CGAL 5.6.3 - Linear and Quadratic Programming Solver

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MPSFormat Concept Reference
Linear and Quadratic Programming Solver Reference » Concepts

## Definition

MPS is a commonly used file format for storing linear and quadratic programs according to the concepts `QuadraticProgram`, `LinearProgram`, `NonnegativeQuadraticProgram`, and `NonnegativeLinearProgram`, see also `https://en.wikipedia.org/wiki/MPS_(format)`.

CGAL supports a large subset of this format, but there are MPS files around that we cannot read (for example, files that encode integrality constraints on the variables). Also, there might be some other MPS-based solvers that will not be able to read the MPS files written by CGAL, since we do not strictly adhere to the very rigid layout requirements of the original MPS format.

Let's look at an example first. The quadratic program

\[ \begin{array}{lrcl} \mbox{minimize} & x^2 + 4(y-4)^2 &(=& x^2 + 4y^2 - 32y + 64) \\ \mbox{subject to} & x + y &\leq& 7 \\ & -x + 2y &\leq& 4 \\ & x &\geq& 0 \\ & y &\geq& 0 \\ & y &\leq& 4 \end{array} \]

has the following description in MPS format.

    NAME first_qp
    ROWS
    N obj
    L c0
    L c1
    COLUMNS
    x0 c0 1
    x0 c1 -1
    x1 obj -32
    x1 c0 1
    x1 c1 2
    RHS
    rhs obj -64
    rhs c0 7
    rhs c1 4
    BOUNDS
    UP BND x1 4
    QMATRIX
    x0 x0 2
    x1 x1 8

Here comes a semiformal description of the format in general.

NAME Section

This (mandatory) section consists of a single line starting with `NAME`. Everything starting from the first non-whitespace after that until the end of the line constitutes the name of the problem.

ROWS Section

In the (mandatory) `ROW` section, you find one line for every constraint, where the letter `L` indicates relation \( \leq\), letter `G` stands for \( \geq\), and `E` for \( =\). In addition, there is a row for the linear objective function (indicated by letter `N`). In that section, names are assigned to the constraints (here: `c0, c1`) and the objective function (here: `obj`). An MPS file may encode several linear objective functions by using several rows starting with `N`, but we ignore all but the first.

COLUMNS Section

The (mandatory) `COLUMNS` section encodes the constraint matrix \( A\) and the linear objective function vector \( c\). Every line consists of one or two sequences of three tokens \( j i val\), where \( j\) is the name of a variable (here, we have variables `x0,x1`), \( i\) is the name of a constraint or the objective function, and \( val\) is the value \( A_{ij}\) (if \( i\) names a constraint), or \( c_j\) (if \( i\) names the linear objective function). Values for pairs \( (i,j)\) that are not specified in this section default to \( 0\). Otherwise, for every pair \( (i,j)\), the last specified value determines \( A_{ij}\) or \( c_j\).

RHS Section

This (mandatory) section encodes the right-hand side vector \( b\) and the constant term \( c_0\) in the objective function. The first token in every line is an identifier (here: `rhs`). An MPS file may encode several right-hand sides \( b\) by using several such identifiers, but we ignore all lines having an identifier different from that of the first line.

The right-hand side identifier is succeeded by one or two sequences of tokens \( i val\), where \( i\) names a constraint or the linear objective function, and \( val\) specifies the value \( b_i\) (if \( i\) names a constraint), or \( -c_0\) (if \( i\) names the linear objective function). Values that are not specified in this section default to \( 0\). Otherwise, for every \( i\), the last specified value determines \( b_{i}\) or \( -c_0\).

BOUNDS Section

This (optional) section encodes the lower and upper bound vectors \( l\) and \( u\) for the variables. The default bounds for any variable \( x_j\) are \( 0\leq x_j\leq \infty\); the `BOUNDS` section is used to override these defaults. In particular, if there is no `BOUNDS` section, the program is nonnegative and actually a model of the concept `NonnegativeQuadraticProgram` or `NonnegativeLinearProgram`.

The first token in every line is succeeded by an (optional) identifier (here: `BND`). An MPS file may encode several bound vectors \( l\) and \( u\) by using several such identifiers, but we ignore all lines having an identifier different from that of the first line. The first token \( t\) itself determines the type of the bound, and the token \( j\) after the bound identifier names the variable to which the bound applies In case of bound types `FX`, `LO`, and `UP`, there is another token \( val\) that specifies the bound value. Here is how bound type and value determine a bound for variable \( x_j\). There may be several bound specifications for a single variable, and they are processed in order of appearance.

bound type | resulting bound
--- | ---
FX | \(x_j = val\) ( \(x_j\) becomes a fixed variable)
LO | \(x_j \geq val\) (upper bound remains unchanged)
UP | \(x_j \leq val\) (lower bound remains unchanged, except if \(val<0\); then, a zero lower bound is reset to \(-\infty\))
FR | \(-\infty \leq x_j\leq\infty\) (previous bounds are discarded)
MI | \(x_j\geq -\infty\) (upper bound remains unchanged)
PL | \(x_j\leq \infty\) (lower bound remains unchanged)

QMATRIX / QUADOBJ / DMATRIX Section

This (optional) section encodes the quadratic objective function matrix \( D\). Every line is a sequence \( i j val\) of three tokens, where both \( i\) and \( j\) name variables, and \( val\) is the value \( 2D_{i,j}\) (in case of `QMATRIX` or `QUADOBJ`), or \( D_{ij}\) (in case of `DMATRIX`).

In case of `QMATRIX` and `DMATRIX`, all nonzero entries must be specified: if there is a line \( i j val\), then there must also be a line \( j i val\), since \( D\) is required to be symmetric. In case of `QUADOBJ`, only the entries of \( 2D\) on or below the diagonal must be specified, entries above the diagonal are deduced from symmetry. It is not allowed to specify two or more different nonzero values for an unordered pair \( \{i,j\}\).

If this section is missing or does not contain nonzero values, the program is a model of the concept `LinearProgram`.

Miscellaneous

Our MPS format also supports an (optional) `RANGES` section, but we don't explain this here.

See also
  `CGAL::Quadratic_program_from_mps<NT>`