Minisymposium “Discrete Optimization”

 

Prof. Dr. Rüdiger Schultz (University Duisburg-Essen)
“Reminiscences of stochastic integer programming”

This talk looks back, not for 50, but 30 to 35 years, of research on bringing together integers with stochastic optimization models. The variety of mathematical recipes met along the  way is a  striking example  for  fruitful inner-mathematical interaction. Often, but not always, this is application-driven, and, considerably, goes beyond stochastics and optimization.

 

Prof. Dr. Jens Brunner (University Augsburg)
“Machine learning supported stabilized column generation with an application for the cutting stock problem”

This work presents a new method to determine parameters for a stabilized column generation algorithm. In most studies, the parameters for stabilized column generation are determined by numerical tests, i.e., the same problem is solved several times with different settings. We develop a neural network to estimate optimal dual values for a common stabilization technique for column generation in the case of cutting stock problems. In our extensive computational study, we show that the neural network is able to predict the parameters in an efficient manner. Additionally, the tests reveal that the estimated parameters dominate traditional update mechanisms.

 

Dr. Miroslav Pistek (The Czech Academy of Sciences, Prague)
“Stochastic Co-dominance”

It is well-known that the stochastically dominating lottery has a higher chance for the higher outcomes than the lottery that is dominated. However, the relation “higher probability of higher outcome” is not the relation of the (first-order) stochastic dominance; we propose to call this relation stochastic co-dominance. The so-called “Steinhaus-Trybula paradox” showing that stochastic co-dominance is not transitive is probably the reason why such an elementary concept is not widely used. We show that this paradox may be naturally resolved by searching for a maximal probability measure within the closed convex hull of the set of probability measures in question. Note that a similar result has been previously known only for finite sets. This may be used in, e.g., financial applications, once risky portfolios are dealt with. Often one may not find a stochastically non-dominated portfolio, nonetheless there exists at least one stochastically co-dominating probabilistic measure on the set of all available portfolios.

 

Dr. Bismark Singh (FAU Erlangen-Nürnberg)
“Approximating two-stage chance-constrained programs with classical probability bounds”

We consider a joint-chance constraint (JCC) as a union of sets, and approximate this union using bounds from classical probability theory. When these bounds are used in an optimization model constrained by the JCC, we obtain corresponding upper and lower bounds on the optimal objective function value. We compare the strength of these bounds against each other under different sampling schemes, and observe that a larger correlation between the uncertainties tends to result in more computationally challenging optimization models. We also  provide various reformulations of the corresponding mathematical models for these approximations.