# Rounding on the standard simplex: regular grids for global optimization

## Author

Listed:
• Immanuel Bomze

()

• Stefan Gollowitzer

()

• E. Yıldırım

()

## Abstract

Given a point on the standard simplex, we calculate a proximal point on the regular grid which is closest with respect to any norm in a large class, including all $$\ell ^p$$ ℓ p -norms for $$p\ge 1$$ p ≥ 1 . We show that the minimal $$\ell ^p$$ ℓ p -distance to the regular grid on the standard simplex can exceed one, even for very fine mesh sizes in high dimensions. Furthermore, for $$p=1$$ p = 1 , the maximum minimal distance approaches the $$\ell ^1$$ ℓ 1 -diameter of the standard simplex. We also put our results into perspective with respect to the literature on approximating global optimization problems over the standard simplex by means of the regular grid. Copyright Springer Science+Business Media New York 2014

## Suggested Citation

• Immanuel Bomze & Stefan Gollowitzer & E. Yıldırım, 2014. "Rounding on the standard simplex: regular grids for global optimization," Journal of Global Optimization, Springer, vol. 59(2), pages 243-258, July.
• Handle: RePEc:spr:jglopt:v:59:y:2014:i:2:p:243-258
DOI: 10.1007/s10898-013-0126-2
as

File URL: http://hdl.handle.net/10.1007/s10898-013-0126-2

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## References listed on IDEAS

as
1. de Klerk, E., 2006. "The Complexity of Optimizing over a Simplex, Hypercube or Sphere : A Short Survey," Discussion Paper 2006-85, Tilburg University, Center for Economic Research.
2. de Klerk, E. & den Hertog, D. & Elabwabi, G., 2008. "On the complexity of optimization over the standard simplex," European Journal of Operational Research, Elsevier, vol. 191(3), pages 773-785, December.
3. Etienne Klerk, 2008. "The complexity of optimizing over a simplex, hypercube or sphere: a short survey," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 16(2), pages 111-125, June.
4. de Klerk, E. & Laurent, M. & Parrilo, P., 2006. "A PTAS for the minimization of polynomials of fixed degree over the simplex," Other publications TiSEM 603897c9-179e-43e4-9e83-6, Tilburg University, School of Economics and Management.
5. L. Casado & I. García & B. Tóth & E. Hendrix, 2011. "On determining the cover of a simplex by spheres centered at its vertices," Journal of Global Optimization, Springer, vol. 50(4), pages 645-655, August.
6. de Klerk, E., 2008. "The complexity of optimizing over a simplex, hypercube or sphere : A short survey," Other publications TiSEM 485b6860-cf1d-4cad-97b8-2, Tilburg University, School of Economics and Management.
Full references (including those not matched with items on IDEAS)

### Keywords

Rounding; Regular grid; Approximation; Maximin distance; Proximal point;

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