QP Example

Consider solving the following Quadratic Programming (QP) problem:

Objective:

Minimize the quadratic function:

\[z = x_1 - 2x_2 + 4x_3 + x_1^2 + 2x_2^2 + 3x_3^2 + x_1 x_3\]

Subject to constraints:

\[\begin{split}\begin{aligned} 3x_1 + 4x_2 - 2x_3 &\leq 10 \\ -3x_1 + 2x_2 + x_3 &\geq 2 \\ 2x_1 + 3x_2 + 4x_3 &= 5 \\ 0 \leq x_1 &\leq 5 \\ 1 \leq x_2 &\leq 5 \\ 0 \leq x_3 &\leq 5 \end{aligned}\end{split}\]

General QP Formulation

The general form of a Quadratic Programming (QP) problem can be expressed as:

Objective: Minimize

\[\frac{1}{2} x^T Q x + c^T x\]

Subject to:

\[ \begin{align}\begin{aligned}Ax = b,\\Gx \leq h,\\lb \leq x \leq ub\end{aligned}\end{align} \]

So:

\[\begin{split}x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \quad Q = \begin{bmatrix} 2 & 0 & 1 \\ 0 & 4 & 0 \\ 1 & 0 & 6 \end{bmatrix}, \quad c = \begin{bmatrix} 1 \\ -2 \\ 4 \end{bmatrix}\end{split}\]

\[A = \begin{bmatrix} 2 & 3 & 4 \end{bmatrix}, \quad b = \begin{bmatrix} 5 \end{bmatrix}\]

\[\begin{split}G = \begin{bmatrix} 3 & 4 & -2 \\ 3 & -2 & -1 \end{bmatrix}, h = \begin{bmatrix} 10 \\ -2 \end{bmatrix}\end{split}\]

\[\begin{split}lb = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, ub = \begin{bmatrix} 5 \\ 5 \\ 5 \end{bmatrix}\end{split}\]

where: - $Q$ is a symmetric matrix representing the quadratic coefficients. - $c$ is a vector representing the linear coefficients. - $A$, $G$, $lb$, and $ub$ are defined as per the problem constraints.

ipc_dict = {
    "model": {
        "Q": {
            "row": [0, 0, 2, 1, 2],
            "col": [0, 2, 0, 1, 2],
            "val": [2, 1, 1, 4, 6]
        },
        "c": [1, -2, 4],
        "A": {
            "row": [0, 0, 0],
            "col": [0, 1, 2],
            "val": [2, 3, 4]
        },
        "b": [5],
        "G": {
            "row": [0, 0, 0, 1, 1, 1],
            "col": [0, 1, 2, 0, 1, 2],
            "val": [3, 4, -2, 3, -2, -1]
        },
        "h": [10, -2],
        "lb": [0, 1, 0],
        "ub": [5, 5, 5],
        "osense": "min"
    },
    "model_name": "QP-tutorial",
    "engine": "julia",
    "solver": {
        "solver_name": "HiGHS",
        "solver_type": "QP",
        "solver_params": {}
    }
}

import pyarrow as pa
import requests

# Convert the dictionary to an Arrow table for IPC serialization
pa_arrays = [pa.array([v]) for v in ipc_dict.values()]
table = pa.Table.from_arrays(pa_arrays, names=list(ipc_dict.keys()))

# Serialize the table to IPC stream bytes
sink = pa.BufferOutputStream()
with pa.ipc.new_stream(sink, table.schema) as writer:
    writer.write(table)
ipc_bytes = sink.getvalue().to_pybytes()

# Send the IPC stream bytes to the server
headers = {"Content-Type": "application/vnd.apache.arrow.stream"}
response = requests.post("http://localhost:8000/compute", data=ipc_bytes, headers=headers)

# Check if the response is successful
if response.status_code != 200:
    print(f"Error: {response.status_code} - {response.text}")
# response is ipc stream
reader = pa.ipc.open_stream(response.content)
result_table = reader.read_all()
result_dict = {name: result_table.column(name).to_pylist() for name in result_table.column_names}
print(result_dict)

headers = {"Content-Type": "application/json"}
response = requests.post("http://localhost:8000/computeJSON", json=ipc_dict, headers=headers)
print(response.json())