Approximation Algorithms for Distributionally Robust Stochastic Optimization

Two-stage stochastic optimization is a widely used framework for modeling uncertainty, where we have a probability distribution over possible realizations of the data, called scenarios, and decisions are taken in two stages: we take first-stage actions knowing only the underlying distribution and before a scenario is realized, and may take additional second-stage recourse actions after a scenario is realized. The goal is typically to minimize the total expected cost. A common criticism levied at this model is that the underlying probability distribution is itself often imprecise. To address this, an approach that is quite versatile and has gained popularity in the stochastic-optimization literature is the two-stage distributionally robust stochastic model: given a collection D of probability distributions, our goal now is to minimize the maximum expected total cost with respect to a distribution in D. There has been almost no prior work however on developing approximation algorithms for distributionally robust problems where the underlying scenario collection is discrete, as is the case with discrete-optimization problems. We provide frameworks for designing approximation algorithms in such settings when the collection D is a ball around a central distribution, defined relative to two notions of distance between probability distributions: Wasserstein metrics (which include the L_1 metric) and the L_infinity metric. Our frameworks yield efficient algorithms even in settings with an exponential number of scenarios, where the central distribution may only be accessed via a sampling oracle. For distributionally robust optimization under a Wasserstein ball, we first show that one can utilize the sample average approximation (SAA) method (solve the distributionally robust problem with an empirical estimate of the central distribution) to reduce the problem to the case where the central distribution has a polynomial-size support, and is represented explicitly. This follows because we argue that a distributionally robust problem can be reduced in a novel way to a standard two-stage stochastic problem with bounded inflation factor, which enables one to use the SAA machinery developed for two-stage stochastic problems. Complementing this, we show how to approximately solve a fractional relaxation of the SAA problem (i.e., the distributionally robust problem obtained by replacing the original central distribution with its empirical estimate). Unlike in two-stage {stochastic, robust} optimization with polynomially many scenarios, this turns out to be quite challenging. We utilize a variant of the ellipsoid method for convex optimization in conjunction with several new ideas to show that the SAA problem can be approximately solved provided that we have an (approximation) algorithm for a certain max-min problem that is akin to, and generalizes, the k-max-min problem (find the worst-case scenario consisting of at most k elements) encountered in two-stage robust optimization. We obtain such an algorithm for various discrete-optimization problems; by complementing this via rounding algorithms that provide local (i.e., per-scenario) approximation guarantees, we obtain the first approximation algorithms for the distributionally robust versions of a variety of discrete-optimization problems including set cover, vertex cover, edge cover, facility location, and Steiner tree, with guarantees that are, except for set cover, within O(1)-factors of the guarantees known for the deterministic version of the problem. For distributionally robust optimization under an L_infinity ball, we consider a fractional relaxation of the problem, and replace its objective function with a proxy function that is pointwise close to the true objective function (within a factor of 2). We then show that we can efficiently compute approximate subgradients of the proxy function, provided that we have an algorithm for the problem of computing the t worst scenarios under a given first-stage decision, given an integer t. We can then approximately minimize the proxy function via a variant of the ellipsoid method, and thus obtain an approximate solution for the fractional relaxation of the distributionally robust problem. Complementing this via rounding algorithms with local guarantees, we obtain approximation algorithms for distributionally robust versions of various covering problems, including set cover, vertex cover, edge cover, and facility location, with guarantees that are within O(1)-factors of the guarantees known for their deterministic versions

The Effect of Physical Application Parameters on Herbicide Efficacy and Droplet Size.

One of the largest challenges in agriculture is weed management. Improper or sub-optimal application techniques can cause decreased weed control and increased environmental contamination. Effective weed management is highly correlated with the product and the application method. Herbicide performance are affected by environmental conditions; they influence the physiology and growth of a plant and as well the herbicide performance. Among all environmental factors, rain shortly after herbicide application is one of the most harmful issues to the performance of the herbicide. Droplet size is a key factor in pesticide applications in regards to both drift and efficacy. Droplet size can be altered by several application parameters, such as the nozzle type, pressure, orifice size and spray solution. Droplet size is a key component in pesticide application with respect to overall application efficacy and off-target movement. As tank mix ingredients can significantly influence the resulting droplet size, agitation systems are critical to ensuring proper mixing of all components and overall performance. Sitting time, a period where the tank is held in a non-agitated state, potentially affects droplet size as well. The objectives of this research were: 1) understand the influence of nozzle spacing, boom height, nozzle type, on weed control, also expand the scientific knowledge on aforementioned parameters. 2) Evaluate the effect of rainfall after herbicide application on weed control, following certain intervals in order to understand the wash off effect. 3) Analyze the impact of nozzle type, application speed and pressure on weed control, in order to contribute to a more reliable recommendation of such parameters. This research highlights the impact of parameters regulated by the sprayer on weed control and allow a better understanding of how non-chemical parameters affect the efficacy on weed management, as well as a greater understanding on absorption and evaporation of herbicide plus losses of application efficacy. The results will clarify some of the most concerning question on one of the most complex process in agriculture. Adviser: Greg R. Kruge

