6.8 Integer Optimization¶
An optimization problem where one or more of the variables are constrained to integer values is called a (mixed) integer optimization problem. MOSEK supports integer variables in combination with linear, quadratic and quadratically constrtained and conic problems (except semidefinite). See the previous tutorials for an introduction to how to model these types of problems.
6.8.1 Basic linear example¶
We use the example
to demonstrate how to set up and solve a problem with integer variables. It has the structure of a linear optimization problem except for integrality constraints on the variables. Therefore, only the specification of the integer constraints requires something new compared to the linear optimization problem discussed previously.
First, the integrality constraints are imposed using the function MSK_putvartype or one of its bulk analogues:
for (j = 0; j < numvar && r == MSK_RES_OK; ++j)
r = MSK_putvartype(task, j, MSK_VAR_TYPE_INT);
Next, the example demonstrates how to set various useful parameters of the mixed-integer optimizer. See Sec. 13.5 (The Mixed-Integer optimizer) for details.
/* Set max solution time */
r = MSK_putdouparam(task,
MSK_DPAR_MIO_MAX_TIME,
60.0);
The complete source for the example is listed Listing 6.13. Please note that when we fetch the solution then the integer solution is requested by using MSK_SOL_ITG. No dual solution is defined for integer optimization problems.
6.8.2 Specifying an initial solution (hot-start)¶
It is a common strategy to provide a starting feasible point (if one is known in advance) to the mixed-integer solver. This is known as hot-start or warm-start. The feasible point may come from the user’s prior knowledge of the model, a heuristic, or a solution from a preceding solve if the problem was modified so that the solution remains feasible. Using hot-start allows the solver to skip worse solutions and potentially get closer to the optimum faster.
There are two modes for MOSEK to utilize an initial solution.
A complete solution. MOSEK will first try to check if the current value of the primal variable solution is a feasible point. The solution can either come from a previous solver call or can be entered by the user, however the full solution with values for all variables (both integer and continuous) must be provided. This check is always performed and does not require any extra action from the user. The outcome of this process can be inspected via information items
MSK_IINF_MIO_INITIAL_FEASIBLE_SOLUTIONandMSK_DINF_MIO_INITIAL_FEASIBLE_SOLUTION_OBJ, and via theInitial feasible solution objectiveentry in the log.A partial integer solution. MOSEK can also try to construct a feasible solution by fixing integer variables to the values provided by the user (rounding if necessary) and optimizing over the remaining continuous variables. In this setup the user must provide initial values for all integer variables. This action is only performed if the parameter
MSK_IPAR_MIO_CONSTRUCT_SOLis switched on. The outcome of this process can be inspected via information itemsMSK_IINF_MIO_CONSTRUCT_SOLUTIONandMSK_DINF_MIO_CONSTRUCT_SOLUTION_OBJ, and via theConstruct solution objectiveentry in the log.
In the following example we focus on inputting a partial integer solution.
Solution values can be set using MSK_putxx, MSK_putxxslice or similar . If the solution to be used as hot-start happens to come from a previous solve of the same model/task then it will be used by the solver automatically and does not have to be input explicitly through the API again.
here to download.¶ if (r == MSK_RES_OK)
{
/* Assign values to integer variables
(we only set a slice of xx) */
double xxInit[] = {1.0, 1.0, 0.0};
r = MSK_putxxslice(task, MSK_SOL_ITG, 0, 3, xxInit);
}
if (r == MSK_RES_OK)
{
/* Request constructing the solution from integer variable values */
r = MSK_putintparam(task, MSK_IPAR_MIO_CONSTRUCT_SOL, MSK_ON);
}
The log output from the optimizer will in this case indicate that the inputted values were used to construct an initial feasible solution:
Construct solution objective : 1.950000000000e+01
The same information can be obtained from the API:
MSK_getintinf(task, MSK_IINF_MIO_CONSTRUCT_SOLUTION, &constr);
MSK_getdouinf(task, MSK_DINF_MIO_CONSTRUCT_SOLUTION_OBJ, &constr_obj);
printf("Construct solution utilization: %d\nConstruct solution objective: %.3f\n", constr, constr_obj);
6.8.3 Basic conic example¶
Integer variables can also be used arbitrarily in conic problems (except semidefinite). We refer to the previous tutorials for how to set up a conic optimization problem. Here we present sample code that sets up a simple optimization problem:
The canonical conic formulation of (6.29) suitable for Optimizer API for C is
int main(int argc, char *argv[])
