Docs - maximize acquisition functions (#540)

* we maximize acquisition functions

* small error

* argmax

Author: bertiqwerty

docs/index.md

@@ -37,31 +37,31 @@ and a set of equations define additional experimental constraints, e.g.
 In the context of multi-objective optimization BoFire allows to define a vector-valued optimization problem
 
 $$
-\min_{x \in \mathbb{X}} s(y(x))
+\argmax_{x \in \mathbb{X}} s(y(x))
 $$
 
 where
 
-* $x \in \mathbb{X}$ is again the experimental design space
+* $\mathbb{X}$ is again the experimental design space
 * $y = \{y_1, \ldots y_M\}$ are known functions describing your experimental outputs and
-* $s = \{s_1, \ldots s_M\}$ are the objectives to be minimized, e.g. $s_1$ is the identity function if $y_1$ is to be minimized.
+* $s = \{s_1, \ldots s_M\}$ are the objectives to be maximized. For instance, $s_1$ is the identity function if $y_1$ is to be maximized.
 
-Since the objectives are in general conflicting, there is no point $x$ that simultaneously optimizes all objectives.
+Since the objectives are usually conflicting, there is no point $x$ that simultaneously optimizes all objectives.
 Instead the goal is to find the Pareto front of all optimal compromises.
 
 A decision maker can then explore these compromises to get a deep understanding of the problem and make the best informed decision.
 
 ## Bayesian optimization
 
 In the context of Bayesian optimization we want to simultaneously learn the unknown function $y(x)$ (exploration), while focusing the experimental effort on promising regions (exploitation).
-This is done by using the experimental data to fit a probabilistic model $p(y|x, {data})$ that estimates the distribution of possible outcomes for $y$.
+This is done by using the experimental data to fit a probabilistic model $p(y|x, \mathrm{data})$ that estimates the distribution of possible outcomes for $y$.
 An acquisition function $a$ then formulates the desired trade-off between exploration and exploitation
 
 $$
-\min_{x \in \mathbb{X}} a(s(p_y(x)))
+\argmax_{x \in \mathbb{X}} a(s(p_y(x)))
 $$
 
-and the minimizer $x_\mathrm{opt}$ of this acquisition function determines the next experiment $y(x)$ to run.
+and the maximizer $x_\mathrm{opt}$ of this acquisition function determines the next experiment $y(x)$ to run.
 
 When there are multiple competing objectives, the task is again to find a suitable approximation of the Pareto front.