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.