15.5 The PTF Format

The PTF format is a human-readable, natural text format that supports all linear, conic and mixed-integer features.

15.5.1 The overall format

The format is indentation based, where each section is started by a head line and followed by a section body with deeper indentation that the head line. For example:

Header line
   Body line 1
   Body line 1
   Body line 1

Section can also be nested:

Header line A
   Body line in A
   Header line A.1
      Body line in A.1
      Body line in A.1
   Body line in A

The indentation of blank lines is ignored, so a subsection can contain a blank line with no indentation. The character # defines a line comment and anything between the # character and the end of the line is ignored.

In a PTF file, the first section must be a Task section. The order of the remaining section is arbitrary, and sections may occur multiple times or not at all.

MOSEK will ignore any top-level section it does not recognize.

15.5.1.1 Names

In the description of the format we use following definitions for name strings:

NAME: PLAIN_NAME | QUOTED_NAME
PLAIN_NAME: [a-zA-Z_] [a-zA-Z0-9_-.!|]
QUOTED_NAME: "'" ( [^'\\\r\n] | "\\" ( [\\rn] | "x" [0-9a-fA-F] [0-9a-fA-F] ) )* "'"

15.5.1.2 Expressions

An expression is a sum of terms. A term is either a linear term (a coefficient and a variable name, where the coefficient can be left out if it is 1.0), or a matrix inner product.

An expression:

EXPR: EMPTY | ( [+-] TERM )*
TERM: LINEAR_TERM | MATRIX_TERM

A linear term

LINEAR_TERM: FLOAT? NAME

A matrix term

MATRIX_TERM: "<" ( [+-] FLOAT? NAME)* ";" NAME ">"

Here the right-hand name is the name of a (semidefinite) matrix variable, and the left-hand side is a sum of symmetric matrices. The actual matrices are defined in a separate section.

Expressions can span multiple lines by giving subsequent lines a deeper indentation.

For example following two section are equivalent:

# Everything on one line:
+ x1 + x2 + x3 + x4

# Split into multiple lines:
+ x1
  + x2
  + x3
  + x4

15.5.2 Task section

The first section of the file must be a Task. The text in this section is not used and may contain comments, or meta-information from the writer or about the content.

Format:

Task NAME
   Anything goes here...

NAME is a the task name.

15.5.3 Objective section

The Objective section defines the objective name, sense and function. The format:

"Objective" NAME?
   ( "Minimize" | "Maximize" ) EXPR

For example:

Objective 'obj'
   Minimize + x1 + 0.2 x2 + < M1 ; X1 >

15.5.4 Constraints section

The constraints section defines a series of constraints. A constraint defines a term \(A\cdot x + b\in K\). For linear constraints A is just one row, while for conic constraints it can be multiple rows. If a constraint spans multiple rows these can either be written inline separated by semi-colons, or each expression in a separete sub-section.

Simple linear constraints:

"Constraints"
  NAME? "[" [-+] (FLOAT | "inf") (";" [-+] (FLOAT | "inf") )? "]" EXPR

If the brackets contain two values, they are used as upper and lower bounds. It they contain one value the constraint is an equality.

For example:

Constraints
  # Ranged constraint
  'c1' [0;10] + x1 + x2 + x3
  # Fixed constraint, expression equals to 0
  [0] + x1 + x2 + x3
  # Nonnegative constraint
  [0;+inf] + x1 + x2 + x3

Constraint blocks put the expression either in a subsection or inline. The cone type (domain) is written in the brackets, and MOSEK currently supports following types:

  • Major (primal) cones:

    • QUAD(N) or SOC(N): Second order cone of dimension N.

    • RQUAD(N) or RSOC(N): Rotated second order cone of dimension N.

    • PEXP: Primal exponential cone of dimension 3.

    • PPOW(N,P): Primal power cone of dimension N with parameter P (float between 0 and 1).

    • PPOW(N;ALPHA): Primal power cone of dimension N with exponent sequence ALPHA (comma-separated list of floats).

    • PGEOMEAN(N): Primal geometric mean cone of domension N.

    • SVECPSD(N): Vectorized symmetric positive semidefinite cone of dimension N (N must be of the form D*(D+1)/2).

  • Dual cones:

    • DEXP: Dual exponential cone of dimension 3.

    • DPOW(N,P): Dual power cone of dimension N with parameter P (float between 0 and 1).

    • DPOW(N;ALPHA): Dual power cone of dimension N with exponent sequence ALPHA (comma-separated list of floats).

