A modern branch and bound algorithm that uses the NAG optimization modelling suite (see Section 3.1 in the E04 Chapter Introduction) has been introduced. This new routine h02bkf has access to all of the suite facilities and uses the same interface as all compatible solvers.
h02bbf solves ‘zero-one’, ‘general’, ‘mixed’ or ‘all’ integer programming problems using a branch and bound method. The routine may also be used to find either the first integer solution or the optimum integer solution. It is not intended for large sparse problems.
The routine may be called by the names h02bbf or nagf_mip_ilp_dense.
3Description
h02bbf is capable of solving certain types of integer programming (IP) problems using a branch and bound (B&B) method, see Taha (1987). In order to describe these types of integer programs and to briefly state the B&B method, we define the following Linear Programming (LP) problem:
If, in (1), it is required that (some or) all the variables take integer values, then the integer program is of type (mixed or) all general IP problem. If additionally, the integer variables are restricted to take only $0$–$1$ values (i.e., ${l}_{j}=0$ and ${u}_{j}=1$) then the integer program is of type (mixed or all) zero-one IP problem.
The B&B method applies directly to these integer programs. The general idea of B&B (for a full description see Dakin (1965) or Mitra (1973)) is to solve the problem without the integral restrictions as an LP problem (first node). If in the optimal solution an integer variable ${x}_{k}$ takes a noninteger value ${x}_{k}^{*}$, two LP sub-problems are created by branching, imposing ${x}_{k}\le \left[{x}_{k}^{*}\right]$ and ${x}_{k}\ge \left[{x}_{k}^{*}\right]+1$ respectively, where $\left[{x}_{k}^{*}\right]$ denotes the integer part of ${x}_{k}^{*}$. This method of branching continues until the first integer solution (bound) is obtained. The hanging nodes are then solved and investigated in order to prove the optimality of the solution. At each node, an LP problem is solved using e04mff/e04mfa.
4References
Dakin R J (1965) A tree search algorithm for mixed integer programming problems Comput. J.8 250–255
Mitra G (1973) Investigation of some branch and bound strategies for the solution of mixed integer linear programs Math. Programming4 155–170
Taha H A (1987) Operations Research: An Introduction Macmillan, New York
5Arguments
1: $\mathbf{itmax}$ – IntegerInput/Output
On entry: an upper bound on the number of iterations for each LP problem.
On exit: unchanged if on entry ${\mathbf{itmax}}>0$, else ${\mathbf{itmax}}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}(50,5\times ({\mathbf{n}}+{\mathbf{m}}))$.
2: $\mathbf{msglvl}$ – IntegerInput
On entry: the amount of printout produced by h02bbf, as indicated below (see Section 5.1 for a description of the printed output). All output is written to the current advisory message unit (as defined by x04abf).
Value
Definition
0
No output.
1
The final IP solution only.
5
One line of output for each node investigated and the final IP solution.
10
The original LP solution (first node), one line of output for each node investigated and the final IP solution.
3: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of variables.
Constraint:
${\mathbf{n}}>0$.
4: $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of general linear constraints.
Constraint:
${\mathbf{m}}\ge 0$.
5: $\mathbf{a}({\mathbf{lda}},*)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array a
must be at least
${\mathbf{n}}$ if ${\mathbf{m}}>0$ and at least $1$ if ${\mathbf{m}}=0$.
On entry: the
$\mathit{i}$th row of a must contain the coefficients of the $\mathit{i}$th general constraint, for $\mathit{i}=1,2,\dots ,m$.
If ${\mathbf{m}}=0$ then the array a is not referenced.
6: $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which h02bbf is called.
7: $\mathbf{bl}\left({\mathbf{n}}+{\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput
8: $\mathbf{bu}\left({\mathbf{n}}+{\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: bl must contain the lower bounds and bu the upper bounds, for all the constraints in the following order. The first $n$ elements of each array must contain the bounds on the variables, and the next $m$ elements the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e., ${l}_{j}=-\infty $), set ${\mathbf{bl}}\left(j\right)\le -{\mathbf{bigbnd}}$ and to specify a nonexistent upper bound (i.e., ${u}_{j}=+\infty $), set ${\mathbf{bu}}\left(j\right)\ge {\mathbf{bigbnd}}$. To specify the $j$th constraint as an equality, set ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)=\beta $, say, where $\left|\beta \right|<{\mathbf{bigbnd}}$.
