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Results tagged with nonlinear-optimization
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user 18739
Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.
14
votes
Accepted
Can a cubic polynomial in two real variables have three saddle points?
The cubic $x^3 - xy^2 - 2x^2 + x$ has critical points in $(1,0)$, $(0,-1)$, $(0,1)$ and $(1/3, 0)$. The determinant of the Hessian matrix is $-4(3x^2 + y^2 - 2x)$. It assumes the values $-4$, $-4$, $- …
- 22.5k
1
vote
How many strict local minima can a quintic polynomial in two real variables have?
The paper Critical points of real polynomials, subdivisions of Newton polyhedra and topology of real algebraic hypersurfaces by Shustin
claims to decide which triples $(m,n,s)$ are possible in all deg …
- 22.5k
35
votes
Accepted
Can a real quartic polynomial in two variables have more than 4 isolated local minima?
Recent addition: Inspired by DimaPasechnik's and Matt F.'s comments about sum of squares decompositions, I tried the following very natural idea: Try to find $f$ of the form $f(x,y)=A(x,y)^2+B(x,y)^2$ …
- 22.5k