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Optimization and Control

New submissions

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New submissions for Tue, 20 Mar 18

[1]  arXiv:1803.06363 [pdf, ps, other]
Title: Geometric Adaptive Control for a Quadrotor UAV with Wind Disturbance Rejection
Subjects: Optimization and Control (math.OC)

This paper presents a geometric adaptive control scheme for a quadrotor unmanned aerial, where the effects of unknown, unstructured disturbances are mitigated by a multilayer neural network that is adjusted online. The stability of the proposed controller is analyzed with Lyapunov stability theory on the special Euclidean group, and it is shown that the tracking errors are uniformly ultimately bounded with an ultimate bound that can be abridged arbitrarily. A mathematical model of wind disturbance on the quadrotor dynamics is presented, and it is shown that the proposed adaptive controller is capable of mitigating the effects of wind disturbances successfully.

[2]  arXiv:1803.06380 [pdf, ps, other]
Title: Distributed Optimization for Second-Order Multi-Agent Systems with Dynamic Event-Triggered Communication
Subjects: Optimization and Control (math.OC)

In this paper, we propose a fully distributed continuous-time algorithm to solve the distributed optimization problem for second-order multi-agent systems. The optimization objective function is a sum of private cost functions associated to the individual agents and the interaction between agents is described by a weighted undirected graph. We show the exponential convergence of the proposed algorithm if the underlying graph is connected and the private cost functions are strongly convex and have locally Lipschitz gradients. Moreover, to reduce the overall need of communication, we then propose a dynamic event-triggered communication scheme that is free of Zeno behavior. It is shown that the exponential convergence is achieved if the private cost functions are globally Lipschitz. Numerical simulations are provided to illustrate the effectiveness of the theoretical results.

[3]  arXiv:1803.06482 [pdf, other]
Title: Asynchronous Distributed Method of Multipliers for Constrained Nonconvex Optimization
Subjects: Optimization and Control (math.OC)

This paper addresses a class of constrained optimization problems over networks in which local cost functions and constraints can be nonconvex. We propose an asynchronous distributed optimization algorithm, relying on the centralized Method of Multipliers, in which each node wakes up in an uncoordinated fashion and performs either a descent step on a local Augmented Lagrangian or an ascent step on the local multiplier vector. These two phases are regulated by a distributed logic-AND, which allows nodes to understand when the descent on the (whole) Augmented Lagrangian is sufficiently small. We show that this distributed algorithm is equivalent to a block coordinate descent algorithm for the minimization of the Augmented Lagrangian followed by an update of the whole multiplier vector. Thus, the proposed algorithm inherits the convergence properties of the Method of Multipliers.

[4]  arXiv:1803.06496 [pdf, ps, other]
Title: A simple algorithm for Max Cut
Comments: Submitted to Conference on March 17, 2018
Subjects: Optimization and Control (math.OC); Combinatorics (math.CO); Numerical Analysis (math.NA)

Based on an explicit equivalent continuous optimization problem, we propose a simple continuous iterative algorithm for Max Cut, which converges to a local optimum in finite steps. The inner subproblem is solved analytically and thus no optimization solver is called. Preliminary results on G-set demonstrate the performance. In particular, the ratio between the best cut values achieved by the simple algorithm without any local breakout techniques and the best known ones is of at least $0.986$.

[5]  arXiv:1803.06523 [pdf, ps, other]
Title: Stochastic model-based minimization of weakly convex functions
Comments: 9 pages
Subjects: Optimization and Control (math.OC); Learning (cs.LG)

We consider an algorithm that successively samples and minimizes stochastic models of the objective function. We show that under weak-convexity and Lipschitz conditions, the algorithm drives the expected norm of the gradient of the Moreau envelope to zero at the rate $O(k^{-1/4})$. Our result yields new complexity guarantees for the stochastic proximal point algorithm on weakly convex problems and for the stochastic prox-linear algorithm for minimizing compositions of convex functions with smooth maps. Moreover, our result also recovers the recently obtained complexity estimate for the stochastic proximal subgradient method on weakly convex problems.

