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Beyond LP and MILP Problems ⊖

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Business Optimization Using Mathematical Programming

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 307))

Abstract

This chapter mentions several optimization problems which go beyond linear and mixed integer linear optimization. The focus is rather on motivation. Therefore, it is not intended to cover these topics in complete depth, but the reader should at least be aware that modeling real-world problems is not restricted to linear models. In fractional programming we show how to transform the problem to linear programming, and successive linear programming as a special solution technique of nonlinear optimization. Next, we briefly discuss stochastic optimization. For quadratic programming, which is again a special case of nonlinear optimization, we provide an equivalent formulation based on special ordered sets. Nonlinear optimization is covered in more detail in the next chapter followed by separate chapters on deterministic global optimization in practice and polylithic modeling and solution approaches.

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Notes

  1. 1.

    Of course, they are not really coefficients but rather functions multiplied onto the variable.

  2. 2.

    A solution is called a local optimum if there exists no better feasible solution in its “vicinity.”

  3. 3.

    A solution is a global optimum if there exists no better feasible solution.

  4. 4.

    In refinery industry the expression quality or quality constraints is commonly used. The quality itself can be measured in dimensionless quantities such as concentration (tons/ton) or in appropriate physical units (take viscosity as an example).

  5. 5.

    In principle the methods work with an arbitrary number of variables — but if problems get larger, convergence problems and the significance of good initial guesses increase.

  6. 6.

    Historically, the term quadratic programming (QP) refers to a quadratic objective function and linear constraints. Nowadays, sometimes, the more general problem with a quadratic objective function and quadratic constraints is also referred to as quadratic programming. Sometimes it is named QPQC. If the QP or QPQC involves integer variables, one also sees the naming MIQCQP. Thus, the advice is: Always check carefully for the definition when authors refer to quadratic programming.

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  1. Department of Astronomy, University of Florida, Gainesville, FL, USA

    Josef Kallrath

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Kallrath, J. (2021). Beyond LP and MILP Problems ⊖. In: Business Optimization Using Mathematical Programming. International Series in Operations Research & Management Science, vol 307. Springer, Cham. https://doi.org/10.1007/978-3-030-73237-0_11

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