Optimization in Banach space: Find functions that minimize the supremum of a convex operator.
Let $D subset mathbb{R}^n$ be compact. Denote by $C(D, mathbb{R}^n)$ the space of continuous functions from $D$ to $mathbb{R}^n$.
Let $K$ be a real, symmetric, positive-definite $n times n$ matrix and let $k in C(D, mathbb{R}^n)$.
We consider the following convex "functional":
$$mathcal{F}:C(D, mathbb{R}^n) rightarrow C(D
)$$
$$mathcal{F}[b](x) := b(x)^T K b(x) - 2 k(x)^T b(x)$$
where $^T$ denotes transpose.
We are interested in the quantity
$$M_b := sup_{x in D} mathcal{F}[b](x)$$
We are looking for the function $b in C(D, mathbb{R}^n)$ that minimizes this quantity, under the constraint that $b(x)$ lies on the probability simplex $Delta$ for all $x$.
$$inf_{substack{b in C(D, mathbb{R}^n)} \ b in Delta} ~ ~ Big(sup_{x in D} mathcal{F}[b](x)Big)$$
Formally, the constraint $b in Delta$ is:
$$forall x in D: sum_{i = 1}^n b_i(x) = 1 text{ and } b_i(x) geq 0 text{ for i=1, ...,n} $$
Where, for $b in C(D, mathbb{R}^n)$, we write: $b(x) = begin{pmatrix}
d_1(x)\
vdots\
d_n(x)
end{pmatrix}$
I am looking for any reference that could help me solve this problem.
More precisely, references to any article / book related to the subject would already help. I don't even know that the name of this field of mathematics is.
Remarks:
- I am interested in numerical or analytical solutions to the above problem.
- An approximate solution would also be OK.
- I know the solution to the unconstrained version of the problem.
- The current setting is to look for solutions in the class of continuous functions, but that setting can be relaxed: in the end, piecewise continous is enough.
optimization banach-spaces calculus-of-variations variational-analysis
asked Nov 29 '18 at 10:30
Cédric TravellettiCédric Travelletti
1095
add a comment |
Let $D subset mathbb{R}^n$ be compact. Denote by $C(D, mathbb{R}^n)$ the space of continuous functions from $D$ to $mathbb{R}^n$.
Let $K$ be a real, symmetric, positive-definite $n times n$ matrix and let $k in C(D, mathbb{R}^n)$.
We consider the following convex "functional":
$$mathcal{F}:C(D, mathbb{R}^n) rightarrow C(D
)$$
$$mathcal{F}[b](x) := b(x)^T K b(x) - 2 k(x)^T b(x)$$
where $^T$ denotes transpose.
We are interested in the quantity
$$M_b := sup_{x in D} mathcal{F}[b](x)$$
We are looking for the function $b in C(D, mathbb{R}^n)$ that minimizes this quantity, under the constraint that $b(x)$ lies on the probability simplex $Delta$ for all $x$.
$$inf_{substack{b in C(D, mathbb{R}^n)} \ b in Delta} ~ ~ Big(sup_{x in D} mathcal{F}[b](x)Big)$$
Formally, the constraint $b in Delta$ is:
$$forall x in D: sum_{i = 1}^n b_i(x) = 1 text{ and } b_i(x) geq 0 text{ for i=1, ...,n} $$
Where, for $b in C(D, mathbb{R}^n)$, we write: $b(x) = begin{pmatrix}
d_1(x)\
vdots\
d_n(x)
end{pmatrix}$
I am looking for any reference that could help me solve this problem.
More precisely, references to any article / book related to the subject would already help. I don't even know that the name of this field of mathematics is.
Remarks:
- I am interested in numerical or analytical solutions to the above problem.
- An approximate solution would also be OK.
- I know the solution to the unconstrained version of the problem.
- The current setting is to look for solutions in the class of continuous functions, but that setting can be relaxed: in the end, piecewise continous is enough.
optimization banach-spaces calculus-of-variations variational-analysis
asked Nov 29 '18 at 10:30
Cédric TravellettiCédric Travelletti
1095
add a comment |
Let $D subset mathbb{R}^n$ be compact. Denote by $C(D, mathbb{R}^n)$ the space of continuous functions from $D$ to $mathbb{R}^n$.
