LP files

Knitro can import linear and quadratic optimization problems for the standard LP (Linear Programming) file format. LP files should be ASCII-encoded text files.

Basic Structure

An LP file is structured into several sections, each identified by a case-insensitive keyword.

  1. MINIMIZE or MAXIMIZE: Defines the objective function.

  2. SUBJECT TO (or ST): Lists the constraints.

  3. BOUNDS: Sets variable bounds (optional).

  4. BINARY: Declares binary variables (optional).

  5. INTEGER: Declares integer variables (optional).

  6. RANGES: Specifies range constraints (optional).

  7. END: Marks the end of the file.

Example

Here is a simple example illustrating the structure of an LP file:

MINIMIZE
 cost: x + 4 y + 9 z

SUBJECT TO
 constraint1: x + y <= 5
 constraint2: x + z >= 10
 constraint3: y - z = 7

BOUNDS
 0 <= x <= 4
 -1 <= y <= 1
 z free

END

Objective Function Section

To include an objective function, the LP file must contain this section, which starts with a keyword such as MINIMIZE, MAXIMIZE, MIN, or MAX. This is followed by an optional label and the objective expression itself.

MINIMIZE
 [label:] expression

The objective expression can be a combination of constant terms and linear terms (e.g., coeff * var). If a variable appears without a coefficient, the coefficient is assumed to be 1.

MINIMIZE
 cost: 2 x + 3.5 y - z + 10

MAXIMIZE
 profit: revenue - expenses

Constraints Section

The constraints section follows the objective and begins with a keyword like SUBJECT TO, ST, SUCH THAT, or S.T.. Each constraint is defined on a new line with the following format:

[label:] expression relation rhs

Where:

  • label: An optional name for the constraint.

  • expression: A linear combination of variables.

  • relation: One of <=, >=, or =.

  • rhs: The numerical right-hand side value.

SUBJECT TO
 capacity: 2 x + 3 y <= 100
 demand: x + y >= 50
 balance: x - y = 0
 c4: 0.5 a + 1.2 b + 0.8 c <= 75.5

Variable Bounds Section

The optional BOUNDS (or BOUND) section allows you to define lower and upper limits for variables. If a variable’s bounds are not specified, it is treated as unbounded (free), ranging from negative to positive infinity.

Bounds can be specified in several ways:

lower <= variable <= upper
variable >= lower
variable <= upper
variable = value
variable free

The keyword free explicitly declares a variable as unbounded.

BOUNDS
 0 <= x <= 100
 y >= -50
 z <= 25
 w = 10
 v free

Declaring Variable Types

Binary Variables

To declare binary variables, list them in the BINARY section (also BIN or BINARIES).

BINARY
 x1 x2 x3

Integer Variables

To declare general integer variables, list them under the INTEGER section. Synonyms like INT, INTEGERS, GENERAL, GENERALS, or GEN are also accepted.

INTEGER
 y1 y2 y3

Complete Example

This example combines multiple sections, including variable type declarations:

MINIMIZE
 objective: 3 x + 2 y + z

SUBJECT TO
 resource1: 2 x + y <= 100
 resource2: x + 3 y + z <= 150
 demand: x + y >= 25

BOUNDS
 x >= 0
 0 <= y <= 50
 z free

BINARY
 x

INTEGER
 y

END

Range Constraints Section

The optional RANGES (or RANGE) section modifies existing constraints to create double-sided inequalities, effectively placing the constraint’s expression within an interval.

Each line in the RANGES section references a constraint by its label and provides a new lower and upper bound for its expression.

lhs <= constraint_label <= rhs

For example, consider these initial constraints:

SUBJECT TO
 con1: x + y <= 10
 con2: 2 x + 3 y >= 15

You can apply ranges to them as follows:

RANGES
 5 <= con1 <= 10
 15 <= con2 <= 25

This effectively transforms the original constraints into:

  • For con1: 5 <= x + y <= 10

  • For con2: 15 <= 2 x + 3 y <= 25

Note that the bounds specified in the RANGES section can override the original bounds of the referenced constraints. The lower bound will replace the original lower bound if it is more restrictive, and similarly for the upper bound.

Quadratic Extensions

Knitro’s LP parser extends the standard format to support quadratic terms in both the objective function and constraints. Quadratic expressions must be enclosed in square brackets [ ]. A quadratic term is represented by either a squared variable (e.g., x^2) or a product of two different variables (e.g., x*y) with an associated coefficient (if the coefficient is 1, it can be omitted). Optionally, a division operator can be added at the end of the square bracket expression to indicate a potential division of the coefficients by a non-zero constant, which can be useful if the quadratic matrix is given within a particular convention.

Quadratic Objective

MINIMIZE
 obj: x + 2 y + [ x^2 + 2 x*y + 3 y^2 ] / 2

Quadratic Constraints

SUBJECT TO
 qc1: x + y + [ x^2 + 2 y^2 - x*y ] <= 10

Unsupported Features

The LP format can describe features that Knitro’s parser does not support. Including any of the following will result in a parsing error:

  • Special Ordered Sets (SOS)

  • Semi-continuous variables

  • Piecewise linear functions

  • Indicator or logical constraints

  • Non-linear functions beyond quadratic terms.

Parser Rules

  • Variable, Constraint, and Objective Names: Names are case-sensitive and must start with a letter. They can contain letters, digits, underscores, and the symbols !#$%&()|~. Spaces and operators (-, +, *, /, ^, =, >, <, ;) are not allowed as they may interfere with parsing. When exporting, only the linear and quadratic components of the problem are written to the file; any general non-linear structures are omitted. Furthermore, if any variable, constraint, or objective names contain special characters (+-*/^=><;) or whitespace, these characters will be replaced with (@). If a variable does not have any name or its name is empty, Knitro will assign it a default name like x1, x2, etc. Similarly, unnamed constraints will be labeled as c1, c2, etc. The indexing of these default names is the same as the indexing used internally by Knitro.

  • Numerical Values: Standard integer and floating-point numbers are supported, including scientific notation (e.g., 1.5e-3). Infinity can be represented as +inf, -inf, +infinity, or -infinity.

  • Comments: Use \ for single-line comments. For block comments, enclose the text between /* and */.

\ This is a single-line comment
MINIMIZE
 obj: x + y  \ Minimize total cost
/* This is a block comment
   spanning multiple lines */

Using LP Files in Knitro

Loading an LP File (C API)

Use the KN_read_problem function to load an LP file into a Knitro context.

/*---- LOAD LP FILE ----*/
if (KN_read_problem (kc, "problem.lp", NULL) != 0) {
    printf("Error loading LP file\n");
    exit(-1);
}

Writing an LP File (C API)

Use the KN_write_problem function to export the current problem in a Knitro context to the LP format.

/*---- WRITE LP FILE ----*/
if (KN_write_problem (kc, "output.lp", NULL) != 0) {
    printf("Error writing LP file\n");
    exit(-1);
}