6.5 Conic Exponential Optimization¶

The structure of a typical conic optimization problem is

$\begin{split}\begin{array}{lccccl} \mbox{minimize} & & & c^T x+c^f & & \\ \mbox{subject to} & l^c & \leq & A x & \leq & u^c, \\ & l^x & \leq & x & \leq & u^x, \\ & & & Fx+g & \in & \D, \end{array}\end{split}$

(see Sec. 12 (Problem Formulation and Solutions) for detailed formulations). We recommend Sec. 6.2 (From Linear to Conic Optimization) for a tutorial on how problems of that form are represented in MOSEK and what data structures are relevant. Here we discuss how to set-up problems with the primal/dual exponential cones.

MOSEK supports two exponential cones, namely:

• Primal exponential cone:

$\EXP = \left\lbrace x \in \real^3: x_0 \geq x_1 \exp(x_2/x_1),\ x_0,x_1\geq 0 \right\rbrace.$
• Dual exponential cone:

$\EXP^* = \left\lbrace s \in \real^3: s_0 \geq -s_2 e^{-1} \exp(s_1/s_2),\ s_2\leq 0,s_0\geq 0 \right\rbrace.$

For example, consider the following constraint:

$(x_4, x_0, x_2) \in \EXP$

which describes a convex cone in $$\real^3$$ given by the inequalities:

$x_4 \geq x_0\exp(x_2/x_0),\ x_0,x_4\geq 0.$

For other types of cones supported by MOSEK, see Sec. 15.10 (Supported domains) and the other tutorials in this chapter. Different cone types can appear together in one optimization problem.

6.5.1 Example CEO1¶

Consider the following basic conic exponential problem which involves some linear constraints and an exponential inequality:

(6.15)$\begin{split}\begin{array} {lrcl} \mbox{minimize} & x_0 + x_1 & & \\ \mbox{subject to} & x_0+x_1+x_2 & = & 1, \\ & x_0 & \geq & x_1\exp(x_2/x_1), \\ & x_0, x_1 & \geq & 0. \end{array}\end{split}$

The affine conic form of (6.15) is:

(6.16)$\begin{split}\begin{array} {lrcl} \mbox{minimize} & x_0 + x_1 & & \\ \mbox{subject to} & x_0+x_1+x_2 & = & 1, \\ & Ix & \in & \EXP, \\ & x & \in & \real^3. \end{array}\end{split}$

where $$I$$ is the $$3\times 3$$ identity matrix.

Setting up the linear part

The linear parts (constraints, variables, objective) are set up using exactly the same methods as for linear problems, and we refer to Sec. 6.1 (Linear Optimization) for all the details. The same applies to technical aspects such as defining an optimization task, retrieving the solution and so on.

Setting up the conic constraints

In order to append the conic constraints we first input the sparse identity matrix $$\afef$$ as indicated by (6.16).

The affine conic constraint is then appended using the function MSK_appendacc, with the primal exponential domain and the list of $$\afef$$ rows, in this case consisting of all rows in their natural order.

r = MSK_putafefentrylist(task, f_nnz, afeidx, varidx, f_val);

if (r == MSK_RES_OK)
r = MSK_appendprimalexpconedomain(task, &domidx);

if (r == MSK_RES_OK)
r = MSK_appendaccseq(task, domidx, numafe, 0, NULL);

The first argument selects the domain, which must be appended before being used, and must have the dimension matching the number of affine expressions appearing in the constraint. Variants of this method are available to append multiple ACCs at a time. It is also possible to define the matrix $$\afef$$ using a variety of methods (row after row, column by column, individual entries, etc.) similarly as for the linear constraint matrix $$A$$.

For a more thorough exposition of the affine expression storage (AFE) matrix $$\afef$$ and vector $$\afeg$$ see Sec. 6.2 (From Linear to Conic Optimization).