[6]  arXiv:1803.06549 [pdf, ps, other]
Title: Low-Order Control Design using a Reduced-Order Model with a Stability Constraint on the Full-Order Model
Subjects: Optimization and Control (math.OC)

We consider low-order controller design for large-scale linear time-invariant dynamical systems with inputs and outputs. Model order reduction is a popular technique, but controllers designed for reduced-order models may result in unstable closed-loop plants when applied to the full-order system. We introduce a new method to design a fixed-order controller by minimizing the $L_\infty$ norm of a reduced-order closed-loop transfer matrix function subject to stability constraints on the closed-loop systems for both the reduced-order and the full-order models. Since the optimization objective and the constraints are all nonsmooth and nonconvex we use a sequential quadratic programming method based on quasi-Newton updating that is intended for this problem class, available in the open-source software package GRANSO. Using a publicly available test set, the controllers obtained by the new method are compared with those computed by the HIFOO (H-Infinity Fixed-Order Optimization) toolbox applied to a reduced-order model alone, which frequently fail to stabilize the closed-loop system for the associated full-order model.

[7]  arXiv:1803.06566 [pdf, other]
Title: Computing the Best Approximation Over the Intersection of a Polyhedral Set and the Doubly Nonnegative Cone
Subjects: Optimization and Control (math.OC)

This paper introduces an efficient algorithm for computing the best approximation of a given matrix onto the intersection of linear equalities, inequalities and the doubly nonnegative cone (the cone of all positive semidefinite matrices whose elements are nonnegative). In contrast to directly applying the block coordinate descent type methods, we propose an inexact accelerated (two-)block coordinate descent algorithm to tackle the four-block unconstrained nonsmooth dual program. The proposed algorithm hinges on the efficient semismooth Newton method to solve the subproblems, which have no closed form solutions since the original four blocks are merged into two larger blocks. The $O(1/k^2)$ iteration complexity of the proposed algorithm is established. Extensive numerical results over various large scale semidefinite programming instances from relaxations of combinatorial problems demonstrate the effectiveness of the proposed algorithm.

[8]  arXiv:1803.06573 [pdf, ps, other]
Title: On the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradient
Authors: Xingyu Zhou
Subjects: Optimization and Control (math.OC)

We provide a simple proof for the Fenchel duality between strong convexity and Lipschitz continuous gradient. To this end, we first establish equivalent conditions of convexity for a general function that may not be differentiable. By utilizing these equivalent conditions, we can directly obtain equivalent conditions for strong convexity and Lipschitz continuous gradient. Based on these results, we can easily prove Fenchel duality. Beside this main result, we also identify several conditions that are implied by strong convexity or Lipschitz continuous gradient, but are not necessarily equivalent to them. This means that these conditions are more general than strong convexity or Lipschitz continuous gradient themselves.

[9]  arXiv:1803.06600 [pdf, ps, other]
Title: Optimizing the Efficiency of First-order Methods for Decreasing the Gradient of Smooth Convex Functions
Subjects: Optimization and Control (math.OC)

This paper optimizes the step coefficients of first-order methods for smooth convex minimization in terms of the worst-case convergence bound (i.e., efficiency) of the decrease of the gradient norm. This work is based on the performance estimation problem approach. The corresponding worst-case gradient bound of the optimized method is optimal up to a constant for large-dimensional smooth convex minimization problems. This paper then illustrates that the resulting method, named OGM-G, has a computationally efficient form that is similar to the optimized gradient method (OGM).

[10]  arXiv:1803.06603 [pdf, other]
Title: Energy-aware networked control systems under temporal logic specifications
Subjects: Optimization and Control (math.OC)

In recent years, event and self-triggered control have been proposed as energy-aware control strategies to expand the life-time of battery powered devices in Networked Control Systems (NCSs). In contrast to the previous works in which their control objective is to achieve stability, this paper presents a novel energy-aware control scheme for achieving high level specifications, or more specifically, temporal logic specifications. Inspired by the standard hierarchical strategy that has been proposed in the field of formal control synthesis paradigm, we propose a new abstraction procedure for jointly synthesizing control and communication strategies, such that the communication reduction in NCSs and the satisfaction of the temporal logic specifications are guaranteed. The benefits of the proposal are illustrated through a numerical example.

