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

Synthesis, characterization and photophysical properties of new cyclometallated platinum(II) complexes with pyrazolonate ancillary ligand

New cyclometalated platinum(II) complexes with pyrazolonate ancillary ligand (ppy)Pt(pmip) (1) and (dfppy)Pt(pmip) (2) (ppy = 2-phenylpyridine, dfppy = (4,6-difluorophenyl)pyridine, Hpmip = 1-phenyl-3-methyl-4-isobutyryl-5- pyrazolone) were synthesized and structurally characterized. Both compounds revealed square-planar geometry. The crystal cell of 1 was found to contain the monomer molecules of platinum compound whereas dimer molecules of 2 with short Pt⋯Pt contacts of 3.2217(3) were observed in the crystal cell of 2. Photophysical properties of 1 and 2 were investigated in detail. The highly resolved photoluminesence spectra of the platinum complexes in solution contain emission bands in the region of 470-550 nm attributed to monomer compounds 1 and 2. The triplet-state energies of 1 and 2 obtained from DFT calculations agree very well with the experimental data. In the crystalline state complex 2 revealed excimer emission as a structureless broad band at ca. 584 nm related to dimer molecules of platinum compound presented in the crystals. © 2013 Elsevier B.V. All rights reserved

Synthesis, crystal structures and luminescent properties of the copper(I) pyrazolonate complexes

© 2014 Elsevier B.V. All rights reserved. New copper(I) complexes with pyrazolonate ligands [Cu(Pri-PMP)(DPEphos)] (1) and [Cu(But-PMP)(DPEphos)] (2) (Pri-PMP = 1-phenyl-3-methyl-4-isobutyryl-5-pyrazolonato, But-PMP = 1-phenyl-3-methyl-(2,2-dimethylpropan-1-oyl)-5-pyrazolonato; DPEphos = bis[2-(diphenylphosphino)-pheny]ether) were synthesized and structurally characterized. An unusual η1 coordination of pyrazolonate ligand to the copper atom was found in complex 2. Photo- and electroluminescent properties of the synthesized compounds were investigated. In crystalline form compounds 1 and 2 revealed dual emission consisting of the bands at 445-450 and 485-488 nm which were assigned to transitions from the S1 and T1 states. DFT and TD DFT calculations as well as electrochemical studies correlate with the photophysical data. Synthesized copper(I) complexes generated electroluminescence of yellowish-orange (1) and yellow (2) colors with the maximum luminance of 286 and 39 cd/m2, respectively

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

RYDBERG TRANSITIONS ORIGINATING AT METAL-LOCALIZED ORBITALS IN LARGE POLYATOMIC MOLECULES: A GAS-PHASE SPECTROSCOPIC STUDY OF TRANSITION- METAL SANDWICH COMPLEXES

Author Institution: G.A. Razuvaev Institute of Organometallic Chemistry, Russian Academy of SciencesThe transition-metal sandwich compounds with an occupied totally symmetric valence molecular orbital (MO) derived from the metal $d_{z}(2)$ appear to represent the first example of polyatomic molecules revealing in the photoabsorption spectra well-resolved Rydberg transitions from a metal-localized MO. The gas-phase UV absorption spectra of ($\eta^{6}$-Arene)$_{2}M$ (M = V, Cr, Mo, W), ($\eta^{5}-C_{5}H_{5})_{2}M$ (M = V, Fe, Ru, Os), ($\eta^{6}$-Arene)($\eta^{5}-C_{5}H_{5}$)Mn, ($\eta^{7}-C_{7}H_{7}$)M (M = V, Nb, Cr, Mo, W) show sharp bands arising from the transitions originating at the metal $d_{z}{2}$ orbital and terminating at Rydberg ns, np and nd levels. Due to the low $d_{z}{2}$ ionisation energies, even higher Rydberg excitations in this compounds lie as a rule, below $50000 cm^{-1}$ making it easy to investigate them using standard UV-visible spectrometers. Rydbberg features disappear on going to the condenced media resulting in a dramatic difference between the gas phase and solution-phase spectra of sandwich complexes. All Rydberg bands observed have been assigned on the basis of corresponding term values and effective quantum numbers. These parameters appear to change very little on going from one sandwich to another. For bisarene and mixed sandwiches, many-membered Rydberg np series converging on the $d_{z}{2}$ ionization limit have been revealed and the corresponding ionization potentials have been calculated with high accuracy using the Rydberg formula. Clear examples of molecular symmetry influence on the Rydberg structure, Rydberg/valence mixing and vibronic coupling in Rydberg states will be presented

On a class of data-driven combinatorial optimization problems under uncertainty: a distributionally robust approach

In this study we analyze linear combinatorial optimization problems where the cost vector is not known a priori, but is only observable through a finite data set. In contrast to the related studies, we presume that the number of observations with respect to particular components of the cost vector may vary. The goal is to find a procedure that transforms the data set into an estimate of the expected value of the objective function (which is referred to as a prediction rule) and a procedure that retrieves a candidate decision (which is referred to as a prescription rule). We aim at finding the least conservative prediction and prescription rules, which satisfy some specified asymptotic guarantees. We demonstrate that the resulting vector optimization problems admit a weakly optimal solution, which can be obtained by solving a particular distributionally robust optimization problem. Specifically, the decision-maker may optimize the worst-case expected loss across all probability distributions with given component-wise relative entropy distances from the empirical marginal distributions. Finally, we perform numerical experiments to analyze the out-of-sample performance of the proposed solution approach