Mixed Integer Linear Programming

Consider the following example,

Given system constraints:

2x + 4y >= 230
3x + 2y <= 190
x >= 0, x is integer
y >= 0, y is continuious

Maximize objective function:

f(x) = 5x + 3y

You need to find numbers for x and y in such a way that it satisfies constraints and maximizes the objective function.

problem_data = {}

Set Constraint Matrix

If the constraints are:

2x + 4y >= 230
3x + 2y <= 190

Constraints are depicted in CSR format. The constraints can be transformed to the CSR matrix as follows:

offsets = [0, 2, 4]
indices = [0, 1, 0, 1]
coefficients = [2.0, 4.0, 3.0, 2.0]

problem_data["csr_constraint_matrix"] = {
    "offsets" : offsets,
    "indices" : indices,
    "values"  :coefficients
}

The offsets indicate the length of the constraint and indices indicate variables.

Set Constraint Bounds

If the constraints are as follows:

2x + 4y >= 230
3x + 2y <= 190

You need to define upper_bounds and lower_bounds of all the constraints, each value signifies the upper or lower bound of each constraint respective to its index.

upper_bounds = ["inf", 190.0]
lower_bounds = [230.0, "ninf"]

problem_data["constraint_bounds"] = {
   "upper_bounds" : upper_bounds,
   "lower_bounds" : lower_bounds
}

inf - infinity and ninf - negative infinity are used when there is no explict upper or lower bound.

Set Variable Bounds

Variables:

x >= 0
y >= 0

Define the variable bounds similar to constraint bounds.

var_upper_bounds = ["inf", "inf"]
var_lower_bounds = [0.0, 0.0]

problem_data["variable_bounds"] = {
   "upper_bounds" : var_upper_bounds,
   "lower_bounds" : var_lower_bounds
}

Set Objective Data

Objective:

f(x) = 5x + 3y

Pass coefficents for objective data and also set whether it needs to be maximized or minimized.

objective_coefficients = [5.0, 3.0]
maximize = True

problem_data["objective_data"] = {
    "coefficients" : objective_coefficients,
    "scalability_factor" : 1.0,
    "offset" : 0.0
}

problem_data["maximize"] = maximize

Set Variable Names

This is optional, but it helps users to navigate the result.

problem_data["variable_names"] = ["x", "y"]

Set Variable Types

Set variable types, I - Integer and C - Continuous

problem_data["variable_types"] = ["I", "C"]

Set Solver Configuration

The solver configuration can be fine-tuned for optimization and runtimes.

solver_config  = {
    "time_limit" : 1.0
}

problem_data["solver_config"] = solver_config

Solve the Problem

For managed service, cuOpt endpoints can be triggered as shown in the thin client example for managed service.

For self-hosted, cuOpt endpoints can be triggered as shown in the thin client example for self-hosted.

Use this data and invoke the cuOpt endpoint, which would return values for x and y.

The following example is using a locally hosted server:

from cuopt_sh_client import CuOptServiceSelfHostClient
import json

# If cuOpt is not running on localhost:5000, edit ip and port parameters
cuopt_service_client = CuOptServiceSelfHostClient(
    ip="localhost",
    port=5000,
    timeout_exception=False
)

repoll_tries = 50

def repoll(solution, repoll_tries):
    # If solver is still busy solving, the job will be assigned a request id and response is sent back in the
    # following format {"reqId": <REQUEST-ID>}. Solver needs to be rei-polled for response using this <REQUEST-ID>.

    if "reqId" in solution and "response" not in solution:
        req_id = solution["reqId"]
        for i in range(repoll_tries):
            solution = cuopt_service_client.repoll(req_id, response_type="dict")
            if "reqId" in solution and "response" in solution:
                break;

            # Sleep for a second before requesting
            time.sleep(1)

    return solution

solution = cuopt_service_client.get_LP_solve(problem_data, response_type="dict")

solution = repoll(solution, repoll_tries)

print(json.dumps(solution, indent=4))

Status - 2 corresponds to Feasible solution is available.

{
    "response": {
        "solver_response": {
            "status": 2,
            "solution": {
                "problem_category": 1,
                "primal_solution": [
                    37.0,
                    39.49999999148369
                ],
                "dual_solution": null,
                "primal_objective": 303.49999997445104,
                "dual_objective": null,
                "solver_time": 1.004452673,
                "vars": {
                    "x": 37.0,
                    "y": 39.49999999148369
                },
                "lp_statistics": {
                    "reduced_cost": null
                },
                "milp_statistics": {
                    "mip_gap": -303.49999997445104,
                    "solution_bound": -2.0,
                    "presolve_time": 0.043964768,
                    "max_constraint_violation": 0.0,
                    "max_int_violation": 0.0,
                    "max_variable_bound_violation": 0.0
                }
            }
        },
        "total_solve_time": 1.0543067455291748
    },
    "reqId": "56927e27-913c-4531-b709-04eb9d91b00b"
}