# 6.1 Linear Optimization¶

The simplest optimization problem is a purely linear problem. A linear optimization problem (see also Sec. 12.1 (Linear Optimization)) is a problem of the following form:

Minimize or maximize the objective function

$\sum_{j=0}^{n-1} c_j x_j + c^f$

subject to the linear constraints

$l_k^c \leq \sum_{j=0}^{n-1} a_{kj} x_j \leq u_k^c,\quad k=0,\ldots ,m-1,$

and the bounds

$l_j^x \leq x_j \leq u_j^x, \quad j=0,\ldots ,n-1.$

The problem description consists of the following elements:

• $$m$$ and $$n$$ — the number of constraints and variables, respectively,

• $$x$$ — the variable vector of length $$n$$,

• $$c$$ — the coefficient vector of length $$n$$

$\begin{split}c = \left[ \begin{array}{c} c_0 \\ \vdots \\ c_{n-1} \end{array} \right],\end{split}$
• $$c^f$$ — fixed term in the objective,

• $$A$$ — an $$m\times n$$ matrix of coefficients

$\begin{split}A = \left[ \begin{array}{ccc} a_{0,0} & \cdots & a_{0,(n-1)} \\ \vdots & \cdots & \vdots \\ a_{(m-1),0} & \cdots & a_{(m-1),(n-1)} \end{array} \right],\end{split}$
• $$l^c$$ and $$u^c$$ — the lower and upper bounds on constraints,

• $$l^x$$ and $$u^x$$ — the lower and upper bounds on variables.

Please note that we are using $$0$$ as the first index: $$x_0$$ is the first element in variable vector $$x$$.

## 6.1.1 Example LO1¶

The following is an example of a small linear optimization problem:

(6.1)$\begin{split}\begin{array} {lccccccccl} \mbox{maximize} & 3 x_0 & + & 1 x_1 & + & 5 x_2 & + & 1 x_3 & & \\ \mbox{subject to} & 3 x_0 & + & 1 x_1 & + & 2 x_2 & & & = & 30, \\ & 2 x_0 & + & 1 x_1 & + & 3 x_2 & + & 1 x_3 & \geq & 15, \\ & & & 2 x_1 & & & + & 3 x_3 & \leq & 25, \end{array}\end{split}$

under the bounds

$\begin{split}\begin{array}{ccccc} 0 & \leq & x_0 & \leq & \infty , \\ 0 & \leq & x_1 & \leq & 10, \\ 0 & \leq & x_2 & \leq & \infty ,\\ 0 & \leq & x_3 & \leq & \infty . \end{array}\end{split}$

Solving the problem

To solve the problem above we go through the following steps:

1. (Optionally) Creating an environment.

4. Optimization.

5. Extracting the solution.

Below we explain each of these steps.

Creating an environment.

The user can start by creating a MOSEK environment, but it is not necessary if the user does not need access to other functionalities, license management, additional routines, etc. Therefore in this tutorial we don’t create an explicit environment.

We create an empty task object. A task object represents all the data (inputs, outputs, parameters, information items etc.) associated with one optimization problem.

    /* Create the optimization task. */

/* Directs the log task stream to the 'printstr' function. */
if (r == MSK_RES_OK)


We also connect a call-back function to the task log stream. Messages related to the task are passed to the call-back function. In this case the stream call-back function writes its messages to the standard output stream. See Sec. 7.4 (Input/Output).

Before any problem data can be set, variables and constraints must be added to the problem via calls to the functions MSK_appendcons and MSK_appendvars.

    /* Append 'numcon' empty constraints.
The constraints will initially have no bounds. */
if (r == MSK_RES_OK)

/* Append 'numvar' variables.
The variables will initially be fixed at zero (x=0). */
if (r == MSK_RES_OK)


New variables can now be referenced from other functions with indexes in $$\idxbeg, \ldots, \idxend{\mathtt{numvar}}$$ and new constraints can be referenced with indexes in $$\idxbeg, \ldots , \idxend{\mathtt{numcon}}$$. More variables and/or constraints can be appended later as needed, these will be assigned indexes from $$\mathtt{numvar}$$/$$\mathtt{numcon}$$ and up. Optionally one can add names.

Setting the objective.

Next step is to set the problem data. We first set the objective coefficients $$c_j = \mathtt{c[j]}$$. This can be done with functions such as MSK_putcj or MSK_putclist.

      /* Set the linear term c_j in the objective.*/
if (r == MSK_RES_OK)


Setting bounds on variables

For every variable we need to specify a bound key and two bounds according to Table 6.1.

Table 6.1 Bound keys as defined in the enum MSKboundkeye.

