Introduction

The primary purpose of opti is to define tasks or problems in a number of closely related fields, including experimental design, multiobjective optimization and decision making and Bayesian optimization.

Opti specifications are json serializable for use in RESTful APIs and are to a large extent agnostic to the specific methods and frameworks in which the problems are solved.

Experimental design

In the context of experimental design opti allows to define a design space

\[ \mathbb{X} = x_1 \otimes x_2 \ldots \otimes x_D \]

where the design parameters may take values depending on their type and domain, e.g.

• continuous: \(x_1 \in [0, 1]\)
• discrete: \(x_2 \in \{1, 2, 5, 7.5\}\)
• categorical: \(x_3 \in \{A, B, C\}\)

and a set of equations define additional experimental constraints, e.g.

• linear equality: \(\sum x_i = 1\)
• linear inequality: \(2 x_1 \leq x_2\)
• non-linear inequality: \(\sum x_i^2 \leq 1\)
• n-choose-k: only \(k\) out of \(n\) parameters can take non-zero values.

Multiobjective optimization

In the context of multiobjective optimization opti allows to define a vector-valued optimization problem

\[ \min_{x \in \mathbb{X}} s(y(x)) \]

where

• \(x \in \mathbb{X}\) 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.

Since the objectives are in general conflicting, there is no point \(x\) that simulataneously 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 posible outcomes for \(y\). An acquisition function \(a\) then formulates the desired trade-off between exploration and exploitation

\[ \min_{x \in \mathbb{X}} a(s(p_y(x))) \]

and the minimizer \(x_\mathrm{opt}\) of this acquisition function. determines the next experiment \(y(x)\) to run. When are multiple competing objectives, the task is again to find a suitable approximation of the Pareto front.