[11]  arXiv:1803.06676 [pdf, other]
Title: Scenario-Based Uncertainty Set for Two-Stage Robust Energy and Reserve Scheduling: A Data-Driven Approach
Subjects: Optimization and Control (math.OC)

Two-stage robust unit commitment (RUC) models have been widely used for day-ahead energy and reserve scheduling under high renewable integration. The current state of the art relies on budget-constrained polyhedral uncertainty sets to control the conservativeness of the solutions. The associated lack of interpretability and parameter specification procedures, as well as the high computational burden exhibited by available exact solution techniques call for new approaches. In this work, we use an alternative scenario-based framework whereby uncertain renewable generation is characterized by a polyhedral uncertainty set relying on the direct specification of its vertexes. Moreover, we present a simple, yet efficient, adaptive data-driven procedure to dynamically update the uncertainty set vertexes with observed daily renewable-output profiles. Within this setting, the proposed data-driven RUC ensures protection against the convex hull of realistic scenarios empirically capturing the complex and time-varying intra-day spatial and temporal interdependencies among units. The resulting counterpart features advantageous properties from a computational perspective and can be effectively solved by the column-and-constraint generation algorithm until $\epsilon$-global optimality. Out-of-sample experiments reveal that the proposed approach is capable of producing efficient solutions in terms of cost and robustness while keeping the model tractable and scalable.

[12]  arXiv:1803.06705 [pdf, other]
Title: Hierarchical Predictive Control Algorithms for Optimal Design and Operation of Microgrids
Comments: To appear in "Power Systems Computation Conference", Dublin, Ireland
Subjects: Optimization and Control (math.OC); Systems and Control (cs.SY)

In recent years, microgrids, i.e., disconnected distribution systems, have received increasing interest from power system utilities to support the economic and resiliency posture of their systems. The economics of long distance transmission lines prevent many remote communities from connecting to bulk transmission systems and these communities rely on off-grid microgrid technology. Furthermore, communities that are connected to the bulk transmission system are investigating microgrid technologies that will support their ability to disconnect and operate independently during extreme events. In each of these cases, it is important to develop methodologies that support the capability to design and operate microgrids in the absence of transmission over long periods of time. Unfortunately, such planning problems tend to be computationally difficult to solve and those that are straightforward to solve often lack the modeling fidelity that inspires confidence in the results. To address these issues, we first develop a high fidelity model for design and operations of a microgrid that include component efficiencies, component operating limits, battery modeling, unit commitment, capacity expansion, and power flow physics; the resulting model is a mixed-integer quadratically-constrained quadratic program (MIQCQP). We then develop an iterative algorithm, referred to as the Model Predictive Control (MPC) algorithm, that allows us to solve the resulting MIQCQP. We show, through extensive computational experiments, that the MPC-based method can scale to problems that have a very long planning horizon and provide high quality solutions that lie within 5\% of optimal.

[13]  arXiv:1803.06742 [pdf, ps, other]
Title: Inventory Control with Modulated Demand and a Partially Observed Modulation Process
Subjects: Optimization and Control (math.OC)

We consider a periodic review inventory control problem having an underlying modulation process that affects demand and that is partially observed by the uncensored demand process and a novel additional observation data (AOD) process. Letting $K$ be the reorder cost, we present a condition, A1, which is a generalization of the Veinott attainability assumption, that guarantees the existence of an optimal myopic base stock policy if $K = 0$ and the existence of an optimal $(s, S)$ policy if $K > 0$, where both policies depend on the belief function of the modulation process. Assuming A1 holds, we show that (i) when $K = 0$, the value of the optimal base stock level is constant within regions of the belief space and that these regions can be described by a finite set of linear inequalities and (ii) when $K > 0$, the values of $s$ and $S$ and upper and lower bounds on these values are constant within regions of the belief space and that these regions can be described by a finite set of linear inequalities. Computational procedures for $K \geq 0$ are outlined, and results for the $K = 0$ case are presented when A1 does not hold. Special cases of this inventory control problem include problems considered in the Markov-modulated demand and Bayesian updating literatures.

[14]  arXiv:1803.06765 [pdf, ps, other]
Title: Sparse Regularization via Convex Analysis
Authors: Ivan Selesnick
Journal-ref: IEEE Transactions on Signal Processing, vol. 65, no. 17, pp. 4481-4494, 2017
Subjects: Optimization and Control (math.OC)

Sparse approximate solutions to linear equations are classically obtained via L1 norm regularized least squares, but this method often underestimates the true solution. As an alternative to the L1 norm, this paper proposes a class of non-convex penalty functions that maintain the convexity of the least squares cost function to be minimized, and avoids the systematic underestimation characteristic of L1 norm regularization. The proposed penalty function is a multivariate generalization of the minimax-concave (MC) penalty. It is defined in terms of a new multivariate generalization of the Huber function, which in turn is defined via infimal convolution. The proposed sparse-regularized least squares cost function can be minimized by proximal algorithms comprising simple computations.

