Recently, a researcher from Princeton University built a mathematical foundation named minima distribution for global optimization [1](https://arxiv.org/pdf/1812.03457.pdf). In this work, a relationship between any continuous function f{\displaystyle f}
on a compact set Ω⊂Rn{\displaystyle \Omega \subset \mathbb {R} ^{n}}
and its global minima f∗{\displaystyle f^{*}}
has been strictly established. As a typical case, it follows that
limk→∞∫Ωf(x)m(k)(x)dx=f∗, where m(k)(x)=e−kf(x)∫Ωe−kf(x)dx;{\displaystyle \lim _{k\to \infty }\int _{\Omega }f(x)m^{(k)}(x)\mathrm {d} x=f^{*},~~{\textrm {where}}~~m^{(k)}(x)={\frac {e^{-kf(x)}}{\int _{\Omega }e^{-kf(x)}\mathrm {d} x}};}
meanwhile,
limk→∞m(k)(x)={1μ(X∗),x∈X∗,0,x∈Ω−X∗,{\displaystyle \lim _{k\to \infty }m^{(k)}(x)=\left\{{\begin{array}{cl}{\frac {1}{\mu (X^{*})}},&x\in X^{*},\\0,&x\in \Omega -X^{*},\end{array}}\right.}
where μ(X∗){\displaystyle \mu (X^{*})}
is the n{\displaystyle n}
-dimensional Lebesgue measure of the set of minimizers X∗∈Ω{\displaystyle X^{*}\in \Omega }
. And if f{\displaystyle f} is not a constant on Ω{\displaystyle \Omega }
, the monotonic relationship
∫Ωf(x)m(k)(x)dx>∫Ωf(x)m(k+Δk)(x)dx>f∗{\displaystyle \int _{\Omega }f(x)m^{(k)}(x)\mathrm {d} x>\int _{\Omega }f(x)m^{(k+\Delta k)}(x)\mathrm {d} x>f^{*}}
holds for all k∈R{\displaystyle k\in \mathbb {R} }
and Δk>0{\displaystyle \Delta k>0}
, which implies a series of monotonous containment relationships, and one of them is, for examples,
Ω⊃Df(k)⊃Df(k+Δk)⊃X∗, where Df(k)={x∈Ω:f(x)⩽∫Ωf(t)m(k)(t)dt}.{\displaystyle \Omega \supset D_{f}^{(k)}\supset D_{f}^{(k+\Delta k)}\supset X^{*},~~{\textrm {where}}~~D_{f}^{(k)}=\left\{x\in \Omega :f(x)\leqslant \int _{\Omega }f(t)m^{(k)}(t)\mathrm {d} t\right\}.}
And we define a minima distribution to be a weak limit mf,Ω{\displaystyle m_{f,\Omega }}
such that the identity
∫Ωmf,Ω(x)φ(x)dx=limk→∞∫Ωm(k)(x)φ(x)dx{\displaystyle \int _{\Omega }m_{f,\Omega }(x)\varphi (x)\mathrm {d} x=\lim _{k\to \infty }\int _{\Omega }m^{(k)}(x)\varphi (x)\mathrm {d} x}
holds for every smooth function φ{\displaystyle \varphi }
with compact support in Ω{\displaystyle \Omega }. Here are two immediate properties of mf,Ω{\displaystyle m_{f,\Omega }}:
As a comparison, the well-known relationship between any differentiable convex function and its minima is strictly established by the gradient. If f{\displaystyle f} is differentiable on a convex set D{\displaystyle D}
, then f{\displaystyle f} is convex if and only if
f(y)⩾f(x)+∇f(x)(y−x), ∀x,y∈D;{\displaystyle f(y)\geqslant f(x)+\nabla f(x)(y-x),~~\forall x,y\in D;}
thus, ∇f(x∗)=0{\displaystyle \nabla f(x^{*})=0}
implies that f(y)⩾f(x∗){\displaystyle f(y)\geqslant f(x^{*})}
holds for all y∈D{\displaystyle y\in D}
, i.e., x∗{\displaystyle x^{*}}
is a global minimizer of f{\displaystyle f} on D{\displaystyle D}.
http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1140/c/dmdico.1140.pdf
