Titlebook: NumericalInfinities and Infinitesimals in Optimization; Yaroslav D. Sergeyev,Renato De Leone Book 2022 The Editor(s) (if applicable) and T

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The Role of , in Nonlinear Programming and Exact Penalty Methodsned optimization problem can be transformed in an “equivalent” unconstrained problem. In this chapter we show how . can be utilized in constructing exact differentiable penalty functions for the case of only equality constraints, the general case of equality and inequality constraints, and quadratic

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Krylov-Subspace Methods for Quadratic Hypersurfaces: A Grossone–based Perspectivelinear systems. We preliminarily explore the relationship between the Conjugate Gradient (CG) method and the Lanczos process, along with their specific role of yielding tridiagonal matrices which retain large information on the original linear system matrix. Then, we show that on one hand there is n

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Multi-objective Lexicographic Mixed-Integer Linear Programming: An Infinity Computer Approach one, by using the Grossone Methodology. Then we provide a simplex-like algorithm, called GrossSimplex, able to solve the original LMOLP problem using a single run of the algorithm (its theoretical correctness is also provided). In the second part, we tackle a Mixed-Integer Lexicographic Multi-Objec

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The Grossone-Based Diagonal Bundle Methodlass of unconstrained nonsmooth optimization methods based on a variable metric approach, where the use of the infinity computing techniques allows one to numerically deal with quantities which can take arbitrarily small or large values, as a consequence of nonsmoothness. In particular, choosing a d

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Exact Numerical Differentiation on the Infinity Computer and Applications in Global OptimizationA novel way to efficiently compute . derivatives (the word “exact” means here with respect to the accuracy of the implementation of .(.)) is presented in this Chapter. It uses a new kind of a supercomputer—the Infinity Computer—able to work numerically with different finite, infinite, and infinitesi