[15]  arXiv:1803.06804 [pdf, ps, other]
Title: Stochastic maximum principle, dynamic programming principle, and their relationship for fully coupled forward-backward stochastic control systems
Subjects: Optimization and Control (math.OC)

In this paper, we consider stochastic optimal control problems for fully coupled forward-backward stochastic control systems with a nonconvex control domain. Within the framework of viscosity solution, the relationship between the maximum principle and dynamic programming principle is investigated, and the set inclusions among the value function and the adjoint processes are obtained. Three special cases are studied. In the first case, the value function W is supposed to be smooth. In the second case, the diffusion term {\sigma} of the forward stochastic differential equation does not include the term z. Finally, we study the local case in which the control domain is convex.

[16]  arXiv:1803.06864 [pdf, other]
Title: On the hierarchical structure of Pareto critical sets
Subjects: Optimization and Control (math.OC)

In this article we show that the boundary of the Pareto critical set of an unconstrained multiobjective optimization problem (MOP) consists of Pareto critical points of subproblems considering subsets of the objective functions. If the Pareto critical set is completely described by its boundary (e.g. if we have more objective functions than dimensions in the parameter space), this can be used to solve the MOP by solving a number of MOPs with fewer objective functions. If this is not the case, the results can still give insight into the structure of the Pareto critical set. This technique is especially useful for efficiently solving many-objective optimization problems by breaking them down into MOPs with a reduced number of objective functions.

[17]  arXiv:1803.06928 [pdf]
Title: Optimization Based Solutions for Control and State Estimation in Non-holonomic Mobile Robots: Stability, Distributed Control, and Relative Localization
Comments: This a preprint of Mohamed Said's PhD thesis
Subjects: Optimization and Control (math.OC); Robotics (cs.RO)

Interest in designing, manufacturing, and using autonomous robots has been rapidly growing during the most recent decade. The main motivation for this interest is the wide range of potential applications these autonomous systems can serve in. The applications include, but are not limited to, area coverage, patrolling missions, perimeter surveillance, search and rescue missions, and situational awareness. In this thesis, the area of control and state estimation in non-holonomic mobile robots is tackled. Herein, optimization based solutions for control and state estimation are designed, analyzed, and implemented to such systems. One of the main motivations for considering such solutions is their ability of handling constrained and nonlinear systems such as non-holonomic mobile robots. Moreover, the recent developments in dynamic optimization algorithms as well as in computer processing facilitated the real-time implementation of such optimization based methods in embedded computer systems.

[18]  arXiv:1803.06929 [pdf, other]
Title: Stochastic filtering and optimal control of pure jump Markov processes with noise-free partial observation
Comments: arXiv admin note: text overlap with arXiv:1707.07202
Subjects: Optimization and Control (math.OC)

We consider an infinite horizon optimal control problem for a pure jump Markov process $X$, taking values in a complete and separable metric space $I$, with noise-free partial observation. The observation process is defined as $Y_t = h(X_t)$, $t \geq 0$, where $h$ is a given map defined on $I$. The observation is noise-free in the sense that the only source of randomness is the process $X$ itself. The aim is to minimize a discounted cost functional. In the first part of the paper we write down an explicit filtering equation and characterize the filtering process as a Piecewise Deterministic Process. In the second part, after transforming the original control problem with partial observation into one with complete observation (the separated problem) using filtering equations, we prove the equivalence of the original and separated problems through an explicit formula linking their respective value functions. The value function of the separated problem is also characterized as the unique fixed point of a suitably defined contraction mapping.

[19]  arXiv:1803.06984 [pdf, other]
Title: Robust Optimization and Control for Electricity Generation and Transmission
Subjects: Optimization and Control (math.OC)

In this paper we present the theoretical results of solving a robust optimization problem for the power system under uncertainty. Solving the deterministic alternating current optimal power flow (ACOPF) problem has been considered a hard problem since 1960s because the optimization problem is nonlinear and highly nonconvex. Linear approximation of the AC power flow system (DC approximation) has been deployed in the industry but does not guarantee a physically feasible system configuration. In recently years, different convex relaxation schemes of the ACOPF problem have been researched, and under some assumptions, a physically feasible solution can be recovered. Based on these convex relaxation schemes, we construct a robust convex optimization problem with recourse to solve for optimal controllable injections (fossil fuel, nuclear etc.) in electricity power systems under uncertainty (renewable energy generation, demand fluctuation, etc.). We propose a cutting-plane method to solve this robust optimization problem and prove its convergence property. Extensive experiment results indicate that the robust convex relaxation of the ACOPF problem will provide a tight lower bound, and for the test cases where the nominal relaxation is tight, the convex relaxation solution can also be used for the non-convex robust ACOPF problem.