    • DGEOMEAN(N): Dual geometric mean cone of domension N.

  • Linear cones:

    • FREE(N) The free (unbounded) cone of dimension N.

    • POSITIVE(N) The non-negative cone of dimension N.

    • NEGATIVE(N) The non-positive cone of dimension N.

    • ZERO(N) The zero-cone of dimension N.

See Sec. 14.9 (Supported domains) for definitions of the parameters.

"Constraints"
  NAME? "[" DOMAIN "]" EXPR_LIST

For example:

Constraints
   'K1' [PPOW(5;3,1)]
      + x1 + x2
      + x2 + x3
      + 1.0
      + x1
      + x3
   'K2' [RQUAD(3)]
      + x1 + x2
      + x2 + x3
      + x3 + x1

15.5.5 Variables section

Any variable used in an expression must be defined in a variable section. The variable section defines each variable domain.

"Variables"
   NAME "[" [-+] (FLOAT | "inf") (";" [-+] (FLOAT | "inf") )? "]"
   NAME "[" "PSD" (INT) "]"

For example, a linear variable

Variables
   # Nonnegative variable
   x1 [0;inf]
   # Ranged variable
   x2 [0;1]
   # Fixed variable
   x3 [5.0]
   # 5-dimensional symmetric matrix variable
   X [PSD(5)]

15.5.6 Integer section

This section contains a list of variables that are integral. For example:

Integer
   x1 x2 x3

15.5.7 SymmetricMatrixes section

This section defines the symmetric matrixes used for matrix coefficients in matrix inner product terms. The section lists named matrixes, each with a size and a number of non-zeros. Only non-zeros in the lower triangular part should be defined.

"SymmetricMatrixes"
   NAME "SYMMAT" "(" INT ")"  ( "(" INT "," INT "," FLOAT ")" )*
   ...

For example:

SymmetricMatrixes
   M1 SYMMAT(3) (0,0,1.0) (1,1,2.0) (2,1,0.5)
   M2 SYMMAT(3)
      (0,0,1.0)
      (1,1,2.0)
      (2,1,0.5)

15.5.8 Solutions section

Each subsection defines a solution. A solution defines for each constraint and for each variable exactly one primal value and either one (for conic domains) or two (for linear domains) dual values. The values follow the same logic as in the MOSEK C API. A primal and a dual solution status defines the meaning of the values primal and dual (solution, certificate, unknown, etc.)

The format is this:

"Solutions"
   "Solution" WHICHSOL
      "ProblemStatus" PROSTA PROSTA?
   "SolutionStatus" SOLSTA SOLSTA?
   "Objective" FLOAT FLOAT_OR_NONE
   "Variables"
      # Linear variable status: level, slx, sux
      NAME "[" STATUS "]" FLOAT FLOAT_OR_NONE FLOAT_OR_NONE
   "Constraints"
      # Linear variable status: level, slx, sux
      NAME "[" STATUS "]" FLOAT FLOAT_OR_NONE FLOAT_OR_NONE
      # Conic constraint status: level, doty
      NAME
         "[" STATUS "]" FLOAT FLOAT_OR_NONE

Nonexistent values (for example, dual values for an integer solution) are replaced with a single dot (.):

FLOAT_OR_NONE = FLOAT | .

Following values for WHICHSOL are supported:

  • interior Interior solution, the result of an interior-point solver.

  • basic Basic solution, as produced by a simplex solver.

  • integer Integer solution, the solution to a mixed-integer problem. This does not define a dual solution.

Following values for PROSTA are supported:

  • unknown The problem status is unknown

  • feasible The problem has been proven feasible

  • infeasible The problem has been proven infeasible

  • illposed The problem has been proved to be ill posed

  • infeasible_or_unbounded The problem is infeasible or unbounded

Following values for SOLSTA are supported:

  • unknown The solution status is unknown

  • feasible The solution is feasible

  • optimal The solution is optimal

  • infeas_cert The solution is a certificate of infeasibility

  • illposed_cert The solution is a certificate of illposedness

Following values for STATUS are supported:

