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 \( Q = 0 \) and \( I = K = \emptyset \), then the model 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. If possible, consider reformulating the model into a convex problem. In order to prevent the accidental creation of non-convex models, theNonConvexparameter must be set to 2 to enable optimization for problems with non-convex constraints or objective. For full details, see theQuadratic Constraintssection of the reference manual.
Gurobi has built-in functionality for creating piecewise-linear objectives and constraints, which can represent or approximate many separable non-convex functions. For more details, see the创eral ConstraintsandFunction Constraintssections of the documentation.
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Do you mean with “solving” that optimal solutions are guaranteed found if the model adheres to the provided canonical form?
Gurobi searches for globally optimal solutions to models of the above form, subject to tolerances like therelative optimality gap. However, Gurobi, just like any other solver, is not necessarily guaranteed to find globally optimal solution to arbitrary problems of this form. For example, some problems, like theopen MIPLIB 2017 problems, have yet to be solved by any software.
You can alwaystry out Gurobifor free to see how well it performs on your model.Academic userscan access the full version of Gurobi at no cost, and commercial users can request afree evaluation license.
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