On the complexity of optimization over the standard simplex
We review complexity results for minimizing polynomials over the standard simplex and unit hypercube. In addition, we derive new results on the computational complexity of approximating the minimum of some classes of functions (including Lipschitz continuous functions) on the standard simplex. The main tools used in the analysis are Bernstein approximation and Lagrange interpolation on the simplex combined with an earlier result by de Klerk et al. [A PTAS for the minimization of polynomials of fixed degree over the simplex, Theoretical Computer Science 361 (2-3) (2006) 210-225].
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- de Klerk, E. & Elfadul, G.E.E. & den Hertog, D., 2006. "Optimization of Univariate Functions on Bounded Intervals by Interpolation and Semidefinite Programming," Discussion Paper 2006-26, Tilburg University, Center for Economic Research.
- Carlo Acerbi & Dirk Tasche, 2002. "Expected Shortfall: A Natural Coherent Alternative to Value at Risk," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 31(2), pages 379-388, 07.
- 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.
- A.M. Bagirov & A.M. Rubinov, 2000. "Global Minimization of Increasing Positively Homogeneous Functions over the Unit Simplex," Annals of Operations Research, Springer, vol. 98(1), pages 171-187, December.
- Bertsimas, Dimitris & Lauprete, Geoffrey J. & Samarov, Alexander, 2004. "Shortfall as a risk measure: properties, optimization and applications," Journal of Economic Dynamics and Control, Elsevier, vol. 28(7), pages 1353-1381, April.
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