[20]  arXiv:1803.07033 [pdf, other]
Title: Natural gradient via optimal transport I
Subjects: Optimization and Control (math.OC); Information Theory (cs.IT)

We study a natural Wasserstein gradient flow on manifolds of probability distributions with discrete sample spaces. We derive the Riemannian structure for the probability simplex from the dynamical formulation of the Wasserstein distance on a weighted graph. We pull back the geometric structure to the parameter space of any given probability model, which allows us to define a natural gradient flow there. In contrast to the natural Fisher-Rao gradient, the natural Wasserstein gradient incorporates a ground metric on sample space. We discuss implementations following the forward and backward Euler methods. We illustrate the analysis on elementary exponential family examples.

[21]  arXiv:1803.07043 [pdf, other]
Title: Projective Splitting with Forward Steps: Asynchronous and Block-Iterative Operator Splitting
Subjects: Optimization and Control (math.OC); Learning (cs.LG)

This work is concerned with the classical problem of finding a zero of a sum of maximal monotone operators. For the projective splitting framework recently proposed by Combettes and Eckstein, we show how to replace the fundamental subproblem calculation using a backward step with one based on two forward steps. The resulting algorithms have the same kind of coordination procedure and can be implemented in the same block-iterative and potentially distributed and asynchronous manner, but may perform backward steps on some operators and forward steps on others. Prior algorithms in the projective splitting family have used only backward steps. Forward steps can be used for any Lipschitz-continuous operators provided the stepsize is bounded by the inverse of the Lipschitz constant. If the Lipschitz constant is unknown, a simple backtracking linesearch procedure may be used. For affine operators, the stepsize can be chosen adaptively without knowledge of the Lipschitz constant and without any additional forward steps. We close the paper by empirically studying the performance of several kinds of splitting algorithms on the lasso problem.

[22]  arXiv:1803.07053 [pdf, ps, other]
Title: Attack-Resilient H2, H-infinity, and L1 State Estimator
Subjects: Optimization and Control (math.OC)

This paper considers the secure state estimation problem for noisy systems in the presence of sparse sensor integrity attacks. We show a fundamental limitation: that is, 2r-detectability is necessary for achieving bounded estimation errors, where r is the number of attacks. This condition is weaker than the 2r-observability condition typically assumed in the literature. Conversely, we propose a real-time state estimator that achieves the fundamental limitation. The proposed state estimator is inspired by robust control and FDI: that is, it consists of local Luenberger estimators, local residual detectors, and a global fusion process. We show its performance guarantees for H2, H-infinity, and L1 systems. Finally, numerical examples show that it has relatively low estimation errors among existing algorithms and average computation time for systems with a sufficiently small number of compromised sensors.

Cross-lists for Tue, 20 Mar 18

[23]  arXiv:1803.05999 (cross-list from cs.LG) [pdf, other]
Title: Escaping Saddles with Stochastic Gradients
Subjects: Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)

We analyze the variance of stochastic gradients along negative curvature directions in certain non-convex machine learning models and show that stochastic gradients exhibit a strong component along these directions. Furthermore, we show that - contrary to the case of isotropic noise - this variance is proportional to the magnitude of the corresponding eigenvalues and not decreasing in the dimensionality. Based upon this observation we propose a new assumption under which we show that the injection of explicit, isotropic noise usually applied to make gradient descent escape saddle points can successfully be replaced by a simple SGD step. Additionally - and under the same condition - we derive the first convergence rate for plain SGD to a second-order stationary point in a number of iterations that is independent of the problem dimension.

[24]  arXiv:1803.06377 (cross-list from cs.SI) [pdf, other]
Title: Spread of Information with Confirmation Bias in Cyber-Social Networks
Subjects: Social and Information Networks (cs.SI); Multiagent Systems (cs.MA); Systems and Control (cs.SY); Optimization and Control (math.OC)

This paper provides a model to investigate information spreading over cyber-social network of agents communicating with each other. The cyber-social network considered here comprises individuals and news agencies. Each individual holds a belief represented by a scalar. Individuals receive information from news agencies that are closer to their belief, confirmation bias is explicitly incorporated into the model. The proposed dynamics of cyber-social networks is adopted from DeGroot-Friedkin model, where the individual's opinion update mechanism is a convex combination of his innate opinion, his neighbors' opinions at the previous time step (obtained from the social network), and the opinions passed along by news agencies from cyber layer which he follows. The characteristics of the interdependent social and cyber networks are radically different here: the social network relies on trust and hence static while the news agencies are highly dynamic since they are weighted as a function of the distance between an individual state and the state of news agency to account for confirmation bias. The conditions for convergence of the aforementioned dynamics to a unique equilibrium are characterized. The estimation and exact computation of the steady-state values under non-linear and linear state-dependent weight functions are provided. Finally, the impact of polarization in the opinions of news agencies on the public opinion evolution is numerically analyzed in the context of the well-known Krackhardt's advice network.

