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Breakthrough New Capability

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.

 

Meet the Experts

Dr. Edward Rothberg

Chief Executive Officer and Co-founder

Dr. Rothberg has served in senior leadership positions in optimization software companies for more than twenty years. Prior to his role as Gurobi CEO, Dr. Rothberg held the Gurobi COO position since co-founding Gurobi in 2008, and prior to that he led the ILOG CPLEX team. Dr. Edward Rothberg has a BS in Mathematical and Computational Science from Stanford University, and an MS and PhD in Computer Science, also from Stanford University. Dr. Rothberg has published numerous papers in the fields of linear algebra, parallel computing, and mathematical programming. He is one of the world's leading experts in sparse Cholesky factorization and computational linear, integer, and quadratic programming. He is particularly well known for his work in parallel sparse matrix factorization, and in heuristics for mixed integer programming.

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