On the analysis of stochastic optimization and variational inequality problems

Uncertainty has a tremendous impact on decision making. The more connected we get, it seems, the more sources of uncertainty we unfold. For example, uncertainty in the parameters of price and cost functions in power, transportation, communication and financial systems have stemmed from the way these networked systems operate and also how they interact with one another. Uncertainty influences the design, regulation and decisions of participants in several engineered systems like the financial markets, electricity markets, commodity markets, wired and wireless networks, all of which are ubiquitous. This poses many interesting questions in areas of understanding uncertainty (modeling) and dealing with uncertainty (decision making). This dissertation focuses on answering a set of fundamental questions that pertain to dealing with uncertainty arising in three major problem classes: [(1)] Convex Nash games; [(2)] Variational inequality problems and complementarity problems; [(3)] Hierarchical risk management problems in financial networks. Accordingly, this dissertation considers the analysis of a broad class of stochastic optimization and variational inequality problems complicated by uncertainty and nonsmoothness of objective functions. Nash games and variational inequalities have assumed practical relevance in industry and business settings because they are natural models for many real-world applications. Nash games arise naturally in modeling a range of equilibrium problems in power markets, communication networks, market-based allocation of resources etc. where as variational inequality problems allow for modeling frictional contact problems, traffic equilibrium problems etc. Incorporating uncertainty into convex Nash games leads us to stochastic Nash games. Despite the relevance of stochastic generalizations of Nash games and variational inequalities, answering fundamental questions regarding existence of equilibria in stochastic regimes has proved to be a challenge. Amongst other reasons, the main challenge arises from the nonlinearity arising from the presence of the expectation operator. Despite the rich literature in deterministic settings, direct application of deterministic results to stochastic regimes is not straightforward. The first part of this dissertation explores such fundamental questions in stochastic Nash games and variational inequality problems. Instead of directly using the deterministic results, by leveraging Lebesgue convergence theorems we are able to develop a tractable framework for analyzing problems in stochastic regimes over a continuous probability space. The benefit of this approach is that the framework does not rely on evaluation of the expectation operator to provide existence guarantees, thus making it amenable to tractable use. We extend the above framework to incorporate nonsmoothness of payoff functions as well as allow for stochastic constraints in models, all of which are important in practical settings. The second part of this dissertation extends the above framework to generalizations of variational inequality problems and complementarity problems. In particular, we develop a set of almost-sure sufficiency conditions for stochastic variational inequality problems with single-valued and multi-valued mappings. We extend these statements to quasi-variational regimes as well as to stochastic complementarity problems. The applicability of these results is demonstrated in analysis of risk-averse stochastic Nash games used in Nash-Cournot production distribution models in power markets by recasting the problem as a stochastic quasi-variational inequality problem and in Nash-Cournot games with piecewise smooth price functions by modeling this problem as a stochastic complementarity problem. The third part of this dissertation pertains to hierarchical problems in financial risk management. In the financial industry, risk has been traditionally managed by the imposition of value-at-risk or VaR constraints on portfolio risk exposure. Motivated by recent events in the financial industry, we examine the role that risk-seeking traders play in the accumulation of large and possibly infinite risk. We proceed to show that when traders employ a conditional value-at-risk (CVaR) metric, much can be said by studying the interaction between value at risk (VaR) (a non-coherent risk measure) and conditional value at risk CVaR (a coherent risk measure based on VaR). Resolving this question requires characterizing the optimal value of the associated stochastic, and possibly nonconvex, optimization problem, often a challenging problem. Our study makes two sets of contributions. First, under general asset distributions on a compact support, traders accumulate finite risk with magnitude of the order of the upper bound of this support. Second, when the supports are unbounded, under relatively mild assumptions, such traders can take on an unbounded amount of risk despite abiding by this VaR threshold. In short, VaR thresholds may be inadequate in guarding against financial ruin

Traumatic events: exploring associations with maternal depression, infant bonding, and oxytocin in Latina mothers

Abstract Background Childhood and adulthood traumatic experiences negatively impact maternal-infant bonding and increase risk of postpartum depression (PPD). Lower oxytocin levels have also been associated with PPD and compromised mother-infant bonding. Despite advances in these areas of investigation, much of the research has not included Latinas, who are important because they have high rates of fertility, traumatic events, and PPD. Methods To address gaps identified in the literature, we explored associations between traumatic life events, PPD, and bonding subscale scores (e.g., Impaired Bonding, Rejection and Anger, Anxiety about Care) in a sample of 28 Latinas. We also examined associations between these factors and oxytocin (OT). Wilcoxon signed-rank tests were employed to examine differences in subscale scores over time. Kruskal–Wallis one-way analysis of variance was used to examine differences in bonding subscale scores and OT by maternal depression status and traumatic events. We also explored interaction effects of traumatic events and OT AUC on bonding subscale scores. Results Women with PPD at 8 weeks had significantly higher Rejection and Anger subscale scores (p = 0.054) than non-PPD women, where higher scores represent more compromised bonding. Significant differences in Rejection and Anger (p = 0.042) and Anxiety about Care (p = 0.005) by adulthood traumatic histories were observed at 8 weeks postpartum. There was also a significant difference in Anxiety about Care scores at 4 weeks postpartum (p = 0.024) and Impaired Bonding at 8 weeks postpartum (p = 0.041) by trauma events involving an infant. There was a significant interaction between OT and childhood sexual abuse on Impaired Bonding (p = 0.038). Conclusion We observed differential responses in bonding subscale scores by traumatic histories. Women who experienced a trauma involving an infant had higher compromised bonding scores, whereas those with adulthood traumatic histories, such as intimate partner violence, had lower scores. We also found an interaction between childhood trauma and oxytocin levels on bonding scores, suggesting a physiological response to early abuse that can have implications on mothers’ bonding perceptions. These preliminary results suggest the need for additional research on the long-term emotional and physiological effects of traumatic events occurring prior to parturition