15.2 The MPS File Format¶
MOSEK supports the standard MPS format with some extensions. For a detailed description of the MPS format see the book by Nazareth [Naz87].
15.2.1 MPS File Structure¶
The version of the MPS format supported by MOSEK allows specification of an optimization problem of the form
where
\(x \in \real^n\) is the vector of decision variables.
\(A \in \real^{m\times n}\) is the constraint matrix.
\(l^c \in \real^m\) is the lower limit on the activity for the constraints.
\(u^c \in \real^m\) is the upper limit on the activity for the constraints.
\(l^x \in \real^n\) is the lower limit on the activity for the variables.
\(u^x \in \real^n\) is the upper limit on the activity for the variables.
\(q: \real^n \rightarrow \real\) is a vector of quadratic functions. Hence,
\[q_i(x) = \half x^T Q^i x\]where it is assumed that \(Q^i = (Q^i)^T\). Please note the explicit \(\half\) in the quadratic term and that \(Q^i\) is required to be symmetric. The same applies to \(q_0\).
\(\K\) is a convex cone.
\(\mathcal{J} \subseteq \{1,2,\ldots ,n\}\) is an index set of the integerconstrained variables.
\(c\) is the vector of objective coefficients.
An MPS file with one row and one column can be illustrated like this:
* 1 2 3 4 5 6
*23456789012345678901234567890123456789012345678901234567890
NAME [name]
OBJSENSE
[objsense]
OBJNAME [objname]
ROWS
? [cname1]
COLUMNS
[vname1] [cname1] [value1] [cname2] [value2]
RHS
[name] [cname1] [value1] [cname2] [value2]
RANGES
[name] [cname1] [value1] [cname2] [value2]
QSECTION [cname1]
[vname1] [vname2] [value1] [vname3] [value2]
QMATRIX
[vname1] [vname2] [value1]
QUADOBJ
[vname1] [vname2] [value1]
QCMATRIX [cname1]
[vname1] [vname2] [value1]
BOUNDS
?? [name] [vname1] [value1]
CSECTION [kname1] [value1] [ktype]
[vname1]
ENDATA
Here the names in capitals are keywords of the MPS format and names in brackets are custom defined names or values. A couple of notes on the structure:
Fields: All items surrounded by brackets appear in fields. The fields named “
valueN
” are numerical values. Hence, they must have the format[+]XXXXXXX.XXXXXX[[eE][+]XXX]
where
X = [0123456789].
Sections: The MPS file consists of several sections where the names in capitals indicate the beginning of a new section. For example,
COLUMNS
denotes the beginning of the columns section.Comments: Lines starting with an
*
are comment lines and are ignored by MOSEK.Keys: The question marks represent keys to be specified later.
Extensions: The sections
QSECTION
andCSECTION
are specific MOSEK extensions of the MPS format. The sectionsQMATRIX
,QUADOBJ
andQCMATRIX
are included for sake of compatibility with other vendors extensions to the MPS format.The standard MPS format is a fixed format, i.e. everything in the MPS file must be within certain fixed positions. MOSEK also supports a free format. See Sec. 15.2.5 (The Free MPS Format) for details.
Linear example lo1.mps
A concrete example of a MPS file is presented below:
* File: lo1.mps
NAME lo1
OBJSENSE
MAX
ROWS
N obj
E c1
G c2
L c3
COLUMNS
x1 obj 3
x1 c1 3
x1 c2 2
x2 obj 1
x2 c1 1
x2 c2 1
x2 c3 2
x3 obj 5
x3 c1 2
x3 c2 3
x4 obj 1
x4 c2 1
x4 c3 3
RHS
rhs c1 30
rhs c2 15
rhs c3 25
RANGES
BOUNDS
UP bound x2 10
ENDATA
Subsequently each individual section in the MPS format is discussed.
15.2.1.1 NAME
(optional)¶
In this section a name ([name]
) is assigned to the problem.
15.2.1.2 OBJSENSE
(optional)¶
This is an optional section that can be used to specify the sense of the objective function. The OBJSENSE
section contains one line at most which can be one of the following:
MIN
MINIMIZE
MAX
MAXIMIZE
It should be obvious what the implication is of each of these four lines.
15.2.1.3 OBJNAME
(optional)¶
This is an optional section that can be used to specify the name of the row that is used as objective function.
objname
should be a valid row name.
15.2.1.4 ROWS
¶
A record in the ROWS
section has the form
? [cname1]
where the requirements for the fields are as follows:
Field 
Starting Position 
Max Width 
required 
Description 