Constraints:
${\mathbf{bl}}\left(\mathit{j}\right)\le {\mathbf{bu}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{m}}$;
if ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)=\beta $, $\left|\beta \right|<{\mathbf{bigbnd}}$.
On entry: indicates which are the integer variables in the problem. For example, if ${x}_{\mathit{j}}$ is an integer variable then ${\mathbf{intvar}}\left(\mathit{j}\right)$ must be set to $1$, and $0$ otherwise.
Constraints:
${\mathbf{intvar}}\left(\mathit{j}\right)=0$ or $1$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}$;
${\mathbf{intvar}}\left(\mathit{j}\right)=1$ for at least one value of $\mathit{j}$.
10: $\mathbf{cvec}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the coefficients ${c}_{j}$ of the objective function $F\left(x\right)={c}_{1}{x}_{1}+{c}_{2}{x}_{2}+\cdots +{c}_{n}{x}_{n}$. The routine attempts to find a minimum of $F\left(x\right)$. If a maximum of $F\left(x\right)$ is desired,
${\mathbf{cvec}}\left(\mathit{j}\right)$ should be set to $-{c}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n$, so that the routine will find a minimum of $-F\left(x\right)$.
11: $\mathbf{maxnod}$ – IntegerInput
On entry: the maximum number of nodes that are to be searched in order to find a solution (optimum integer solution). If ${\mathbf{maxnod}}\le 0$ and ${\mathbf{intfst}}\le 0$, then the B&B tree search is continued until all the nodes have been investigated.
12: $\mathbf{intfst}$ – IntegerInput
On entry: specifies whether to terminate the B&B tree search after the first integer solution (if any) is obtained. If ${\mathbf{intfst}}>0$ then the B&B tree search is terminated at node $k$ say, which contains the first integer solution. For ${\mathbf{maxnod}}>0$ this applies only if $k\le {\mathbf{maxnod}}$.
13: $\mathbf{maxdpt}$ – IntegerInput
On entry: the maximum depth of the B&B tree used for branch and bound.
14: $\mathbf{toliv}$ – Real (Kind=nag_wp)Input/Output
On entry: the integer feasibility tolerance; i.e., an integer variable is considered to take an integer value if its violation does not exceed toliv. For example, if the integer variable ${x}_{j}$ is near unity then ${x}_{j}$ is considered to be integer only if $(1-{\mathbf{toliv}})\le {x}_{j}\le (1+{\mathbf{toliv}})$.
On exit: unchanged if on entry ${\mathbf{toliv}}>0.0$, else ${\mathbf{toliv}}={10}^{\mathrm{-5}}$.
15: $\mathbf{tolfes}$ – Real (Kind=nag_wp)Input/Output
On entry: the maximum acceptable absolute violation in each constraint at a ‘feasible’ point (feasibility tolerance); i.e., a constraint is considered satisfied if its violation does not exceed tolfes.
On exit: unchanged if on entry ${\mathbf{tolfes}}>0.0$, else ${\mathbf{tolfes}}=\sqrt{\epsilon}$ (where $\epsilon $ is the machine precision).
16: $\mathbf{bigbnd}$ – Real (Kind=nag_wp)Input/Output
On entry: the ‘infinite’ bound size in the definition of the problem constraints. More precisely, any upper bound greater than or equal to bigbnd will be regarded as $+\infty $ and any lower bound less than or equal to $-{\mathbf{bigbnd}}$ will be regarded as $-\infty $.
On exit: unchanged if on entry ${\mathbf{bigbnd}}>0.0$, else ${\mathbf{bigbnd}}={10}^{20}$.
17: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: an initial estimate of the original LP solution.
On exit: with ${\mathbf{ifail}}={\mathbf{0}}$, x contains a solution which will be an estimate of either the optimum integer solution or the first integer solution, depending on the value of intfst. If ${\mathbf{ifail}}={\mathbf{9}}$, then x contains a solution which will be an estimate of the best integer solution that was obtained by searching maxnod nodes.
