1,720,986 research outputs found
Passivity-based Analysis of the ADMM Algorithm for Constraint-Coupled Optimization
Full text linkWe propose a novel, system theoretic analysis of the Alternating Direction Method of Multipliers (ADMM) applied to a convex constraint-coupled optimization problem. The resulting algorithm can be interpreted as a linear, discrete-time dynamical system (modeling the multiplier ascent update) in closed loop with a static nonlinearity (representing the minimization of the augmented Lagrangian). When expressed in suitable coordinates, we prove that the discrete-time linear dynamical system has a discrete positive-real transfer function and is interconnected in closed loop with a static, passive nonlinearity. This readily shows that the origin is a stable equilibrium for the feedback interconnection. Finally, we also show global asymptotic stability of the origin for the closed-loop system and, thus, global asymptotic convergence of ADMM to the optimal solution of the optimization problem
Achievement and Fragility of Long-Term Equitability
No full textEquipping current decision-making tools with notions of fairness, equitability, or other ethically motivated outcomes, is one of the top priorities in recent research efforts in machine learning, AI, and optimization. In this paper, we investigate how to allocate limited resources to locally interacting communities in a way to maximize a pertinent notion of equitability. In particular, we look at the dynamic setting where the allocation is repeated across multiple periods (e.g., yearly), the local communities evolve in the meantime (driven by the provided allocation), and the allocations are modulated by feedback coming from the communities themselves. We employ recent mathematical tools stemming from data-driven feedback online optimization, by which communities can learn their (possibly unknown) evolution, satisfaction, as well as they can share information with the deciding bodies. We design dynamic policies that converge to an allocation that maximize equitability in the long term. We further demonstrate our model and methodology with realistic examples of healthcare and education subsidies design in Sub-Saharian countries. One of the key empirical takeaways from our setting is that long-Term equitability is fragile, in the sense that it can be easily lost when deciding bodies weigh in other factors (e.g., equality in allocation) in the allocation strategy. Moreover, a naive compromise, while not providing significant advantage to the communities, can promote inequality in social outcomes
Constraint-Coupled Distributed Optimization: A Relaxation and Duality Approach
Full text linkIn this paper, we consider a general challenging distributed optimization setup arising in several important network control applications. Agents of a network want to minimize the sum of local cost functions, each one depending on a local variable, subject to local and coupling constraints, with the latter involving all the decision variables. We propose a novel fully distributed algorithm based on a relaxation of the primal problem and an elegant exploration of duality theory. Despite its complex derivation, based on several duality steps, the distributed algorithm has a very simple and intuitive structure. That is, each node finds a primal-dual optimal solution pair of a local relaxed version of the original problem and then updates suitable auxiliary local variables. We prove that agents asymptotically compute their portion of an optimal (feasible) solution of the original problem. This primal recovery property is obtained without any averaging mechanism typically used in dual decomposition methods. To corroborate the theoretical results, we show how the methodology applies to an instance of a distributed model-predictive control scheme in a microgrid control scenario
Stability, Linear Convergence, and Robustness of the Wang-Elia Algorithm for Distributed Consensus Optimization
No full textWe revisit an algorithm for distributed consensus optimization proposed in 2010 by J. Wang and N. Elia. By means of a Lyapunov-based analysis, we prove input-to-state stability of the algorithm relative to a closed invariant set composed of optimal equilibria and with respect to perturbations affecting the algorithm's dynamics. In the absence of perturbations, this result implies linear convergence of the local estimates and Lyapunov stability of the optimal steady state. Moreover, we unveil fundamental connections with the wellknown Gradient Tracking and with distributed integral control. Overall, our results suggest that a control theoretic approach can have a considerable impact on (distributed) optimization, especially when robustness is considered
Distributed Primal Decomposition for Large-Scale MILPs
Full text linkThis paper deals with a distributed Mixed-Integer Linear Programming (MILP) set-up arising in several control applications. Agents of a network aim to minimize the sum of local linear cost functions subject to both individual constraints and a linear coupling constraint involving all the decision variables. A key, challenging feature of the considered set-up is that some components of the decision variables must assume integer values. The addressed MILPs are NP-hard, nonconvex and large-scale. Moreover, several additional challenges arise in a distributed framework due to the coupling constraint, so that feasible solutions with guaranteed suboptimality bounds are of interest. We propose a fully distributed algorithm based on a primal decomposition approach and an appropriate tightening of the coupling constraint. The algorithm is guaranteed to provide feasible solutions in finite time. Moreover, asymptotic and finite-time suboptimality bounds are established for the computed solution. Montecarlo simulations highlight the extremely low suboptimality bounds achieved by the algorithm
On-Policy Data-Driven Linear Quadratic Regulator via Combined Policy Iteration and Recursive Least Squares
No full textIn this paper, we address infinite-horizon Linear Quadratic Regulator (LQR) problems for unknown discrete- time systems. As an additional challenge, we address an on- policy setup in which system matrices are identified while controlling the real system with a progressively optimized policy. Specifically, we consider a time-varying control policy that, while applied to the real unknown system, is iteratively refined (based on the most updated estimate of the system matrices) towards the optimal LQR solution. The overall learning procedure combines a recursive least squares method with a direct policy search based on the gradient method. By resorting to Lyapunov-based analysis tools in combination with averaging theory for nonlinear systems, exponential stability for the closed-loop scheme can be proven. Finally, a numerical example showing the effectiveness of the considered strategy corroborates the theoretical findings
Triggered Gradient Tracking for asynchronous distributed optimization
Full text linkThis paper proposes ASYNCHRONOUS TRIGGERED GRADIENT TRACKING, i.e., a distributed optimization algorithm to solve consensus optimization over networks with asynchronous communication. As a building block, we devise the continuous-time counterpart of the recently proposed (discrete-time) distributed gradient tracking called CONTINUOUS GRADIENT TRACKING. By using a Lyapunov approach, we prove exponential stability of the equilibrium corresponding to agents' estimates being consensual to the optimal solution, with arbitrary initialization of the local estimates. Then, we propose two triggered versions of the algorithm. In the first one, the agents continuously integrate their local dynamics and exchange with neighbors their current local variables in a synchronous way. In ASYNCHRONOUS TRIGGERED GRADIENT TRACKING, we propose a totally asynchronous scheme in which each agent sends to neighbors its current local variables based on a triggering condition that depends on a locally verifiable condition. The triggering protocol preserves the linear convergence of the algorithm and avoids the Zeno behavior, i.e., an infinite number of triggering events over a finite interval of time is excluded. By using the stability analysis of CONTINUOUS GRADIENT TRACKING as a preparatory result, we show exponential stability of the equilibrium point holds for both triggered algorithms and any estimate initialization. Finally, the simulations validate the effectiveness of the proposed methods on a data analytics problem, showing also improved performance in terms of inter-agent communication
Distributed Personalized Gradient Tracking with Convex Parametric Models
Full text linkWe present a distributed optimization algorithm for solving online personalized optimization problems over a network of computing and communicating nodes, each of which linked to a specific user. The local objective functions are assumed to have a composite structure and to consist of a known time-varying (engineering) part and an unknown (user-specific) part. Regarding the unknown part, it is assumed to have a known parametric (e.g., quadratic) structure a priori, whose parameters are to be learned along with the evolution of the algorithm. The algorithm is composed of two intertwined components: (i) a dynamic gradient tracking scheme for finding local solution estimates and (ii) a recursive least squares scheme for estimating the unknown parameters via user's noisy feedback on the local solution estimates. The algorithm is shown to exhibit a bounded regret under suitable assumptions. Finally, a numerical example corroborates the theoretical analysis
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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