By Franco Giannessi, Antonino Maugeri, and Panos M. Pardalos
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Extra resources for Equilibrium problems: nonsmooth optimization and variational inequality models
Nonlinear and Mixed-Integer Optimisation. Fundamentals and Applications, Oxford University Press, New York, 1995.  Vaidyanathan, R. : Global optimisation of MINLPs by interval analysis, In: I. E. ), Global Optimisation for Engineering Design, Kluwer Acad. , Dordrecht, 1996, pp. 175–195. Chapter 4 ONE-DIMENSIONAL GLOBAL OPTIMIZATION BASED ON STATISTICAL MODELS James M. Calvin Department of Computer and Information Science New Jersey Institute of Technology Newark, NJ 07102-1982, USA Antanas Žilinskas Institute of Mathematics and Informatics, VMU Akademijos str.
Byrne A number of extended arithmetic types are useful in process simulation: Intervals are an extended type for calculating the results of application of operators to ranges. Vectors and matrices are, in principle, extended types built from arrays of scalar types and rules to manipulate the data. The convex underestimators proposed by McCormick  provide the rules for adding, subtracting and multiplying convex underestimators. A suitable definition of the compound object which maintains the data required at each step makes this an extended type.
This method has the advantage that one set of gradient bounds obtained by AD can be used to construct as many underestimators as necessary, thereby reducing the total number of AD flowsheet evaluations. Each new underestimator requires one evaluation of the real arithmetic flowsheet. Applying the approach taken with the equation based problems of constructing two underestimating planes at x and requires one evaluation of the interval AD flowsheet which provides bounds on the variables in the flowsheet and bounds on the gradients with respect to the independent variables followed by one evaluation with real arithmetic for each of the points x and This gives two underestimating planes and two overestimating planes.
Equilibrium problems: nonsmooth optimization and variational inequality models by Franco Giannessi, Antonino Maugeri, and Panos M. Pardalos