Distributed Multi-Agent Optimization with State-Dependent Communication

We study distributed algorithms for solving global optimization problems in which the objective function is the sum of local objective functions of agents and the constraint set is given by the intersection of local constraint sets of agents. We assume that each agent knows only his own local objective function and constraint set, and exchanges information with the other agents over a randomly varying network topology to update his information state. We assume a state-dependent communication model over this topology: communication is Markovian with respect to the states of the agents and the probability with which the links are available depends on the states of the agents. In this paper, we study a projected multi-agent subgradient algorithm under state-dependent communication. The algorithm involves each agent performing a local averaging to combine his estimate with the other agents' estimates, taking a subgradient step along his local objective function, and projecting the estimates on his local constraint set. The state-dependence of the communication introduces significant challenges and couples the study of information exchange with the analysis of subgradient steps and projection errors. We first show that the multi-agent subgradient algorithm when used with a constant stepsize may result in the agent estimates to diverge with probability one. Under some assumptions on the stepsize sequence, we provide convergence rate bounds on a "disagreement metric" between the agent estimates. Our bounds are time-nonhomogeneous in the sense that they depend on the initial starting time. Despite this, we show that agent estimates reach an almost sure consensus and converge to the same optimal solution of the global optimization problem with probability one under different assumptions on the local constraint sets and the stepsize sequence

Strategic dynamic vehicle routing with spatio-temporal dependent demands

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 51-53).Dynamic vehicle routing problems address the issue of determining optimal routes for a set of vehicles, to serve a given set of demands that arrive sequentially in time. Traditionally, demands are assumed to be generated over time by an exogenous stochastic process. This thesis is concerned with the study of dynamic vehicle routing problems where demands are strategically placed in the space by an agent with selfish interests and physical constraints. In particular, we focus on the following problem: a team of vehicles seek to device dynamic routing policies that minimize the average waiting time of a typical demand, from the moment it is placed in the space until its location is visited; while an adversarial agent operating from a central depot with limited capacity aims at the opposite, strategically choosing the spatio-temporal point process according to which place demands. We model the above problem and its inherent pure conflict of interests as a zero-sum game, and characterize equilibria under heavy load regime. For the analysis we discriminate between two cases: bounded and unbounded domains. In both cases we show that a routing policy based on performing successive TSP tours through outstanding demands and a power-law spatial distribution of demands are optimal, saddle point of the utility function of the game. The latter emerges as the unique solution of maximizing a non-convex nowhere differentiable functional over the infinite-dimensional space of probability densities; the non-convexity is the result of the spatio-temporal dependence induced by the physical constraints imposed on the behavior of the agent, and the non-differentiability is due to the emptiness of the interior of the positive cone of integrable functions. We solve this problem applying Fenchel conjugate duality for partially finite programming in the case of bounded domains; and a direct duality approach exploiting the structure of a concave integral functional part of the objective and the linear equality constraints, for unbounded domains. Remarkably, all the results obtained hold for any domain with a sufficiently smooth boundary, clossedness or connectedness is not needed. We provide numerical simulations to validate the theory.by Diego Feijer.S.M

Strategic Dynamic Vehicle Routing with Spatio-Temporal Dependent Demands

Abstract-We study a dynamic vehicle routing problem where demands are strategically placed in the region by an adversarial agent with unitary capacity operating from a depot. In particular, we focus on the following problem: a system planner seeks to design dynamic vehicle routing policies for a vehicle that minimize the average waiting time of a typical demand, defined as the time difference between the moment the demand is placed in the region until its location is visited by the vehicle; while the agent aims at the opposite, strategically choosing the spatial distribution to place demands. We model the problem as a complete information zero-sum game and characterize an equilibrium in the limiting case where the vehicle travels arbitrarily slower than the agent. We show that such an equilibrium is constituted by a routing policy based on performing successive traveling salesperson tours through outstanding demands and a unique power-law spatial density centered at the depot location

Financial market failures and systemic crises

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2015.Cataloged from PDF version of thesis.Includes bibliographical references (pages 97-103).This thesis contributes to the theoretical literature that studies the macroeconomic implications of financial frictions. It develops frameworks to address different financial market failures, and evaluate preventive policies to mitigate the vulnerability of the economy to costly systemic crises. First, it identifies a credit risk (fire sale) externality that justifies the macroprudential regulation of short-term debt to mitigate the probability of systemic bank runs. Without regulation, banks do not internalize how their funding decisions affects the terms at which other market participants can obtain credit. The formal welfare study conducted, provides a general equilibrium notion of systemic risk that captures both fundamental insolvency and illiquidity risk. It also connects this measure with the optimal Pigouvian (corrective) tax. Second, it shows that liquidity crises may arise as the result of endogenous information panics. It finds that collective ignorance is welfare maximizing but it is fragile, susceptible to self-fulfilling fears about asymmetric information. Adverse selection may thus obtain in equilibrium, sustained by negative aggregate expectations. The mechanism that gives rise to multiple equilibria is robust to the introduction of noisy private signals, and warrants the regulation of information acquisition for rent-seeking (speculative) motives. Finally, it demonstrates the limitations of unconventional credit easing policies to stimulate lending during market-freezes. With inter-temporal investment complementarities, credit to non-financial firms may be curtailed as the result of dynamic coordination failures. Interest rate cuts mitigate coordination risk, but increase the average duration of credit market freezes when the productivity of capital is high. Capital injections in the banking sector, or direct lending to non-financial firms, are completely ineffective, because reductions in deposits from households crowd out government spending. In contrast, government guarantees improve welfare by reducing strategic uncertainty.by Diego Feijer.Ph. D

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We study a zero-sum game formulation of a dynamic vehicle routing problem: a system planner seeks to design dynamic routing policies for a team of vehicles to minimize the average waiting time of demands that are strategically placed in a region by an adversarial agent with unitary capacity operating from a depot. We characterize an equilibrium in the limiting case where vehicles travel arbitrarily slower than the agent (heavy load). We show that such an equilibrium consists of a routing policy based on performing successive TSP tours through outstanding demands and a unique power-law spatial density centered at the depot location