? 
2 
1 
Yes 
Constraint key 

5 
8 
Yes 
Constraint name 
Hence, in this section each constraint is assigned a unique name denoted by [cname1]
. Please note that [cname1]
starts in position 5 and the field can be at most 8 characters wide. An initial key ?
must be present to specify the type of the constraint. The key can have values E
, G
, L
, or N
with the following interpretation:
Constraint type 
\(l_{i}^{c}\) 
\(u_i^c\) 


finite 
\(=l_i^c\) 

finite 
\(\infty\) 

\(\infty\) 
finite 

\(\infty\) 
\(\infty\) 
In the MPS format the objective vector is not specified explicitly, but one of the constraints having the key N
will be used as the objective vector \(c\) . In general, if multiple N
type constraints are specified, then the first will be used as the objective vector \(c\), unless something else was specified in the section OBJNAME
.
15.2.1.5 COLUMNS
¶
In this section the elements of \(A\) are specified using one or more records having the form:
[vname1] [cname1] [value1] [cname2] [value2]
where the requirements for each field are as follows:
Field 
Starting Position 
Max Width 
required 
Description 


5 
8 
Yes 
Variable name 

15 
8 
Yes 
Constraint name 

25 
12 
Yes 
Numerical value 

40 
8 
No 
Constraint name 

50 
12 
No 
Numerical value 
Hence, a record specifies one or two elements \(a_{ij}\) of \(A\) using the principle that [vname1]
and [cname1]
determines \(j\) and \(i\) respectively. Please note that [cname1]
must be a constraint name specified in the ROWS
section. Finally, [value1]
denotes the numerical value of \(a_{ij}\) . Another optional element is specified by [cname2]
, and [value2]
for the variable specified by [vname1]
. Some important comments are:
All elements belonging to one variable must be grouped together.
Zero elements of \(A\) should not be specified.
At least one element for each variable should be specified.
15.2.1.6 RHS
(optional)¶
A record in this section has the format
[name] [cname1] [value1] [cname2] [value2]
where the requirements for each field are as follows:
Field 
Starting Position 
Max Width 
required 
Description 


5 
8 
Yes 
Name of the 

15 
8 
Yes 
Constraint name 

25 
12 
Yes 
Numerical value 

40 
8 
No 
Constraint name 

50 
12 
No 
Numerical value 
The interpretation of a record is that [name]
is the name of the RHS
vector to be specified. In general, several vectors can be specified. [cname1]
denotes a constraint name previously specified in the ROWS
section. Now, assume that this name has been assigned to the \(i\) h constraint and \(v_1\) denotes the value specified by [value1]
, then the interpretation of \(v_1\) is:
Constraint 
\(l_i^c\) 
\(u_i^c\) 


\(v_1\) 
\(v_1\) 

\(v_1\) 


\(v_1\) 


An optional second element is specified by [cname2]
and [value2]
and is interpreted in the same way. Please note that it is not necessary to specify zero elements, because elements are assumed to be zero.
15.2.1.7 RANGES
(optional)¶
A record in this section has the form
[name] [cname1] [value1] [cname2] [value2]
where the requirements for each fields are as follows:
Field 
Starting Position 
Max Width 
required 
Description 


5 
8 
Yes 
Name of the 

15 
8 
Yes 
Constraint name 

25 
12 
Yes 
Numerical value 

40 
8 
No 
Constraint name 

50 
12 
No 
Numerical value 
The records in this section are used to modify the bound vectors for the constraints, i.e. the values in \(l^c\) and \(u^c\) . A record has the following interpretation:[name]
is the name of the RANGE
vector and [cname1]
is a valid constraint name. Assume that [cname1]
is assigned to the \(i\)th constraint and let \(v_1\) be the value specified by [value1]
, then a record has the interpretation:
Constraint type 
Sign of \(v_1\) 
\(l_i^c\) 
\(u_i^c\) 


