Pareto efficiency for robust linear programs was introduced by
Iancu and Trichakis in [9]. We generalize their approach and theoretical results
to robust optimization problems in Euclidean spaces with linear uncertainty.
Additionally, we demonstrate the value of this approach in an exemplary
manner in the area of robust semidefinite programming (SDP). In particular,
we prove that computing a Pareto robustly optimal solution for a robust SDP
is tractable and illustrate the benefit of such solutions at the example of the
maximal eigenvalue problem. Furthermore, we modify the famous algorithm of
Goemans and Williamson [8] in order to compute cuts for the robust max cut
problem that yield an improved approximation guarantee in non-worst-case
scenarios.
Joint model of probabilistic/robust (probust) constraints applied to gas network optimization
(2017)
Optimization tasks under uncertain conditions abound in many
real-life applications. Whereas solution approaches for probabilistic constraints
are often developed in case the uncertainties can be assumed to follow a
certain probability distribution, robust approaches are usually used in case
solutions are sought that are feasible for all realizations of uncertainties within
some pre-defined uncertainty set. As many applications contain different types
of uncertainties that require robust as well as probabilistic treatments, we deal with a class of joint probabilistic/robust constraints as its appears in
optimization problems under uncertainty. Focusing on complex uncertain gas
network optimization problems, we show the relevance of this class of problems
for the task of maximizing free booked capacities in an algebraic model for a
stationary gas network. We furthermore present approaches for their solution.
Finally, we study the problem of controlling a transient system that is governed
by the wave equation. The task consists in determining controls such that a
certain robustness measure remains below some given upper bound, with high
probability.