18: $\mathbf{objmip}$ – Real (Kind=nag_wp)Output
On exit: with ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{9}}$, objmip contains the value of the objective function for the IP solution.
If ${\mathbf{msglvl}}>0$, the amounts of workspace provided and required (with ${\mathbf{maxdpt}}=3\times {\mathbf{n}}/2$) are printed. As an alternative to computing maxdpt, liwork and lrwork from the formulas given above, you may prefer to obtain appropriate values from the output of a preliminary run with the values of maxdpt, liwork and lrwork set to $1$. If however only liwork and lrwork are set to $1$, then the appropriate values of these arguments for the given value of maxdpt will be computed and printed unless ${\mathbf{maxdpt}}<2$. In both cases h02bbf will then terminate with ${\mathbf{ifail}}={\mathbf{6}}$.
23: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $\mathrm{-1}$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
5.1Description of Printed Output
The level of printed output from h02bbf is controlled by you (see the description of msglvl in Section 5).
When ${\mathbf{msglvl}}>0$, the summary printout at the end of execution of h02bbf includes a listing of the status of every variable and constraint. Note that default names are assigned to all variables and constraints. The following describes the printout for each variable.
Varbl
gives the name (V) and index $\mathit{j}$, for $\mathit{j}=1,2,\dots ,n$, of the variable.
State
gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the Feasibility Tolerance, State will be ++ or -- respectively.
Value
is the value of the variable at the final iterate.
Lower Bound
is the lower bound specified for the variable. (None indicates that ${\mathbf{bl}}\left(j\right)\le -{\mathbf{bigbnd}}$.) Note that if ${\mathbf{intvar}}\left(j\right)=1$, then the printed value of Lower Bound for the $j$th variable may not be the same as that originally supplied in ${\mathbf{bl}}\left(j\right)$.
Upper Bound
is the upper bound specified for the variable. (None indicates that ${\mathbf{bu}}\left(j\right)\ge {\mathbf{bigbnd}}$.) Note that if ${\mathbf{intvar}}\left(j\right)=1$, then the printed value of Upper Bound for the $j$th variable may not be the same as that originally supplied in ${\mathbf{bu}}\left(j\right)$.
Lagr Mult
is the value of the Lagrange-multiplier for the associated bound constraint. This will be zero if State is FR or TF. If $x$ is optimal, the multiplier should be non-negative if State is LL, and non-positive if State is UL.
Residual
is the difference between the variable Value and the nearer of its bounds ${\mathbf{bl}}\left(j\right)$ and ${\mathbf{bu}}\left(j\right)$.
The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, ${\mathbf{bl}}\left(j\right)$ and ${\mathbf{bu}}\left(j\right)$ are replaced by ${\mathbf{bl}}\left(n+j\right)$ and ${\mathbf{bu}}\left(n+j\right)$ respectively, and with the following change in the heading.
L Con
gives the name (L) and index $\mathit{j}$, for $\mathit{j}=1,2,\dots ,m$, of the constraint.
When ${\mathbf{msglvl}}>1$, the summary printout at the end of every node during the execution of h02bbf is a listing of the outcome of forcing an integer variable with a noninteger value to take a value within its specified lower and upper bounds.
Node No
is the current node number of the B&B tree being investigated.
Parent Node
is the parent node number of the current node.
Obj Value
is the final objective function value. If a node does not have a feasible solution then No Feas Soln is printed instead of the objective function value. If a node whose optimum solution exceeds the best integer solution so far is encountered (i.e., it does not pay to explore the sub-problem any further), then its objective function value is printed together with a CO (Cut Off).
Varbl Chosen
is the index of the integer variable chosen for branching.
Value Before
is the noninteger value of the integer variable chosen.
Lower Bound
is the lower bound value that the integer variable is allowed to take.
Upper Bound
is the upper bound value that the integer variable is allowed to take.
Value After
is the value of the integer variable after the current optimization.
Depth
is the depth of the B&B tree at the current node.
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases h02bbf may return useful information.
${\mathbf{ifail}}=1$
The problem does not have a feasible integer solution.
