## Contributed Session Mon.2.H 3503

#### Monday, 13:15 - 14:45 h, Room: H 3503

**Cluster 20: Robust optimization** [...]

### Robust nonlinear optimization

**Chair: Daniel Fleischman**

**Monday, 13:15 - 13:40 h, Room: H 3503, Talk 1**

**Martin Mevissen**

Distributionally robust optimization for polynomial optimization problems

**Coauthors: Emanuele Ragnoli, Jia Yuan Yu**

**Abstract:**

In many real-world optimization problems, one faces the dual challenge of hard nonlinear functions in both objective and constraints and uncertainty in some of the problem parameters. Often, samples for each uncertain parameter are given, whereas its actual distribution is unknown. We propose a novel approach for constructing distributionally robust counterparts of a broad class of polynomial optimization problems. The approach aims to use the given samples, not only to approximate the support of the unknown distribution or the first and second order moments, but also its density. We show that polynomial optimization problems with distributional uncertainty sets defined via density estimates are particular instances of the generalized problem of moments with polynomial data and employ Lasserre's hierarchy of SDP relaxations to approximate the distributionally robust solutions. As a result of using distributional uncertainty sets, we obtain a less conservative solution than classical robust optimization. We demonstrate the potential of our approach for a range of polynomial optimization problems including linear regression and water network problems.

**Monday, 13:45 - 14:10 h, Room: H 3503, Talk 2**

**Hans Pirnay**

An algorithm for robust optimization of nonlinear dynamic systems

**Coauthor: Wolfgang Marquardt**

**Abstract:**

Nonlinear model predictive control (NMPC) is an attractive control methodology for chemical processes.

In NMPC, the control is computed by solving a dynamic optimization problem constrained by a differential algebraic model of the underlying process.

These systems are often subject to unknown disturbances, which, if not taken into account, can lead to deterioration of the control quality, and even instability of the control loop.

To overcome this problem, the dynamic optimization problem has to be formulated in a robust way such that feasibility is guaranteed under all circumstances.

This leads to a bi-level formulation with a dynamic lower level problem.

Unfortunately, even simple dynamical models used for NMPC lead to non-convex feasible sets, which tremendously complicates the solution.

In this talk, we present an algorithm for bi-level dynamic optimization in the context of NMPC.

To deal with the non-convexity, recent advances in dynamic global optimization are employed.

In addition, we take advantage of the parametric nature of the lower level problem to speed up the computation and make the algorithm viable for real-time applications.

**Monday, 14:15 - 14:40 h, Room: H 3503, Talk 3**

**Daniel Fleischman**

On the trade-off between robustness and value

**Coauthor: Mike Todd**

**Abstract:**

Linear programming problems may be formed based on data collected with

measurement error, which may make the optimal solution to the problem

with the nominal parameters infeasible for the "real parameters''.

One way to approach this difficulty is by using robust optimization,

where we form an uncertainty set **E** around the nominal parameter

vector, and a solution has to be feasible for any vector in **E**.

One question that arises is how large the uncertainty set **E**

should be. The larger it is, the safer we are, but at the same time,

our solution becomes worse. We study such questions when the uncertainty

set for each constraint is a uniform scale factor times

a fixed ellipsoid, and propose a simple, easy to compute

approximate solution depending on the scale factor.