\(\) 
\(u_i^c+v_1\) 


\(+\) 
\(l_i^c+v_1\) 


\(\) or \(+\) 
\(l_i^c+v_1\) 


\(\) or \(+\) 
\(u_i^cv_1\) 


Another constraint bound can optionally be modified using [cname2]
and [value2]
the same way.
15.2.1.8 QSECTION
(optional)¶
Within the QSECTION
the label [cname1]
must be a constraint name previously specified in the ROWS
section. The label [cname1]
denotes the constraint to which the quadratic terms belong. A record in the QSECTION
has the form
[vname1] [vname2] [value1] [vname3] [value2]
where the requirements for each field are:
Field 
Starting Position 
Max Width 
required 
Description 


5 
8 
Yes 
Variable name 

15 
8 
Yes 
Variable name 

25 
12 
Yes 
Numerical value 

40 
8 
No 
Variable name 

50 
12 
No 
Numerical value 
A record specifies one or two elements in the lower triangular part of the \(Q^i\) matrix where [cname1]
specifies the \(i\) . Hence, if the names [vname1]
and [vname2]
have been assigned to the \(k\)th and \(j\)th variable, then \(Q_{kj}^i\) is assigned the value given by [value1]
An optional second element is specified in the same way by the fields [vname1]
, [vname3]
, and [value2]
.
The example
has the following MPS file representation
* File: qo1.mps
NAME qo1
ROWS
N obj
G c1
COLUMNS
x1 c1 1.0
x2 obj 1.0
x2 c1 1.0
x3 c1 1.0
RHS
rhs c1 1.0
QSECTION obj
x1 x1 2.0
x1 x3 1.0
x2 x2 0.2
x3 x3 2.0
ENDATA
Regarding the QSECTION
s please note that:
Only one
QSECTION
is allowed for each constraint.The
QSECTION
s can appear in an arbitrary order after theCOLUMNS
section.All variable names occurring in the
QSECTION
must already be specified in theCOLUMNS
section.All entries specified in a
QSECTION
are assumed to belong to the lower triangular part of the quadratic term of \(Q\) .
15.2.1.9 QMATRIX/QUADOBJ
(optional)¶
The QMATRIX
and QUADOBJ
sections allow to define the quadratic term of the objective function. They differ in how the quadratic term of the objective function is stored:
QMATRIX
stores all the nonzeros coefficients, without taking advantage of the symmetry of the \(Q\) matrix.QUADOBJ
stores the upper diagonal nonzero elements of the \(Q\) matrix.
A record in both sections has the form:
[vname1] [vname2] [value1]
where the requirements for each field are:
Field 
Starting Position 
Max Width 
required 
Description 


5 
8 
Yes 
Variable name 

15 
8 
Yes 
Variable name 

25 
12 
Yes 
Numerical value 
A record specifies one elements of the \(Q\) matrix in the objective function . Hence, if the names [vname1]
and [vname2]
have been assigned to the \(k\)th and \(j\)th variable, then \(Q_{kj}\) is assigned the value given by [value1]
. Note that a line must appear for each offdiagonal coefficient if using a QMATRIX
section, while only one entry is required in a QUADOBJ
section. The quadratic part of the objective function will be evaluated as \(1/2x^T Q x\).
The example
has the following MPS file representation using QMATRIX
* File: qo1_matrix.mps
NAME qo1_qmatrix
ROWS
N obj
G c1
COLUMNS
x1 c1 1.0
x2 obj 1.0
x2 c1 1.0
x3 c1 1.0
RHS
rhs c1 1.0
QMATRIX
x1 x1 2.0
x1 x3 1.0
x3 x1 1.0
x2 x2 0.2
x3 x3 2.0
ENDATA
or the following using QUADOBJ
* File: qo1_quadobj.mps
NAME qo1_quadobj
ROWS
N obj
G c1
COLUMNS
x1 c1 1.0
x2 obj 1.0
x2 c1 1.0
x3 c1 1.0
RHS
rhs c1 1.0
QUADOBJ
x1 x1 2.0
x1 x3 1.0
x2 x2 0.2
x3 x3 2.0
ENDATA
Please also note that:
A
QMATRIX
/QUADOBJ
section can appear in an arbitrary order after theCOLUMNS
section.All variable names occurring in the
QMATRIX
/QUADOBJ
section must already be specified in theCOLUMNS
section.
15.2.1.10 QCMATRIX
(optional)¶
A QCMATRIX
section allows to specify the quadratic part of a given constraint. Within the QCMATRIX
the label [cname1]
must be a constraint name previously specified in the ROWS
section. The label [cname1]
denotes the constraint to which the quadratic term belongs. A record in the QSECTION
has the form
[vname1] [vname2] [value1]
where the requirements for each field are:
Field 
Starting Position 
Max Width 
required 
Description 


