---name: cuopt-numerical-optimization-apiversion: 26.10.00description: LP, MILP, and QP (beta) with cuOpt — Python, C, and CLI. Use when the user is solving LP, MILP, or QP with any cuOpt interface.license: Apache-2.0metadata: author: NVIDIA cuOpt Teamtags: ---

cuOpt Numerical Optimization API

Model and solve LP, MILP, and QP problems using NVIDIA cuOpt's GPU-accelerated solver.

Interface Selection

Choose the reference for the user's interface:

Interface	When to use	Reference
Python	User is writing Python code	references/python_api.md
C / C++	User is embedding in a C/C++ application	references/c_api.md
CLI	User is solving from MPS files on the command line	references/cli_api.md

If the interface is not yet clear, ask before writing any code.

Already using a modeling language? cuOpt also works as a solver backend for third-party modeling tools — AMPL, GAMS / GAMSPy, PuLP, JuMP, Pyomo, and CVXPY — with near-zero code changes (point the model's solver at cuOpt). CVXPY additionally covers convex QP and, in beta, QCQP / SOCP. Prefer this when the user already has a model in one of these tools rather than porting it to the cuOpt API. See Third-Party Modeling Languages.

Choosing LP vs MILP vs QP

Decide from the objective and variables:

If the objective is...	And variables are...	Use
Linear (sum of c_i * x_i)	All continuous	LP
Linear	Some integer or binary	MILP
Has squared (x*x) or cross (x*y) terms	Continuous (integer QP not supported)	QP (beta)

Prefer LP when the problem allows it. LP solves faster and has stronger optimality guarantees. Use MILP only when the problem logically requires whole numbers or yes/no decisions. Use QP only when the objective is genuinely quadratic (variance, squared error, kinetic energy).

• Use LP when every quantity can meaningfully be fractional: flows, proportions, rates, dollars, hours, tonnes of material, etc.
• Use MILP when the problem mentions counts of discrete entities, yes/no choices, or either/or decisions (e.g. open a facility or not, assign a person to a shift, number of trucks).
• Use QP when the objective minimizes variance, squared error, or any expression with x*x or x*y terms (portfolio optimization, least squares, regularized regression).

Integer vs Continuous from Wording

Problem wording / concept	Variable type	Examples
Discrete entities (counts)	INTEGER	Workers, cars, trucks, machines, pilots, facilities, units to manufacture
Yes/no or on/off	INTEGER (binary, lb=0 ub=1)	Open a facility, run a machine, assign a person to a shift
Amounts that can be fractional	CONTINUOUS	Tonnes, litres, dollars, hours, kWh, proportion of capacity
Rates or fractions	CONTINUOUS	Utilization, percentage, share of budget

Rule of thumb: "How many things" → INTEGER. "How much" → CONTINUOUS.

QP Rules (all interfaces)

• MINIMIZE only — the solver rejects MAXIMIZE for quadratic objectives. To maximize f(x), minimize -f(x) and negate the reported objective value.
• Continuous variables only — integer QP is not supported.
• Q should be positive semi-definite for a convex, well-posed problem.
• Beta — API may evolve; treat as production-capable for typical convex QP.

Dual Values

Duals and reduced costs are available for LP and QP only:

• MILP — no duals (integer optima are not continuous).
• Quadratic constraints — duals unavailable even for LP/QP; all values return NaN.
• PDLP warmstart — LP only; MILP solves do not accept a PDLP warmstart.

Common Issues (all interfaces)

Problem	Likely cause	Fix
Infeasible	Conflicting constraints	Check constraint logic and bounds
Unbounded	Missing bounds	Add variable bounds
Slow solve	Large problem	Set time limit; increase gap tolerance
QP rejected with MAXIMIZE	QP only supports MINIMIZE	Negate the objective; negate the result
QP returns non-optimal	Q not PSD or badly scaled	Check Q is PSD; rescale variables

Solver Settings (concepts)

Setting	Purpose
time_limit	Stop after N seconds
mip_relative_gap	Stop MILP when within X% of optimal
mip_absolute_tolerance	Absolute MIP gap stop
log_to_console	Enable solver logging

Syntax varies by interface — see the interface reference file.