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Goldstein-Price Function

The Goldstein-Price function is the bivariate polynomial function

f(x,y)=[1+(x+y+1)^2(19-14x+3x^2-14y+6xy+3y^2)]×[30+(2x-3y)^2(18-32x+12x^2+48y-36xy+27y^2)].
(1)

It is commonly evaluated on the square [-2,2]^2, where it has several local minima and a global minimum f(0,-1)=3 (Surjanovic and Bingham 2013).

The Goldstein-Price function is used as a test function for global optimization algorithms. Maaz and Strzeboński (2025) used a modification involving radicals, f(x,y)-sqrt(x-1)-sqrt(y-1) on x>=1, y>=1 to compare two reformulations as polynomial optimization problems.

See also

Global Minimum, Global Optimization, Polynomial Optimization, Rosenbrock Function

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References

Maaz, M. and Strzeboński, A. W. "A New Method for Reducing Algebraic Programs to Polynomial Programs." 12 Feb 2025. https://arxiv.org/abs/2502.08210.Surjanovic, S. and Bingham, D. "Goldstein-Price Function." Virtual Library of Simulation Experiments: Test Functions and Datasets. 2013. https://www.sfu.ca/~ssurjano/goldpr.html.

Cite this as:

Weisstein, Eric W. "Goldstein-Price Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Goldstein-PriceFunction.html

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