[25]  arXiv:1803.06460 (cross-list from q-fin.PM) [pdf, other]
Title: Mean Reverting Portfolios via Penalized OU-Likelihood Estimation
Comments: 7 pages, 6 figures
Subjects: Portfolio Management (q-fin.PM); Optimization and Control (math.OC); Machine Learning (stat.ML)

We study an optimization-based approach to con- struct a mean-reverting portfolio of assets. Our objectives are threefold: (1) design a portfolio that is well-represented by an Ornstein-Uhlenbeck process with parameters estimated by maximum likelihood, (2) select portfolios with desirable characteristics of high mean reversion and low variance, and (3) select a parsimonious portfolio, i.e. find a small subset of a larger universe of assets that can be used for long and short positions. We present the full problem formulation, a specialized algorithm that exploits partial minimization, and numerical examples using both simulated and empirical price data.

[26]  arXiv:1803.06510 (cross-list from stat.ML) [pdf, other]
Title: Hidden Integrality of SDP Relaxation for Sub-Gaussian Mixture Models
Subjects: Machine Learning (stat.ML); Information Theory (cs.IT); Learning (cs.LG); Optimization and Control (math.OC); Statistics Theory (math.ST)

We consider the problem of estimating the discrete clustering structures under Sub-Gaussian Mixture Models. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solutions to the SDP are not integer-valued in general, their estimation errors can be upper bounded in terms of the error of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. To the best of our knowledge, this is the first exponentially decaying error bound for convex relaxations of mixture models, and our results reveal the "global-to-local" mechanism that drives the performance of the SDP relaxation.
A corollary of our results shows that in certain regimes the SDP solutions are in fact integral and exact, improving on existing exact recovery results for convex relaxations. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of $(1-\delta)$ fraction of the points for any $\delta\in(0,1)$. As a special case, we show that under the $d$-dimensional Stochastic Ball Model, SDP achieves non-trivial (sometimes exact) recovery when the center separation is as small as $\sqrt{1/d}$, which complements previous exact recovery results that require constant separation.

[27]  arXiv:1803.06531 (cross-list from cs.SY) [pdf, ps, other]
Title: Topology Estimation using Graphical Models in Multi-Phase Power Distribution Grids
Comments: 12 pages 9 figures
Subjects: Systems and Control (cs.SY); Optimization and Control (math.OC); Machine Learning (stat.ML)

Distribution grid is the medium and low voltage part of a large power system. Structurally, the majority of distribution networks operate radially, such that energized lines form a collection of trees, i.e. forest, with a substation being at the root of any tree. The operational topology/forest may change from time to time, however tracking these changes, even though important for the distribution grid operation and control, is hindered by limited real-time monitoring. This paper develops a learning framework to reconstruct radial operational structure of the distribution grid from synchronized voltage measurements in the grid subject to the exogenous fluctuations in nodal power consumption. To detect operational lines our learning algorithm uses conditional independence tests for continuous random variables that is applicable to a wide class of probability distributions of the nodal consumption and Gaussian injections in particular. Moreover, our algorithm applies to the practical case of unbalanced three-phase power flow. Algorithm performance is validated on AC power flow simulations over IEEE distribution grid test cases.

[28]  arXiv:1803.06687 (cross-list from quant-ph) [pdf, other]
Title: Sub-Riemannian Geodesics on SU(n)/S(U(n-1)xU(1)) and Optimal Control of Three Level Quantum Systems
Subjects: Quantum Physics (quant-ph); Optimization and Control (math.OC)

We study the time optimal control problem for the evolution operator of an n-level quantum system from the identity to any desired final condition. For the considered class of quantum systems the control couples all the energy levels to a given one and is assumed to be bounded in Euclidean norm. From a mathematical perspective, such a problem is a sub-Riemannian K-P problem, whose underlying symmetric space is SU(n)/S(U(n-1) x U(1)). Following the method of symmetry reduction, we consider the action of S(U(n-1) xU(1)) on SU(n) as a conjugation X ---> AXA^{-1}. This allows us to do a symmetry reduction and consider the problem on a quotient space. We give an explicit description of such a quotient space which has the structure of a stratified space. We prove several properties of sub-Riemannian problems with the given structure. We derive the explicit optimal control for the case of three level quantum systems where the desired operation is on the lowest two energy levels (Lambda-systems). We solve this latter problem by reducing it to an integer quadratic optimization problem with linear constraints.