Let $K$ be a real, symmetric, positive-definite $n times n$ matrix and let $k in C(D, mathbb{R}^n)$.
We consider the following convex "functional":
$$mathcal{F}:C(D, mathbb{R}^n) rightarrow C(D
)$$
$$mathcal{F}[b](x) := b(x)^T K b(x) - 2 k(x)^T b(x)$$
where $^T$ denotes transpose.
We are interested in the quantity
$$M_b := sup_{x in D} mathcal{F}[b](x)$$
We are looking for the function $b in C(D, mathbb{R}^n)$ that minimizes this quantity, under the constraint that $b(x)$ lies on the probability simplex $Delta$ for all $x$.
$$inf_{substack{b in C(D, mathbb{R}^n)} \ b in Delta} ~ ~ Big(sup_{x in D} mathcal{F}[b](x)Big)$$
Formally, the constraint $b in Delta$ is:
$$forall x in D: sum_{i = 1}^n b_i(x) = 1 text{ and } b_i(x) geq 0 text{ for i=1, ...,n} $$
Where, for $b in C(D, mathbb{R}^n)$, we write: $b(x) = begin{pmatrix}
d_1(x)\
vdots\
d_n(x)
end{pmatrix}$
I am looking for any reference that could help me solve this problem.
More precisely, references to any article / book related to the subject would already help. I don't even know that the name of this field of mathematics is.
Remarks:
- I am interested in numerical or analytical solutions to the above problem.
- An approximate solution would also be OK.
- I know the solution to the unconstrained version of the problem.
- The current setting is to look for solutions in the class of continuous functions, but that setting can be relaxed: in the end, piecewise continous is enough.
optimization banach-spaces calculus-of-variations variational-analysis
asked Nov 29 '18 at 10:30
Cédric TravellettiCédric Travelletti
1095
Let $D subset mathbb{R}^n$ be compact. Denote by $C(D, mathbb{R}^n)$ the space of continuous functions from $D$ to $mathbb{R}^n$.
Let $K$ be a real, symmetric, positive-definite $n times n$ matrix and let $k in C(D, mathbb{R}^n)$.
We consider the following convex "functional":
$$mathcal{F}:C(D, mathbb{R}^n) rightarrow C(D
)$$
$$mathcal{F}[b](x) := b(x)^T K b(x) - 2 k(x)^T b(x)$$
where $^T$ denotes transpose.
We are interested in the quantity
$$M_b := sup_{x in D} mathcal{F}[b](x)$$
We are looking for the function $b in C(D, mathbb{R}^n)$ that minimizes this quantity, under the constraint that $b(x)$ lies on the probability simplex $Delta$ for all $x$.
$$inf_{substack{b in C(D, mathbb{R}^n)} \ b in Delta} ~ ~ Big(sup_{x in D} mathcal{F}[b](x)Big)$$
Formally, the constraint $b in Delta$ is:
$$forall x in D: sum_{i = 1}^n b_i(x) = 1 text{ and } b_i(x) geq 0 text{ for i=1, ...,n} $$
Where, for $b in C(D, mathbb{R}^n)$, we write: $b(x) = begin{pmatrix}
d_1(x)\
vdots\
d_n(x)
end{pmatrix}$
I am looking for any reference that could help me solve this problem.
More precisely, references to any article / book related to the subject would already help. I don't even know that the name of this field of mathematics is.
Remarks:
- I am interested in numerical or analytical solutions to the above problem.
- An approximate solution would also be OK.
- I know the solution to the unconstrained version of the problem.
- The current setting is to look for solutions in the class of continuous functions, but that setting can be relaxed: in the end, piecewise continous is enough.
optimization banach-spaces calculus-of-variations variational-analysis
optimization banach-spaces calculus-of-variations variational-analysis
asked Nov 29 '18 at 10:30
Cédric TravellettiCédric Travelletti
1095
asked Nov 29 '18 at 10:30
Cédric TravellettiCédric Travelletti
1095
asked Nov 29 '18 at 10:30
Cédric TravellettiCédric Travelletti
1095
asked Nov 29 '18 at 10:30
Cédric TravellettiCédric Travelletti
1095
asked Nov 29 '18 at 10:30
Cédric TravellettiCédric Travelletti
1095
1095
add a comment |
add a comment |
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