Evaluation of biomass and coal co-gasification of brazilian feedstock using a chemical equilibrium model

Coal and biomass are energy sources with great potential for use in Brazil. Coal-biomass cogasification enables the combination of the positive characteristics of each fuel, besides leading to a cleaner use of coal. The present study evaluates the potential of co-gasification of binary coal-biomass blends using sources widely available in Brazil. This analysis employs computational simulations using a reliable thermodynamic equilibrium model. Favorable operational conditions at high temperatures are determined in order to obtain gaseous products suitable for energy cogeneration and chemical synthesis. This study shows that blends with biomass ratios of 5% and equivalence ratios ≤ 0.3 lead to high cold gas efficiencies. Suitable gaseous products for chemical synthesis were identified at biomass ratios ≤ 35% and moisture contents ≥ 40%. Formation of undesirable nitrogen and sulfur compounds was also analyzed

Opening and tuning of band gap by the formation of diamond superlattices in twisted bilayer graphene

We report results of first-principles density functional theory calculations, which introduce a new class of carbon nanostructures formed due to creation of covalent interlayer C-C bonds in twisted bilayer graphene (TBG). This interlayer bonding becomes possible by hydrogenation of the graphene layers according to certain hydrogenation patterns. The resulting relaxed configurations consist of two-dimensional (2D) superlattices of diamondlike nanocrystals embedded within the graphene layers, with the same periodicity as that of the Moiré pattern corresponding to the rotational layer stacking in TBG. The 2D diamond nanodomains resemble the cubic or the hexagonal diamond phase. The detailed structure of these superlattice configurations is determined by parameters that include the twist angle, ranging from 0° to ∼15°, and the number of interlayer C-C bonds formed per unit cell of the superlattice. We demonstrate that formation of such interlayer-bonded finite domains causes the opening of a band gap in the electronic band structure of TBG, which depends on the density and spatial distribution of interlayer C-C bonds. We have predicted band gaps as wide as 1.2 eV and found that the band gap increases monotonically with increasing size of the embedded diamond nanodomain in the unit cell of the superlattice. Such nanostructure formation constitutes a promising approach for opening a precisely tunable band gap in bilayer graphene

Ab initio studies of thermodynamic and electronic properties of phosphorene nanoribbons

We present a density functional theory study of the thermodynamic and electronic properties of phosphorene nanoribbons. We consider a variety of terminations and reconstructions of ribbon edges, both with and without hydrogen passivation, and calculate an ab intio phase diagram that identifies energetically preferred edges as a function of temperature and hydrogen partial pressure. These studies are also accompanied by detailed electronic structure calculations from which we find that ribbons with hydrogenated edges are typically direct gap semiconductors with fundamental gaps that are in excess of phosphorene, the gaps varying inversely with ribbon width. In contrast, ribbons with bare or partially passivated edges either have metallic edges or are semiconducting with band gaps that are smaller than those of their hydrogenated counterparts due to the appearance of midgap edge states. Overall, our studies provide a basis for tailoring the electronic properties of phosphorene nanoribbons by controlling the edge termination via processing conditions (temperature and hydrogen partial pressure) as well as by confinement of carriers via control over ribbon width

Ab initio studies of thermodynamic and electronic properties of phosphorene nanoribbons

We present a density functional theory study of the thermodynamic and electronic properties of phosphorene nanoribbons. We consider a variety of terminations and reconstructions of ribbon edges, both with and without hydrogen passivation, and calculate an ab intio phase diagram that identifies energetically preferred edges as a function of temperature and hydrogen partial pressure. These studies are also accompanied by detailed electronic structure calculations from which we find that ribbons with hydrogenated edges are typically direct gap semiconductors with fundamental gaps that are in excess of phosphorene, the gaps varying inversely with ribbon width. In contrast, ribbons with bare or partially passivated edges either have metallic edges or are semiconducting with band gaps that are smaller than those of their hydrogenated counterparts due to the appearance of midgap edge states. Overall, our studies provide a basis for tailoring the electronic properties of phosphorene nanoribbons by controlling the edge termination via processing conditions (temperature and hydrogen partial pressure) as well as by confinement of carriers via control over ribbon width