With the release of Gurobi 9.0’s addition of a new bilinear solver, the Gurobi Optimizer now supports non-convex quadratic optimization. This groundbreaking new capability allows users to solve problems with non-convex quadratic constraints and objectives – enabling them to find globally optimal solutions to classic bilinear pooling and blending problems and continuous manufacturing problems.

Business Applications

Companies utilizing mathematical optimization are able to apply non-convex quadratic optimization to a number of industries and problems including:

• Pooling problem (blending problem is LP, pooling introduces intermediate pools, which lead to bilinear constraints)

• Petrochemical industry (oil refinery: constraints on ratio of components in tanks)

• Wastewater treatment

• Emissions regulation

• Agricultural / food industry (blending based on pre-mix products)

• Mining

• Energy

• Production planning (constraints on ratio between internal and external workforce)

• Logistics (restrictions from free trade agreements)

• Water distribution (Darcy-Weisbach equation for volumetric flow)

• Engineering design

• Finance

General MINLP:

• For general MINLP, another important building block is the support to get automatic

  piece-wise linearization of certain standard non-linear univariate functions like y =

  exp(x), y = sin(x), or y = log(x).

• Gurobi 9.0 allows to use certain standard non-linear univariate functions like y =

  exp(x) or y = sin(x) in a model. These are automatically approximated using piece-wise

  linear functions.

• Many classes of general MINLPs can be solved by using these non-linear univariate

  functions and approximating multi-variate functions as polynomials. But note that with

  higher degrees of polynomials, the numerics of the problem become more challenging.

Standard Pooling Problem:

Pooling problems are common in the petrochemical refining, wastewater treatment, and mining industries. This problem can be regarded as a generalization of the minimum-cost flow problem and the blending problem. We construct a non-convex mixed-integer quadratically-constrained programming (MIQCP) model of this problem, implement this model in the Gurobi Python API, and compute an optimal solution.