[29]  arXiv:1803.06689 (cross-list from quant-ph) [pdf, ps, other]
Title: Controllability of Symmetric Spin Networks
Subjects: Quantum Physics (quant-ph); Optimization and Control (math.OC)

We consider a network of n spin 1/2 systems which are pairwise interacting via Ising interaction and are controlled by the same electro-magnetic control field. Such a system presents symmetries since the Hamiltonian is unchanged if we permute two spins. This prevents full (operator) controllability in that not every unitary evolution can be obtained. We prove however that controllability is verified if we restrict ourselves to unitary evolutions which preserve the above permutation invariance. For low dimensional cases, n=2 and n=3, we provide an analysis of the Lie group of available evolutions and give explicit control laws to transfer between any two permutation invariant states. This class of states includes highly entangled states such as GHZ states and W states, which are of interest in quantum information.

[30]  arXiv:1803.06921 (cross-list from cs.SY) [pdf, ps, other]
Title: Approximating Flexibility in Distributed Energy Resources: A Geometric Approach
Comments: accepted for presentation at the Power Systems Computations Conference 2018
Subjects: Systems and Control (cs.SY); Optimization and Control (math.OC)

With increasing availability of communication and control infrastructure at the distribution systems, it is expected that the distributed energy resources (DERs) will take an active part in future power systems operations. One of the main challenges associated with integration of DERs in grid planning and control is in estimating the available flexibility in a collection of (heterogeneous) DERs, each of which may have local constraints that vary over time. In this work, we present a geometric approach for approximating the flexibility of a DER in modulating its active and reactive power consumption. The proposed method is agnostic about the type and model of the DERs, thereby facilitating a plug-and-play approach, and allows scalable aggregation of the flexibility of a collection of (heterogeneous) DERs at the distributed system level. Simulation results are presented to demonstrate the performance of the proposed method.

[31]  arXiv:1803.07055 (cross-list from cs.LG) [pdf, other]
Title: Simple random search provides a competitive approach to reinforcement learning
Comments: 22 pages, 5 figures, 9 tables
Subjects: Learning (cs.LG); Artificial Intelligence (cs.AI); Optimization and Control (math.OC); Machine Learning (stat.ML)

A common belief in model-free reinforcement learning is that methods based on random search in the parameter space of policies exhibit significantly worse sample complexity than those that explore the space of actions. We dispel such beliefs by introducing a random search method for training static, linear policies for continuous control problems, matching state-of-the-art sample efficiency on the benchmark MuJoCo locomotion tasks. Our method also finds a nearly optimal controller for a challenging instance of the Linear Quadratic Regulator, a classical problem in control theory, when the dynamics are not known. Computationally, our random search algorithm is at least 15 times more efficient than the fastest competing model-free methods on these benchmarks. We take advantage of this computational efficiency to evaluate the performance of our method over hundreds of random seeds and many different hyperparameter configurations for each benchmark task. Our simulations highlight a high variability in performance in these benchmark tasks, suggesting that commonly used estimations of sample efficiency do not adequately evaluate the performance of RL algorithms.

