Quadratic Programming

Continuous variables, a quadratic objective, linear constraints.

In plain terms

Often the cost of a decision doesn’t grow in a straight line: deviating twice as far from a target hurts four times as much, and risks compound when choices interact. Quadratic programming captures exactly that: smooth trade-offs around a sweet spot, with limits to respect.

The technical picture

Quadratic programming extends LP with a quadratic objective: costs and risks that grow with the square of a decision, and pairwise interactions between variables (portfolio risk, tracking error, control effort) under linear constraints.

Quicopt takes the quadratic objective in its native form through the same modeling interface as every other class: no reformulation, no separate solver to license and integrate.

Mathematical model▾

Minimize a quadratic objective subject to linear constraints over continuous variables.

x \in R^{n} min s.t. \frac{1}{2} x^{⊤} Q x + c^{⊤} x A x \leq b

where Q \in R^{n \times n} is symmetric, c \in R^{n}, A \in R^{m \times n}, b \in R^{m} .

Example

From install to solved model: a small, self-contained example, copy-paste ready.

Install the client

$ pip install "quicopt[mathopt]"

Copy the example

qp.py

from ortools.math_opt.python import mathopt
from quicopt import Client

# A QP: continuous variables, a convex quadratic objective, a linear constraint.
# Unconstrained the optimum would be (1, 2); the constraint x + y <= 2 pushes
# it to (0.5, 1.5) with objective 0.5.
model = mathopt.Model(name="qp")
x = model.add_variable(lb=0.0, name="x")
y = model.add_variable(lb=0.0, name="y")
model.add_linear_constraint(x + y <= 2)
model.minimize((x - 1) * (x - 1) + (y - 2) * (y - 2))

client = Client("https://try.quicoptapi.pgi.fz-juelich.de")
result = client.solve(model)
print(result.display)

Run it

$ python qp.py

What you’ll see

├── status:     optimal
├── feasible:   true
├── objective:  0.4999999825141841
├── x:          x=0.5, y=1.5  (2 variables)
└── solve_time: 0.3984 s

Docs, API reference and more examples live in the Developer Hub →

Benchmark

How Quicopt performs on representative quadratic programs.

Illustrative, pending measurement

QuicoptEstablished solver

Illustrative scaling, to be replaced with measured data.

Measured results for this class are being prepared and will appear here.