4630 553 289 140296 159.6s 0.0% inactive integer columns, restarting Model after restart has 15090 rows, 24338 cols (24338 bin., 0 int., 0 impl., 0 cont.), and 217075 nonzeros 0 0 0 0.00% 0 inf inf 550 0 0 140296 160.5s … 0 0 0 0.00% 0 inf inf 5602 524 260 277323 318.0s 6.2% inactive integer columns, restarting Model after restart has 14185 rows, 22816 cols (22816 bin., 0 int., 0 impl., 0 cont.), and 200423 nonzeros 0 0 0 0.00% 0 inf inf 524 0 0 277323 318.6s … 0 0 0 0.00% 0 inf inf 4683 535 525 408393 505.8s Symmetry detection completed in 3.1s Found 215 generators and 12 full orbitope(s) acting on 730 columns
For information about symmetry detection and orbitopes, see Hojny, Pfetsch, and Schmitt [7] and Pfetsch and Rehn [12]. The software continues to add dynamic constraints as it proceeds with branchandbound steps, described next.
BranchandBound. The branchandbound 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. For a minimization problem, the first upper bound is any feasible solution, and the first lower bound is the solution to the relaxed problem.
During the branchandbound procedure, intlinprog
gives the following iterative display.
Nodes  B&B Tree  Objective Bounds  Dynamic Constraints  Work Proc. InQueue  Leaves Expl.  BestBound BestSol Gap  Cuts InLp Confl.  LpIters Time 72 0 2 1.56% 0 inf inf 4695 535 738 667009 686.9s … T 271 107 51 1.56% 0 449 100.00% 6051 271 1767 792786 776.8s T 279 107 53 1.56% 0 439 100.00% 6053 271 1794 793241 777.5s … L 1223 538 295 1.98% 0 434 100.00% 6984 243 7628 1689k 1580.6s … 1321 539 333 1.98% 0 434 100.00% 7029 243 8628 1898k 1650.6s Restarting search from the root node Model after restart has 13947 rows, 22426 cols (22426 bin., 0 int., 0 impl., 0 cont.), and 194633 nonzeros 1323 0 0 0.00% 0 434 100.00% 243 0 0 1902k 1653.5s 1323 0 0 0.00% 0 434 100.00% 243 76 10 1905k 1655.7s … 1694 173 52 0.00% 0 434 100.00% 9411 318 2220 3205k 2584.3s B 1710 167 55 0.00% 0 433 100.00% 9415 318 2263 3207k 2586.2s 1726 224 56 0.00% 0 433 100.00% 9999 378 2307 3237k 2608.7s
The leftmost column shows a code indicating how the new feasible point was found for that row of the display:
L
— While solving a subMIP problem during primal heuristicsT
— During tree search, while evaluating a nodeB
— During branchingH
— By heuristicsP
— During startup, before solving MIPC
— By central roundingR
— By randomized roundingS
— While solving an LPF
— By Feasibility PumpU
— From an unbounded LP
Finish. The reason that the algorithm stopped and a summary of results appear at the end of the iterative display.
Solving report Status Time limit reached Primal bound 14 Dual bound 0 Gap 100% (tolerance: 0.01%) Solution status feasible 14 (objective) 0 (bound viol.) 1.7763568394e15 (int. viol.) 0 (row viol.) Timing 7200.02 (total) 3.08 (presolve) 0.00 (postsolve) Nodes 5830 LP iterations 9963775 (total) 2657448 (strong br.) 324908 (separation) 2140011 (heuristics) Solver stopped prematurely. Integer feasible point found. Intlinprog stopped because it exceeded the time limit, options.MaxTime = 7200 . The intcon variables are integer within tolerance, options.ConstraintTolerance = 1e06.
Status
— Reason the iterations stoppedPrimal bound
— Upper bound on objective function value for a minimization problem, lower bound for a maximization problemDual bound
— Lower bound on objective function value for a maximization problem, upper bound for a minimization problemGap
— Relative gap between primal and dual boundsSolution status
Status of the solution
Objective function value, labeled
(objective)
Maximum violation of the solution with respect to variable bounds, labeled
(bound viol.)
Maximum violation of the integertype variables from integer values, labeled
(int. viol.)
Maximum violation of the solution with respect to the linear constraints, labeled
(row viol.)
Timing
— Timing of various solution phases in secondsNodes
— Number of nodes exploredLP iterations
— Number of linear program iterations by phase and in total
References
[1] Achterberg, T.,Bixby, R.,
Gu, Z., Rothberg, E., and Weninger, D. Presolve Reductions in Mixed
Integer Programming. INFORMS J. Computing 32(2), 2019, pp. 473–506.
Available at https://doi.org/10.1287/ijoc.2018.0857
.
[2] Achterberg, T., T. Koch
and A. Martin. Branching rules revisited. Operations Research
Letters 33, 2005, pp. 42–54. Available at https://opus4.kobv.de/opus4zib/files/788/ZR0413.pdf
.
[3] Andersen, E. D., and Andersen, K. D. Presolving in linear programming. Mathematical Programming 71, pp. 221–245, 1995.
[4] 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.
[5] Berthold, T. Primal Heuristics for Mixed
Integer Programs. Technischen Universität Berlin, September 2006.
Available at https://opus4.kobv.de/opus4zib/files/1029/Berthold_Primal_Heuristics_For_Mixed_Integer_Programs.pdf
.
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