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Nonlinear programming and Grossone: Quadratic Programing and the role of Constraint Qualifications

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  • De Leone, Renato

Abstract

A novel and interesting approach to infinite and infinitesimal numbers was recently proposed in a series of papers and a book by Sergeyev. This novel numeral system is based on the use of a new infinite unit of measure (the number grossone, indicated by the numeral ①), the number of elements of the set, IN, of natural numbers. Based on the use of ①, De Cosmis and De Leone (2012) have then proposed a new exact differentiable penalty function for constrained optimization problems. In this paper these results are specialized to the important case of quadratic problems with linear constraints. Moreover, the crucial role of Constraint Qualification conditions (well know in constraint minimization literature) is also discussed with reference to the new proposed penalty function.

Suggested Citation

  • De Leone, Renato, 2018. "Nonlinear programming and Grossone: Quadratic Programing and the role of Constraint Qualifications," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 290-297.
    • Handle: RePEc:eee:apmaco:v:318:y:2018:i:c:p:290-297
    DOI: 10.1016/j.amc.2017.03.029
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    References listed on IDEAS

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    1. Lolli, Gabriele, 2015. "Metamathematical investigations on the theory of Grossone," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 3-14.
    2. Sergeyev, Yaroslav D., 2009. "Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3042-3046.
    3. Sergeyev, Yaroslav D., 2007. "Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 50-75.
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    Cited by:

    1. Renato Leone & Giovanni Fasano & Massimo Roma & Yaroslav D. Sergeyev, 2020. "Iterative Grossone-Based Computation of Negative Curvature Directions in Large-Scale Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 554-589, August.
    2. Renato De Leone & Giovanni Fasano & Yaroslav D. Sergeyev, 2018. "Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming," Computational Optimization and Applications, Springer, vol. 71(1), pages 73-93, September.
    3. Falcone, Alberto & Garro, Alfredo & Mukhametzhanov, Marat S. & Sergeyev, Yaroslav D., 2021. "A Simulink-based software solution using the Infinity Computer methodology for higher order differentiation," Applied Mathematics and Computation, Elsevier, vol. 409(C).

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