{
MSKint32t numvar = 3; /* x, y, t */
MSKvariabletypee vart[] = { MSK_VAR_TYPE_INT, MSK_VAR_TYPE_INT };
MSKint32t intsub[] = { 0, 1 };
MSKint32t i, j;
MSKenv_t env = NULL;
MSKtask_t task = NULL;
MSKrescodee r, trm;
r = MSK_makeenv(&env, NULL);
if (r == MSK_RES_OK)
{
r = MSK_maketask(env, 0, 0, &task);
if (r == MSK_RES_OK)
r = MSK_linkfunctotaskstream(task, MSK_STREAM_LOG, NULL, printstr);
if (r == MSK_RES_OK)
r = MSK_appendvars(task, numvar);
if (r == MSK_RES_OK)
r = MSK_putvarboundsliceconst(task, 0, numvar, MSK_BK_FR, -0.0, 0.0);
/* Integrality constraints */
if (r == MSK_RES_OK)
r = MSK_putvartypelist(task, 2, intsub, vart);
/* Objective */
if (r == MSK_RES_OK)
r = MSK_putobjsense(task, MSK_OBJECTIVE_SENSE_MINIMIZE);
if (r == MSK_RES_OK)
r = MSK_putcj(task, 2, 1.0); /* Minimize t */
/* Conic part of the problem */
if (r == MSK_RES_OK)
{
/* Set up the affine expressions */
/* x, x-3.8, y, t, 1.0 */
MSKint64t afeidx[] = {0, 1, 2, 3};
MSKint32t varidx[] = {0, 0, 1, 2};
MSKrealt val[] = {1.0, 1.0, 1.0, 1.0};
MSKrealt g[] = {0.0, -3.8, 0.0, 0.0, 1.0};
MSKint64t domExp, domQuad;
MSKint64t afeidxExp[] = {1, 4, 2};
MSKint64t afeidxQuad[] = {3, 0, 2};
if (r == MSK_RES_OK)
r = MSK_appendafes(task, 5);
if (r == MSK_RES_OK)
r = MSK_putafefentrylist(task,
4,
afeidx,
varidx,
val);
if (r == MSK_RES_OK)
r = MSK_putafegslice(task, 0, 5, g);
// Add constraint (x-3.8, 1, y) \in \EXP
if (r == MSK_RES_OK)
r = MSK_appendprimalexpconedomain(task, &domExp);
if (r == MSK_RES_OK)
r = MSK_appendacc(task, domExp, 3, afeidxExp, NULL);
// Add constraint (t, x, y) \in \QUAD
if (r == MSK_RES_OK)
r = MSK_appendquadraticconedomain(task, 3, &domQuad);
if (r == MSK_RES_OK)
r = MSK_appendacc(task, domQuad, 3, afeidxQuad, NULL);
}
/* Optimize the problem */
if (r == MSK_RES_OK)
r = MSK_optimizetrm(task, &trm);
if (r == MSK_RES_OK)
r = MSK_solutionsummary(task, MSK_STREAM_MSG);
if (r == MSK_RES_OK)
{
MSKrealt xx[] = {0, 0};
r = MSK_getxxslice(task, MSK_SOL_ITG, 0, 2, xx);
if (r == MSK_RES_OK)
printf("x = %.2f, y = %.2f\n", xx[0], xx[1]);
}
if (task) MSK_deletetask(&task);
}
if (env) MSK_deleteenv(&env);
return r;
}
Error and solution status handling were omitted for readability.
6.8.4 Fixed problem and dual values¶
The dual solution is not defined for mixed-integer problems, but in some cases the user may want to obtain some dual information (shadow prices). One typical strategy is to compute shadow price information under the assumption that the combinatorial decisions (integer variable values) do not change, that is:
solve the mixed-integer model (to some feasible solution, not necessarily optimal),
fix all integer variables to their values in the solution,
solve the fixed model as a continuous problem and extract the dual values.
Optimizer API for C facilitates the construction of the fixed model with with MSK_getfixedproblem.
As an example we consider a toy production planning model with two plants with maximum capacities 70, 80 units and a demand of 100 units:
We begin by solving the mixed-integer problem and verifying that it has a feasible solution with \(s_1=s_2=1\) i.e. both plants active with production levels \((x_1,x_2)=(70,30)\). Then, assuming no modifications are made to the problem structure or numerical data, we can immediately construct and solve the fixed integer model and retrieve its solution as shown below.
/* Create and solve the problem with fixed variables */
MOSEKCALL(r, MSK_getfixedproblem(task, &fixTask));
MOSEKCALL(r, MSK_linkfunctotaskstream(fixTask, MSK_STREAM_LOG, NULL, printstr));
MOSEKCALL(r, MSK_optimize(fixTask));
MOSEKCALL(r, MSK_getprosta(fixTask, MSK_SOL_BAS, &fixProsta));
if (r == MSK_RES_OK && fixProsta != MSK_PRO_STA_PRIM_AND_DUAL_FEAS) {
printf("Unsuitable problem status, exiting\n");
return -1;
}
if (r == MSK_RES_OK) {
double xfix[2], y[3];
MOSEKCALL(r, MSK_getxxslice(fixTask, MSK_SOL_BAS, 0, 2, xfix));
MOSEKCALL(r, MSK_getyslice(fixTask, MSK_SOL_BAS, 0, 3, y));
printf("xfix = %.2f, %.2f\n", xfix[0], xfix[1]);
printf("demand dual = %.2f\n", y[0]);
printf("production dual = %.2f, %.2f\n", y[1], y[2]);
}
For instance, the shadow price for the first plant’s production constraint is \(-4\), corresponding to the fact that increasing the capacity of the first plant by 1 unit would allow shifting one production unit form plant 2 to plant 1, reducing the objective cost by \(14-10=4\).
Note that the solution of the mixed-integer problem will typically have small violations, which implies that the fixed model may
be declared (borderline) infeasible. To avoid this effect, the fixing algorithm will by default introduce small perturbations
of bounds in order to make the fixed problem strictly feasible. If needed this can be controlled by setting the parameter MSK_IPAR_FIXING_METHOD in
the original task prior to fixing.
