qp-formulation

QP Formulation

Domain concepts for quadratic programming. No API or interface details here. QP support in cuOpt is currently in beta.

What is QP

• Objective: Quadratic in the variables (e.g. x², x·y terms). Example: portfolio variance xᵀQx.
• Constraints: Linear only. cuOpt does not support quadratic constraints.

Important domain rule: minimize only

QP objectives must be minimization. To maximize a quadratic expression, negate it and minimize; then negate the optimal value.

Required questions (problem formulation)

Ask these if not already clear:

1. Objective — Does it have squared or cross terms (x², x·y)? If purely linear, use LP/MILP instead.
2. Minimize or maximize? — If maximize, user must negate objective and minimize.
3. Convexity — For minimization, the quadratic form (matrix Q) should be positive semi-definite for well-posed problems.
4. Constraints — All linear (no quadratic constraints)?

Typical use cases

• Portfolio optimization (minimize variance subject to return and budget).
• Least squares (minimize ‖Ax − b‖²).
• Other quadratic objectives with linear constraints.