Source code

Listing 6.6 Source code solving problem (6.15). Click here to download.
#include <stdio.h>
#include "mosek.h" /* Include the MOSEK definition file. */

static void MSKAPI printstr(void *handle,
const char str[])
{
printf("%s", str);
} /* printstr */

int main(int argc, const char *argv[])
{
MSKrescodee  r;

const MSKint32t numvar = 3,
numcon = 1;
const MSKint64t numafe = 3,
f_nnz  = 3;

MSKboundkeye bkc = MSK_BK_FX;
double       blc = 1.0;
double       buc = 1.0;

MSKboundkeye bkx[] = {MSK_BK_FR,
MSK_BK_FR,
MSK_BK_FR
};
double       blx[] = {-MSK_INFINITY,
-MSK_INFINITY,
-MSK_INFINITY
};
double       bux[] = { +MSK_INFINITY,
+MSK_INFINITY,
+MSK_INFINITY
};

double       c[]   = {1.0,
1.0,
0.0
};

double    a[]     = {1.0, 1.0, 1.0};
MSKint32t asub[]  = {0, 1, 2};

MSKint64t   afeidx[] = {0, 1, 2};
MSKint32t   varidx[] = {0, 1, 2};
MSKrealt    f_val[]  = {1, 1, 1};

MSKint64t   domidx = 0;

MSKint32t   i, j, csub[3];

MSKenv_t    env  = NULL;
MSKtask_t   task = NULL;

/* Create the mosek environment. */
r = MSK_makeenv(&env, NULL);

if (r == MSK_RES_OK)
{
/* Create the optimization task. */
r = MSK_maketask(env, numcon, numvar, &task);

if (r == MSK_RES_OK)
{
MSK_linkfunctotaskstream(task, MSK_STREAM_LOG, NULL, printstr);

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

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

/* Append 'numafe' affine expressions.
The affine expressions will initially be empty. */
if (r == MSK_RES_OK)
r = MSK_appendafes(task, numafe);

/* Set up the linear part */
if (r == MSK_RES_OK)
r = MSK_putcslice(task, 0, numvar, c);
if (r == MSK_RES_OK)
r = MSK_putarow(task, 0, numvar, asub, a);
if (r == MSK_RES_OK)
r = MSK_putconbound(task, 0, bkc, blc, buc);
if (r == MSK_RES_OK)
r = MSK_putvarboundslice(task, 0, numvar, bkx, blx, bux);

if (r == MSK_RES_OK)
{
r = MSK_putafefentrylist(task, f_nnz, afeidx, varidx, f_val);

if (r == MSK_RES_OK)
r = MSK_appendprimalexpconedomain(task, &domidx);

if (r == MSK_RES_OK)
r = MSK_appendaccseq(task, domidx, numafe, 0, NULL);
}

if (r == MSK_RES_OK)
{
MSKrescodee trmcode;

/* Run optimizer */
r = MSK_optimizetrm(task, &trmcode);

/* Print a summary containing information
about the solution for debugging purposes*/
MSK_solutionsummary(task, MSK_STREAM_MSG);

if (r == MSK_RES_OK)
{
MSKsolstae solsta;

MSK_getsolsta(task, MSK_SOL_ITR, &solsta);

switch (solsta)
{
case MSK_SOL_STA_OPTIMAL:
{
double *xx = NULL;

xx = calloc(numvar, sizeof(double));
if (xx)
{
MSK_getxx(task,
MSK_SOL_ITR,    /* Request the interior solution. */
xx);

printf("Optimal primal solution\n");
for (j = 0; j < numvar; ++j)
printf("x[%d]: %e\n", j, xx[j]);
}
else
{
r = MSK_RES_ERR_SPACE;
}
free(xx);
}
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:
printf("The status of the solution could not be determined. Termination code: %d.\n", trmcode);
break;
default:
printf("Other solution status.");
break;
}
}
else
{
printf("Error while optimizing.\n");
}
}

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. */
MSK_deletetask(&task);
}

/* Delete the environment and the associated data. */
MSK_deleteenv(&env);

return (r);
} /* main */