# 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.