5 
8 
Yes 
Variable name 

15 
8 
Yes 
Variable name 

25 
12 
Yes 
Numerical value 
A record specifies an entry of the \(Q^i\) matrix where [cname1]
specifies the \(i\). Hence, if the names [vname1]
and [vname2]
have been assigned to the \(k\)th and \(j\)th variable, then \(Q_{kj}^i\) is assigned the value given by [value1]
. Moreover, the quadratic term is represented as \(1/2 x^T Q x\).
The example
has the following MPS file representation
* File: qo1.mps
NAME qo1
ROWS
N obj
G c1
L q1
COLUMNS
x1 c1 1.0
x2 obj 1.0
x2 c1 1.0
x3 c1 1.0
RHS
rhs c1 1.0
rhs q1 10.0
QCMATRIX q1
x1 x1 2.0
x1 x3 1.0
x3 x1 1.0
x2 x2 0.2
x3 x3 2.0
ENDATA
Regarding the QCMATRIX
s please note that:
Only one
QCMATRIX
is allowed for each constraint.The
QCMATRIX
s can appear in an arbitrary order after theCOLUMNS
section.All variable names occurring in the
QSECTION
must already be specified in theCOLUMNS
section.QCMATRIX
does not exploit the symmetry of \(Q\): an offdiagonal entry \((i,j)\) should appear twice.
15.2.1.11 BOUNDS
(optional)¶
In the BOUNDS
section changes to the default bounds vectors \(l^x\) and \(u^x\) are specified. The default bounds vectors are \(l^x = 0\) and \(u^x = \infty\) . Moreover, it is possible to specify several sets of bound vectors. A record in this section has the form
?? [name] [vname1] [value1]
where the requirements for each field are:
Field 
Starting Position 
Max Width 
Required 
Description 


2 
2 
Yes 
Bound key 

5 
8 
Yes 
Name of the 

15 
8 
Yes 
Variable name 

25 
12 
No 
Numerical value 
Hence, a record in the BOUNDS
section has the following interpretation:[name]
is the name of the bound vector and [vname1]
is the name of the variable for which the bounds are modified by the record. ??
and [value1]
are used to modify the bound vectors according to the following table:

\(l_j^x\) 
\(u_j^x\) 
Made integer (added to \(\mathcal{J}\)) 


