Gurobi versions 9.0 and later can solve models with linear constraints, quadratic constraints (both convex and non-convex), and second-order cone constraints. This can involve any combination of continuous and integer variables. The canonical form is:

$$\begin{alignat}{3} \min_x\ x^\top Q x + c^\top x &+ d && && \textrm{(Objective)} \\ \textrm{s.t.} \qquad\qquad\enspace Ax &= b && && \textrm{(Linear constraints)} \\x^\top Q_i x + c_i^\top x &\le d_i && \forall i \in I && \textrm{(Quadratic constraints)} \\ \ell \le x &\le u && && \textrm{(Bound constraints)} \\ x_j &\in \mathbb{Z} && \forall j \in J && \textrm{(Integrality constraints)} \\ x_k &\in \textrm{second-order cones} \quad && \forall k \in K && \textrm{(Cone constraints)} \\ x_h &\in \textrm{SOS} && \forall h \in H \qquad && \textrm{(Special ordered set constraints)} \end{alignat}$$

Not all elements are required. For instance, if \( Q =Q_i = 0 \) and \( I = K = H = \emptyset \), this is a mixed-integer linear program (MILP).

Before Gurobi 9.0, the quadratic matrices \( Q \) and \( Q_i \) were required to be positive semi-definite, ensuring that the model is convex. Gurobi 9.0+ supports general non-convex quadratic constraints and objective functions, including bilinear and quadratic equality constraints.

Non-convex models are typically harder to solve than convex models. Often, it is useful to reformulate the model into a convex problem if possible. In order to prevent the accidental creation of non-convex models, the parameter NonConvex must be set to 2 in order to enable optimization with non-convex constraints. For full details, see the Quadratic Constraints section of the Reference Manual.

There are also features for piecewise-linear objectives and constraints, which can represent or approximate many separable non-convex functions. For more details, see the sections on General Constraints and Function Constraints.

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