It was not possible to satisfy all the integer variables to within the integer feasibility tolerance (determined by toliv). Increase toliv and rerun h02bbf.
${\mathbf{ifail}}=2$
The $\mathrm{LP}$ solution is unbounded.
This value of ifail implies that a step as large as bigbnd would have to be taken in order to continue the algorithm (see Section 9).
${\mathbf{ifail}}=3$
The $\mathrm{LP}$ does not have a feasible solution.
It was not possible to satisfy all the constraints to within the feasibility tolerance (determined by tolfes). If the data for the constraints are accurate only to the absolute precision $\sigma $, you should ensure that the value of the feasibility tolerance is greater than $\sigma $. For example, if all elements of $A$ are of order unity and are accurate only to three decimal places, the feasibility tolerance should be at least ${10}^{-3}$ (see Section 9).
${\mathbf{ifail}}=4$
Iteration limit (determined by itmax) reached without finding a solution. (See Section 9.)
${\mathbf{ifail}}=6$
On entry, $i=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{intvar}}\left(i\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{intvar}}=0$ or $1$.
On entry, ${\mathbf{lda}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{m}})$.
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{maxdpt}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{maxdpt}}\ge 2$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}>0$.
On entry, the dimension of iwork is too small: ${\mathbf{liwork}}=\u27e8\mathit{\text{value}}\u27e9$. iwork must be of dimension (at least) $\u27e8\mathit{\text{value}}\u27e9$.
On entry, the dimension of rwork is too small: ${\mathbf{lrwork}}=\u27e8\mathit{\text{value}}\u27e9$. rwork must be of dimension (at least) $\u27e8\mathit{\text{value}}\u27e9$.
On entry, there were $\u27e8\mathit{\text{value}}\u27e9$ infinite or inconsistent bounds given in arrays bl and bu. For further advisory details set ${\mathbf{msglvl}}>2$.
${\mathbf{ifail}}=7$
Search of a branch was terminated due to iteration limit.
The IP solution reported is not the optimum IP solution. In other words, the B&B tree search for at least one of the branches had to be terminated since an LP sub-problem in the branch did not have a solution (see Section 9).
${\mathbf{ifail}}=8$
maxdpt is too small to solve the problem: ${\mathbf{maxdpt}}=\u27e8\mathit{\text{value}}\u27e9$.
The $\mathrm{IP}$ solution returned is the best solution for the number of nodes investigated in the B&B tree.
${\mathbf{ifail}}=10$
No feasible solution was found for the number of nodes investigated in the B&B tree.
${\mathbf{ifail}}=11$
Not enough workspace to solve problem.
Although the workspace sizes are sufficient to meet the documented restriction, they are not sufficiently large to accommodate an internal partition of the workspace that meets the requirements of the problem. Increase the workspace sizes.
The maximum depth of the B&B tree used for branch and bound (determined by maxdpt) is too small. Increase maxdpt (along with liwork and/or lrwork if appropriate) and rerun h02bbf.
Overflow
It may be possible to avoid the difficulty by increasing the magnitude of the feasibility tolerance (tolfes) and rerunning the program. If the message recurs even after this change, see Section 9.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
h02bbf implements a numerically stable active set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
h02bbf is not thread safe and should not be called from a multithreaded user program. Please see Section 1 in FL Interface Multithreading for more information on thread safety.
h02bbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The original LP problem may not have an optimum solution, i.e., h02bbf terminates with ${\mathbf{ifail}}={\mathbf{2}}$, ${\mathbf{3}}$ or ${\mathbf{4}}$ or overflow may occur. In this case, you are recommended to relax the integer restrictions of the problem and try to find the optimum LP solution by using e04mff/e04mfa instead.
In the B&B method, it is possible for an LP sub-problem to terminate without finding a solution. This may occur due to the number of iterations exceeding the maximum allowed. Therefore, the B&B tree search for that particular branch cannot be continued. Thus the returned solution may not be optimal. (${\mathbf{ifail}}={\mathbf{7}}$). For the second and unlikely case, a solution could not be found despite a second attempt at an LP solution.
10Example
This example solves the integer programming problem: