A study of distributionally robust mixed-integer programming with Wasserstein metric: on the value of incomplete data

This study addresses a class of linear mixed-integer programming (MILP) problems that involve uncertainty in the objective function parameters. The parameters are assumed to form a random vector, whose probability distribution can only be observed through a finite training data set. Unlike most of the related studies in the literature, we also consider uncertainty in the underlying data set. The data uncertainty is described by a set of linear constraints for each random sample, and the uncertainty in the distribution (for a fixed realization of data) is defined using a type-1 Wasserstein ball centered at the empirical distribution of the data. The overall problem is formulated as a three-level distributionally robust optimization (DRO) problem. First, we prove that the three-level problem admits a single-level MILP reformulation, if the class of loss functions is restricted to biaffine functions. Secondly, it turns out that for several particular forms of data uncertainty, the outlined problem can be solved reasonably fast by leveraging the nominal MILP problem. Finally, we conduct a computational study, where the out-of-sample performance of our model and computational complexity of the proposed MILP reformulation are explored numerically for several application domains

Magnetically induced ring currents in metallocenothiaporphyrins

The magnetically induced current-density susceptibility tensor (CDT) of the lowest singlet and triplet states of the metallocenothiaporphyrins, where the metal is V, Cr, Mn, Fe, Co, Ni, Mo, Tc, Ru, or Rh, have been studied with the gauge-including magnetically induced currents (GIMIC) method. The compounds containing V, Mn, Co, Tc or Rh were studied as cations because the neutral molecules have an odd number of electrons. The calculations show that the aromatic nature of most of the studied molecules follows the Huckel and Baird rules of aromaticity. CDT calculations on the high-spin states of the neutral metallocenothiaporphyrins with V, Mn, Co, Tc or Rh also shows that these molecules follow a unified extended Huckel and Baird aromaticity orbital-count rule stating that molecules with an odd number of occupied conjugated valence orbitals are aromatic, whereas molecules with an even number of occupied conjugated orbitals are antiaromatic.Peer reviewe

Increasing the Brønsted acidity of Ph2PO2H by the Lewis acid B(C6F5)3. Formation of an eight-membered boraphosphinate ring [Ph2POB(C6F5)2O]2

The Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged for financial support. The theoretical part of this work was supported by the Russian Science Foundation (Project 14-13-00832).Autoprotolysis of the metastable acid (C6F5)3BOPPh2OH, prepared in situ by the reaction of the rather weak Brønsted acid Ph2PO2H with the strong Lewis acid B(C6F5)3, gave rise to the formation of the eight-membered ring [Ph2POB(C6F5)2O]2 and C6F5H. The conjugate base was isolated as stable sodium crown ether salt [Na(15-crown-5)][Ph2PO2B(C6F5)3].Publisher PDFPeer reviewe

On the multi-stage shortest path problem under distributional uncertainty

In this paper we consider an ambiguity-averse multi-stage network game between a user and an attacker. The arc costs are assumed to be random variables that satisfy prescribed first-order moment constraints for some subsets of arcs and individual probability constraints for some particular arcs. The user aims at minimizing its cumulative expected loss by traversing between two fixed nodes in the network, while the attacker maximizes the user's objective function by selecting a distribution of arc costs from the family of admissible distributions. In contrast to most of the previous studies in the related literature, both the user and the attacker can dynamically adjust their decisions at each node of the user's path. By observing the user's decisions, the attacker needs to reveal some additional distributional information associated with the arcs emanated from the current user's position. It is shown that the resulting multi-stage distributionally robust shortest path problem admits a linear mixed-integer programming reformulation (MIP). In particular, we distinguish between acyclic and general graphs by introducing different forms of non-anticipativity constraints. Finally, we perform a numerical study, where the quality of adaptive decisions and computational tractability of the proposed MIP reformulation are explored with respect to several classes of synthetic network instances

A study of distributionally robust mixed-integer programming with Wasserstein metric: on the value of incomplete data

This study addresses a class of linear mixed-integer programming (MILP) problems that involve uncertainty in the objective function parameters. The parameters are assumed to form a random vector, whose probability distribution can only be observed through a finite training data set. Unlike most of the related studies in the literature, we also consider uncertainty in the underlying data set. The data uncertainty is described by a set of linear constraints for each random sample, and the uncertainty in the distribution (for a fixed realization of data) is defined using a type-1 Wasserstein ball centered at the empirical distribution of the data. The overall problem is formulated as a three-level distributionally robust optimization (DRO) problem. First, we prove that the three-level problem admits a single-level MILP reformulation, if the class of loss functions is restricted to biaffine functions. Secondly, it turns out that for several particular forms of data uncertainty, the outlined problem can be solved reasonably fast by leveraging the nominal MILP problem. Finally, we conduct a computational study, where the out-of-sample performance of our model and computational complexity of the proposed MILP reformulation are explored numerically for several application domains