# 16.5 The PTF Format¶

The PTF format is a human-readable, natural text format suuporting all of **MOSEK** optimization problems in conic form, possibly with integer variables and disjunctive constraints.

## 16.5.1 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.

### 16.5.1.1 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] ) )* "'"
```

### 16.5.1.2 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 )*
TERM: LINEAR_TERM | MATRIX_TERM
```

A linear term

```
LINEAR_TERM: FLOAT? NAME
```

A matrix term

```
MATRIX_TERM: "<" FLOAT? NAME ( [+-] 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 matrixes. The actual matrixes 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
```

## 16.5.2 `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.

## 16.5.3 `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 >
```

## 16.5.4 `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
'c1' [0;10] x1 + x2 + x3
[0] 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:

`SOC(N)`

Second order cone of dimension`N`

`RSOC(N)`

Rotated second order cone of dimension`N`

`PSD(N)`

Symmetric positive semidefinite cone of dimension`N`

. This contains`N*(N+1)/2`

elements.`PEXP`

Primal exponential cone of dimension 3`DEXP`

Dual exponential cone of dimension 3`PPOW(N,P)`

Primal power cone of dimension`N`

with parameter`P`

`DPOW(N,P)`

Dual power cone of dimension`N`

with parameter`P`

`ZERO(N)`

The zero-cone of dimension`N`

.

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

For example:

```
Constraints
'K1' [SOC(3)] x1 + x2 ; x2 + x3 ; x3 + x1
'K2' [RSOC(3)]
x1 + x2
x2 + x3
x3 + x1
```

## 16.5.5 `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 "[" DOMAIN "]" NAMES
For example, a linear variable
```

```
Variables
x1 [0;Inf]
```

As with constraints, members of a conic domain can be listed either inline or in a subsection:

```
Variables
k1 [SOC(3)] x1 ; x2 ; x3
k2 [RSOC(3)]
x1
x2
x3
```

## 16.5.6 `Integer`

section¶

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

```
Integer
x1 x2 x3
```

## 16.5.7 `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)
```

## 16.5.8 `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
"Variables"
# Linear variable status: level, slx, sux
NAME "[" STATUS "]" FLOAT (FLOAT FLOAT)?
# Conic variable status: level, snx
NAME
"[" STATUS "]" FLOAT FLOAT?
...
"Constraints"
# Linear variable status: level, slx, sux
NAME "[" STATUS "]" FLOAT (FLOAT FLOAT)?
# Conic variable status: level, snx
NAME
"[" STATUS "]" FLOAT 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

## 16.5.9 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 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]
```

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]'
```