Bound key

Type of bound

Lower bound

Upper bound

MSK_BK_FX

$$u_j = l_j$$

Finite

Identical to the lower bound

MSK_BK_FR

Free

$$-\infty$$

$$+\infty$$

MSK_BK_LO

$$l_j \leq \cdots$$

Finite

$$+\infty$$

MSK_BK_RA

$$l_j \leq \cdots \leq u_j$$

Finite

Finite

MSK_BK_UP

$$\cdots \leq u_j$$

$$-\infty$$

Finite

For instance bkx[0]= MSK_BK_LO means that $$x_0 \geq l_0^x$$. Finally, the numerical values of the bounds on variables are given by

$l_j^x = \mathtt{blx[j]}$

and

$u_j^x = \mathtt{bux[j]}.$

Let us assume we have the bounds on variables stored in the arrays

  const MSKboundkeye bkx[]  = {MSK_BK_LO,     MSK_BK_RA, MSK_BK_LO,     MSK_BK_LO     };
const double       blx[]  = {0.0,           0.0,       0.0,           0.0           };
const double       bux[]  = { +MSK_INFINITY, 10.0,      +MSK_INFINITY, +MSK_INFINITY };


Then we can set them using various functions such MSK_putvarbound, MSK_putvarboundslice, MSK_putvarboundlist, depending on what is most convenient in the given context. For instance:


/* Set the bounds on variable j.
blx[j] <= x_j <= bux[j] */
if (r == MSK_RES_OK)
j,           /* Index of variable.*/
bkx[j],      /* Bound key.*/
blx[j],      /* Numerical value of lower bound.*/
bux[j]);     /* Numerical value of upper bound.*/


Defining the linear constraint matrix.

Recall that in our example the $$A$$ matrix is given by

$\begin{split}A = \left[ \begin{array}{cccc} 3 & 1 & 2 & 0 \\ 2 & 1 & 3 & 1 \\ 0 & 2 & 0 & 3 \end{array} \right].\end{split}$

This matrix is stored in sparse format:

  const MSKint32t    aptrb[] = {0, 2, 5, 7},
aptre[] = {2, 5, 7, 9},
asub[]  = { 0, 1,
0, 1, 2,
0, 1,
1, 2
};
const double       aval[]  = { 3.0, 2.0,
1.0, 1.0, 2.0,
2.0, 3.0,
1.0, 3.0
};


The aptrb, aptre, asub, and aval arguments define the constraint matrix $$A$$ in the column ordered sparse format (for details, see Sec. 15.1.4.2 (Column or Row Ordered Sparse Matrix)).

We now input the linear constraint matrix into the task. This can be done in many alternative ways, row-wise, column-wise or element by element in various orders. See functions such as MSK_putarow, MSK_putarowlist, MSK_putaijlist, MSK_putacol and similar.

        r = MSK_putacol(task,
j,                 /* Variable (column) index.*/
aptre[j] - aptrb[j], /* Number of non-zeros in column j.*/
asub + aptrb[j],   /* Pointer to row indexes of column j.*/
aval + aptrb[j]);  /* Pointer to Values of column j.*/


Setting bounds on constraints

Finally, the bounds on each constraint are set similarly to the variable bounds, using the bound keys as in Table 6.1. This can be done with one of the many functions MSK_putconbound, MSK_putconboundslice, MSK_putconboundlist, depending on the situation.

    /* Set the bounds on constraints.
for i=1, ...,numcon : blc[i] <= constraint i <= buc[i] */
for (i = 0; i < numcon && r == MSK_RES_OK; ++i)
i,           /* Index of constraint.*/
bkc[i],      /* Bound key.*/
blc[i],      /* Numerical value of lower bound.*/
buc[i]);     /* Numerical value of upper bound.*/


Optimization

After the problem is set-up the task can be optimized by calling the function MSK_optimizetrm .

      r = MSK_optimizetrm(task, &trmcode);


Extracting the solution.

After optimizing the status of the solution is examined with a call to MSK_getsolsta.

        if (r == MSK_RES_OK)
MSK_SOL_BAS,
&solsta);


If the solution status is reported as MSK_SOL_STA_OPTIMAL the solution is extracted:

                MSK_getxx(task,
MSK_SOL_BAS,    /* Request the basic solution. */
xx);


The MSK_getxx function obtains the solution. MOSEK may compute several solutions depending on the optimizer employed. In this example the basic solution is requested by setting the first argument to MSK_SOL_BAS. For details about fetching solutions see Sec. 7.2 (Accessing the solution).

Source code

The complete source code lo1.c of this example appears below. See also lo2.c for a version where the $$A$$ matrix is entered row-wise.

