## Section: Research Program

### Modeling problems with complex structures

A classical optimization problem can be formulated as follows:

$\begin{array}{cc}min\hfill & f\left(x\right)\hfill \\ s.\phantom{\rule{0.222222em}{0ex}}t.\phantom{\rule{0.222222em}{0ex}}\hfill & x\in X,\hfill \end{array}$ | (1) |

In this problem, $X$ is the set of feasible solutions. Typically, in mathematical programming, $X$ is defined by a set of constraints. $x$ may be also limited to non-negative integer values.

INOCS team plan to address optimization problem where two types of decision are addressed jointly and are interrelated. More precisely, let us assume that variables $x$ and $y$ are associated with these decisions. A generic model for CS problems is the following:

$\begin{array}{cc}min\hfill & g(x,y)\hfill \\ s.\phantom{\rule{0.222222em}{0ex}}t.\phantom{\rule{0.222222em}{0ex}}\hfill & x\in X,\hfill \\ (x,y)\hfill & \in XY,\hfill \\ y\hfill & \in Y\left(x\right).\hfill \end{array}$ | (2) |

In this model, $X$ is the set of feasible values for $x$. $XY$ is the set of feasible values for $x$ and $y$ jointly. This set is typically modeled through linking constraints. Last, $Y\left(x\right)$ is the set of feasible values for $y$ for a given $x$. In INOCS, we do not assume that $Y\left(x\right)$ has any properties.

The INOCS team plans to model optimization CS problems according to three types of optimization paradigms: large scale complex structures optimization, bilevel optimization and robust/stochastic optimization. These paradigms instantiate specific variants of the generic model.

Large scale complex structures optimization problems can be formulated through the simplest variant of the generic model given above. In this case, it is assumed that $Y\left(x\right)$ does not depend on $x$. In such models, $X$ and $Y$ are associated with constraints on $x$ and on $y$, $XY$ are the linking constraints. $x$ and $y$ can take continuous or integer values. Note that all the problem data are deterministically known.

Bilevel programs allow the modeling of situations in which a decision-maker, hereafter the leader, optimizes his objective by taking explicitly into account the response of another decision maker or set of decision makers (the follower) to his/her decisions. Bilevel programs are closely related to Stackelberg (leader-follower) games as well as to the principal-agent paradigm in economics. In other words, bilevel programs can be considered as demand-offer equilibrium models where the demand is the result of another mathematical problem. Bilevel problems can be formulated through the generic CS model when $Y\left(x\right)$ corresponds to the optimal solutions of a mathematical program defined for a given $x$, i.e. $Y\left(x\right)=arg\; min\left\{h(x,y)|y\in {Y}_{2},(x,y)\in X{Y}_{2}\right\}$ where ${Y}_{2}$ is defined by a set of constraints on $y$, and $X{Y}_{2}$ is associated with the linking constraints.

In robust/stochastic optimization, it is assumed that the data related to a problem are subject to uncertainty. In stochastic optimization, probability distributions governing the data are known, and the objective function involves mathematical expectation(s). In robust optimization, uncertain data take value within specified sets, and the function to optimize is formulated in terms of a min-max objective typically (the solution must be optimal for the worst-case scenario). . A standard modeling of uncertainty on data is obtained by defining a set of possible scenarios that can be described explicitly or implicitly. In stochastic optimization, in addition, a probability of occurrence is associated with each scenario and the expected objective value is optimized.