\(\infty\) 
\(\infty\) 
No 

\(v_1\) 
\(v_1\) 
No 

\(v_1\) 
unchanged 
No 

\(\infty\) 
unchanged 
No 

unchanged 
\(\infty\) 
No 

unchanged 
\(v_1\) 
No 

\(0\) 
\(1\) 
Yes 

\(\lceil v_1 \rceil\) 
unchanged 
Yes 

unchanged 
\(\lfloor v_1 \rfloor\) 
Yes 
Here \(v_1\) is the value specified by [value1]
.
15.2.1.12 CSECTION
(optional)¶
The purpose of the CSECTION
is to specify the conic constraint
in (15.2). It is assumed that \(\K\) satisfies the following requirements. Let
be vectors comprised of parts of the decision variables \(x\) so that each decision variable is a member of exactly one vector \(x^t\), for example
Next define
where \(\K_t\) must have one of the following forms:
\(\real\) set:
\[\K_t = \real^{n^t}.\]Zero cone:
Quadratic cone:
Rotated quadratic cone:
Primal exponential cone:
Primal power cone (with parameter \(0< \alpha< 1\)):
Dual exponential cone:
Dual power cone (with parameter \(0< \alpha< 1\)):
In general, membership in the \(\real\) set is not specified. If a variable is not a member of any other cone then it is assumed to be a member of the \(\real\) cone.
Next, let us study an example. Assume that the power cone
and the rotated quadratic cone
should be specified in the MPS file. One CSECTION
is required for each cone and they are specified as follows:
* 1 2 3 4 5 6
*23456789012345678901234567890123456789012345678901234567890
CSECTION konea 3e1 PPOW
x4
x5
x8
CSECTION koneb 0.0 RQUAD
x7
x3
x1
x0
In general, a CSECTION
header has the format
CSECTION [kname1] [value1] [ktype]
where the requirements for each field are as follows:
Field 
Starting Position 
Max Width 
Required 
Description 


15 
8 
Yes 
Name of the cone 

25 
12 
No 
Cone parameter 

40 
Yes 
Type of the cone. 
The possible cone type keys are:

Members 

Interpretation. 


\(\geq 0\) 
unused 
Zero cone (15.3). 

\(\geq 1\) 
unused 
Quadratic cone (15.4). 

\(\geq 2\) 
unused 
Rotated quadratic cone (15.5). 

\(3\) 
unused 
Primal exponential cone (15.6). 

\(\geq 2\) 
\(\alpha\) 
Primal power cone (15.7). 

\(3\) 
unused 
Dual exponential cone (15.8). 

\(\geq 2\) 
\(\alpha\) 
Dual power cone (15.9). 
A record in the CSECTION
has the format
[vname1]
where the requirements for each field are
Field 
Starting Position 
Max Width 
required 
Description 


5 
8 
Yes 
A valid variable name 
A variable must occur in at most one CSECTION
.
15.2.1.13 ENDATA
¶
This keyword denotes the end of the MPS file.
15.2.2 Integer Variables¶
Using special bound keys in the BOUNDS
section it is possible to specify that some or all of the variables should be integerconstrained i.e. be members of \(\mathcal{J}\). However, an alternative method is available. This method is available only for backward compatibility and we recommend that it is not used. This method requires that markers are placed in the COLUMNS
section as in the example:
COLUMNS
x1 obj 10.0 c1 0.7
x1 c2 0.5 c3 1.0
x1 c4 0.1
* Start of integerconstrained variables.
MARK000 'MARKER' 'INTORG'
x2 obj 9.0 c1 1.0
x2 c2 0.8333333333 c3 0.66666667
x2 c4 0.25
x3 obj 1.0 c6 2.0
MARK001 'MARKER' 'INTEND'
* End of integerconstrained variables.
Please note that special marker lines are used to indicate the start and the end of the integer variables. Furthermore be aware of the following
All variables between the markers are assigned a default lower bound of 0 and a default upper bound of 1. This may not be what is intended. If it is not intended, the correct bounds should be defined in the
BOUNDS
section of the MPS formatted file.MOSEK ignores field 1, i.e.
MARK0001
andMARK001
, however, other optimization systems require them.Field 2, i.e.
MARKER
, must be specified including the single quotes. This implies that no row can be assigned the nameMARKER
.Field 3 is ignored and should be left blank.
Field 4, i.e.
INTORG
andINTEND
, must be specified.It is possible to specify several such integer marker sections within the
COLUMNS
section.
15.2.3 General Limitations¶
An MPS file should be an ASCII file.
15.2.4 Interpretation of the MPS Format¶
Several issues related to the MPS format are not welldefined by the industry standard. However, MOSEK uses the following interpretation:
If a matrix element in the
COLUMNS
section is specified multiple times, then the multiple entries are added together.If a matrix element in a
QSECTION
section is specified multiple times, then the multiple entries are added together.
15.2.5 The Free MPS Format¶
MOSEK supports a free format variation of the MPS format. The free format is similar to the MPS file format but less restrictive, e.g. it allows longer names. However, a name must not contain any blanks.