Listing 6.1 Linear optimization example. Click here to download.
#include <stdio.h>
#include "mosek.h"

/* This function prints log output from MOSEK to the terminal. */
static void MSKAPI printstr(void       *handle,
const char str[])
{
printf("%s", str);
} /* printstr */

int main(int argc, const char *argv[])
{
const MSKint32t    numvar = 4,
numcon = 3;

const double       c[]     = {3.0, 1.0, 5.0, 1.0};
/* Below is the sparse representation of the A
matrix stored by column. */
const MSKint32t    aptrb[] = {0, 2, 5, 7},
aptre[] = {2, 5, 7, 9},
asub[]  = { 0, 1,
0, 1, 2,
0, 1,
1, 2
};
const double       aval[]  = { 3.0, 2.0,
1.0, 1.0, 2.0,
2.0, 3.0,
1.0, 3.0
};

/* Bounds on constraints. */
const MSKboundkeye bkc[]  = {MSK_BK_FX, MSK_BK_LO,     MSK_BK_UP    };
const double       blc[]  = {30.0,      15.0,          -MSK_INFINITY};
const double       buc[]  = {30.0,      +MSK_INFINITY, 25.0         };
/* Bounds on variables. */
const MSKboundkeye bkx[]  = {MSK_BK_LO,     MSK_BK_RA, MSK_BK_LO,     MSK_BK_LO     };
const double       blx[]  = {0.0,           0.0,       0.0,           0.0           };
const double       bux[]  = { +MSK_INFINITY, 10.0,      +MSK_INFINITY, +MSK_INFINITY };
MSKrescodee        r = MSK_RES_OK;
MSKint32t          i, j;

if (r == MSK_RES_OK)
{
/* Create the optimization task. */

/* Directs the log task stream to the 'printstr' function. */
if (r == MSK_RES_OK)

/* Append 'numcon' empty constraints.
The constraints will initially have no bounds. */
if (r == MSK_RES_OK)

/* Append 'numvar' variables.
The variables will initially be fixed at zero (x=0). */
if (r == MSK_RES_OK)

for (j = 0; j < numvar && r == MSK_RES_OK; ++j)
{
/* Set the linear term c_j in the objective.*/
if (r == MSK_RES_OK)

/* Set the bounds on variable j.
blx[j] <= x_j <= bux[j] */
if (r == MSK_RES_OK)
j,           /* Index of variable.*/
bkx[j],      /* Bound key.*/
blx[j],      /* Numerical value of lower bound.*/
bux[j]);     /* Numerical value of upper bound.*/

/* Input column j of A */
if (r == MSK_RES_OK)
j,                 /* Variable (column) index.*/
aptre[j] - aptrb[j], /* Number of non-zeros in column j.*/
asub + aptrb[j],   /* Pointer to row indexes of column j.*/
aval + aptrb[j]);  /* Pointer to Values of column j.*/
}

/* Set the bounds on constraints.
for i=1, ...,numcon : blc[i] <= constraint i <= buc[i] */
for (i = 0; i < numcon && r == MSK_RES_OK; ++i)
i,           /* Index of constraint.*/
bkc[i],      /* Bound key.*/
blc[i],      /* Numerical value of lower bound.*/
buc[i]);     /* Numerical value of upper bound.*/

/* Maximize objective function. */
if (r == MSK_RES_OK)

if (r == MSK_RES_OK)
{
MSKrescodee trmcode;

/* Run optimizer */

/* Print a summary containing information
about the solution for debugging purposes. */

if (r == MSK_RES_OK)
{
MSKsolstae solsta;

if (r == MSK_RES_OK)
MSK_SOL_BAS,
&solsta);
switch (solsta)
{
case MSK_SOL_STA_OPTIMAL:
{
double *xx = (double*) calloc(numvar, sizeof(double));
if (xx)
{
MSK_SOL_BAS,    /* Request the basic solution. */
xx);

printf("Optimal primal solution\n");
for (j = 0; j < numvar; ++j)
printf("x[%d]: %e\n", j, xx[j]);

free(xx);
}
else
r = MSK_RES_ERR_SPACE;

break;
}
case MSK_SOL_STA_DUAL_INFEAS_CER:
case MSK_SOL_STA_PRIM_INFEAS_CER:
printf("Primal or dual infeasibility certificate found.\n");
break;
case MSK_SOL_STA_UNKNOWN:
{
char symname[MSK_MAX_STR_LEN];
char desc[MSK_MAX_STR_LEN];

/* If the solutions status is unknown, print the termination code
indicating why the optimizer terminated prematurely. */

MSK_getcodedesc(trmcode,
symname,
desc);

printf("The solution status is unknown.\n");
printf("The optimizer terminitated with code: %s\n", symname);
break;
}
default:
printf("Other solution status.\n");
break;
}
}
}

if (r != MSK_RES_OK)
{
/* In case of an error print error code and description. */
char symname[MSK_MAX_STR_LEN];
char desc[MSK_MAX_STR_LEN];

printf("An error occurred while optimizing.\n");
MSK_getcodedesc(r,
symname,
desc);
printf("Error %s - '%s'\n", symname, desc);
}

/* Delete the task and the associated data. */