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.


  • 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.

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