  • unknown The value is unknown

  • super_basic The value is super basic

  • at_lower The value is basic and at its lower bound

  • at_upper The value is basic and at its upper bound

  • fixed The value is basic fixed

  • infinite The value is at infinity

15.5.9 Examples

Linear example lo1.ptf

Task ''
    # Written by MOSEK v10.0.13
    # problemtype: Linear Problem
    # number of linear variables: 4
    # number of linear constraints: 3
    # number of old-style A nonzeros: 9
Objective obj
    Maximize + 3 x1 + x2 + 5 x3 + x4
Constraints
    c1 [3e+1] + 3 x1 + x2 + 2 x3
    c2 [1.5e+1;+inf] + 2 x1 + x2 + 3 x3 + x4
    c3 [-inf;2.5e+1] + 2 x2 + 3 x4
Variables
    x1 [0;+inf]
    x2 [0;1e+1]
    x3 [0;+inf]
    x4 [0;+inf]

Conic quadratic example cqo1.ptf

Task ''
    # Written by MOSEK v10.0.17
    # problemtype: Conic Problem
    # number of linear variables: 6
    # number of linear constraints: 1
    # number of  old-style cones: 0
    # number of positive semidefinite variables: 0
    # number of positive semidefinite matrixes: 0
    # number of affine conic constraints: 2
    # number of disjunctive constraints: 0
    # number scalar affine expressions/nonzeros : 6/6
    # number of old-style A nonzeros: 3
Objective obj
    Minimize + x4 + x5 + x6
Constraints
    c1 [1] + x1 + x2 + 2 x3
    k1 [QUAD(3)]
        @ac1: + x4
        @ac2: + x1
        @ac3: + x2
    k2 [RQUAD(3)]
        @ac4: + x5
        @ac5: + x6
        @ac6: + x3
Variables
    x4
    x1 [0;+inf]
    x2 [0;+inf]
    x5
    x6
    x3 [0;+inf]

Power cone example cqo1.ptf

Task ''
Objective ''
    Maximize - x0 + x3 + x4
Constraints
    c0 [2] + x0 + x1 + 5e-1 x2
    C1 [PPOW(3,2e-1)]
        + x0
        + x1
        + x3
    C2 [PPOW(3;4.0,6.0)]
        + x2
        + x5
        + x4
Variables
    x0
    x1
    x2
    x3
    x4 
    x5 [1.0]

Disjunctive example djc1.ptf

Task djc1
Objective ''
    Minimize + 2 'x[0]' + 'x[1]' + 3 'x[2]' + 'x[3]'
Constraints
    @c0 [-10;+inf] + 'x[0]' + 'x[1]' + 'x[2]' + 'x[3]'
    @D0 [OR]
        [AND]
            [NEGATIVE(1)]
                 + 'x[0]' - 2 'x[1]' + 1
            [ZERO(2)]
                 + 'x[2]'
                 + 'x[3]'
        [AND]
            [NEGATIVE(1)]
                 + 'x[2]' - 3 'x[3]' + 2
            [ZERO(2)]
                 + 'x[0]'
                 + 'x[1]'
    @D1 [OR]
        [ZERO(1)]
             + 'x[0]' - 2.5
        [ZERO(1)]
             + 'x[1]' - 2.5
        [ZERO(1)]
             + 'x[2]' - 2.5
        [ZERO(1)]
             + 'x[3]' - 2.5
Variables
    'x[0]'
    'x[1]'
    'x[2]'
    'x[3]'

Semidefinite example sdo1.ptf

Task ''
    # Written by MOSEK v10.0.17
    # problemtype: Conic Problem
    # number of linear variables: 3
    # number of linear constraints: 0
    # number of  old-style cones: 0
    # number of positive semidefinite variables: 1
    # number of positive semidefinite matrixes: 3
    # number of affine conic constraints: 2
    # number of disjunctive constraints: 0
    # number scalar affine expressions/nonzeros : 5/6
    # number of old-style A nonzeros: 0
Objective ''
    Minimize + @x0 + <M0;@X0>
Constraints
    @C0 [ZERO(2)]
        @ac0: + @x0 + < + M1;@X0> - 1
        @ac1: + @x1 + @x2 + < + M2;@X0> - 0.5
    @C1 [QUAD(3)]
        @ac2: + @x0
        @ac3: + @x1
        @ac4: + @x2
Variables
    @x0
    @x1
    @x2
    @X0 [PSD(3)]
SymmetricMatrixes
    M0 SYMMAT(3) (0,0,2) (1,0,1) (1,1,2) (2,1,1) (2,2,2)
    M1 SYMMAT(3) (0,0,1) (1,1,1) (2,2,1)
    M2 SYMMAT(3) (0,0,1) (1,0,1) (1,1,1) (2,0,1) (2,1,1) (2,2,1)