Replacements for Tue, 20 Mar 18

[32]  arXiv:1508.06269 (replaced) [pdf, other]
Title: A systematic process for evaluating structured perfect Bayesian equilibria in dynamic games with asymmetric information
Comments: 36 pages, 3 figures, IEEE Transactions on Automatic Control, vol. PP, no. 99, pp. 1-1, 2018
Subjects: Optimization and Control (math.OC); Computer Science and Game Theory (cs.GT); Systems and Control (cs.SY)
[33]  arXiv:1701.04102 (replaced) [pdf, ps, other]
Title: Two-stage Linear Decision Rules for Multi-stage Stochastic Programming
Subjects: Optimization and Control (math.OC)
[34]  arXiv:1703.03436 (replaced) [pdf, ps, other]
Title: Forward-Backward-Half Forward Algorithm with non Self-Adjoint Linear Operators for Solving Monotone Inclusions
Comments: 34 Pages
Subjects: Optimization and Control (math.OC)
[35]  arXiv:1710.07402 (replaced) [pdf, ps, other]
Title: The Collatz-Wielandt quotient for pairs of nonnegative operators
Authors: Shmuel Friedland
Comments: 26 pages
Subjects: Optimization and Control (math.OC)
[36]  arXiv:1710.07858 (replaced) [pdf, ps, other]
Title: Gradient flows, second order gradient systems and convexity
Subjects: Optimization and Control (math.OC)
[37]  arXiv:1712.06356 (replaced) [pdf, other]
Title: On convergence of infinite matrix products with alternating factors from two sets of matrices
Authors: Victor Kozyakin
Comments: 7 pages, 13 bibliography references, expanded Introduction and Section 4 "Remarks and Open Questions", accepted for publication in Discrete Dynamics in Nature and Society
Subjects: Optimization and Control (math.OC); Rings and Algebras (math.RA)
[38]  arXiv:1802.07529 (replaced) [pdf, ps, other]
Title: Algorithms and Convergence Results of Projection Methods for Inconsistent Feasibility Problems: A Review
Comments: We dedicate this paper to Adi Ben-Israel, our scientific father and grandfather, respectively. Revised version March 19, 2018
Subjects: Optimization and Control (math.OC)
[39]  arXiv:1803.01238 (replaced) [pdf, ps, other]
Title: Dynamic risk measure for BSVIE with jumps and semimartingale issues
Authors: Nacira Agram
Comments: 11 pages
Subjects: Optimization and Control (math.OC)
[40]  arXiv:1803.04750 (replaced) [pdf, other]
Title: Electric Vehicle Charge Scheduling Mechanism to Maximize Cost Efficiency and User Convenience
Comments: 11 pages. accepted by IEEE Transactions on Smart Grid
Subjects: Optimization and Control (math.OC)
[41]  arXiv:1505.03898 (replaced) [pdf, ps, other]
Title: Pinball Loss Minimization for One-bit Compressive Sensing: Convex Models and Algorithms
Comments: 11 pages
Subjects: Information Theory (cs.IT); Numerical Analysis (math.NA); Optimization and Control (math.OC); Machine Learning (stat.ML)
[42]  arXiv:1702.08501 (replaced) [pdf, other]
Title: Formal Synthesis of Control Strategies for Positive Monotone Systems
Comments: To appear in IEEE Transactions on Automatic Control (TAC) (2018), 16 pages, double column
Subjects: Systems and Control (cs.SY); Optimization and Control (math.OC)
[43]  arXiv:1703.02382 (replaced) [pdf, other]
Title: Assessing the Privacy Cost in Centralized Event-Based Demand Response for Microgrids
Subjects: Systems and Control (cs.SY); Cryptography and Security (cs.CR); Optimization and Control (math.OC)
[44]  arXiv:1707.05797 (replaced) [pdf]
Title: Low-complexity implementation of convex optimization-based phase retrieval
Subjects: Information Theory (cs.IT); Optimization and Control (math.OC)
[45]  arXiv:1709.00483 (replaced) [pdf, other]
Title: Iteratively Linearized Reweighted Alternating Direction Method of Multipliers for a Class of Nonconvex Problems
Subjects: Numerical Analysis (cs.NA); Computer Vision and Pattern Recognition (cs.CV); Numerical Analysis (math.NA); Optimization and Control (math.OC); Machine Learning (stat.ML)
[46]  arXiv:1709.01781 (replaced) [pdf, other]
Title: Parameterizations for Ensemble Kalman Inversion
Subjects: Numerical Analysis (math.NA); Optimization and Control (math.OC); Methodology (stat.ME)
[47]  arXiv:1712.01975 (replaced) [pdf, other]
Title: Sparsity Regularization and feature selection in large dimensional data
Subjects: Learning (cs.LG); Numerical Analysis (cs.NA); Optimization and Control (math.OC)
[48]  arXiv:1712.09203 (replaced) [pdf, ps, other]
Title: Algorithmic Regularization in Over-parameterized Matrix Sensing and Neural Networks with Quadratic Activations
Comments: clarified that the sensing matrices can be symmetric wlog and revised the quadratic neural nets section
Subjects: Learning (cs.LG); Data Structures and Algorithms (cs.DS); Optimization and Control (math.OC); Machine Learning (stat.ML)
[49]  arXiv:1803.06299 (replaced) [pdf, other]
Title: On the existence of a scalar pressure field in the Bredinger problem
Authors: Aymeric Baradat
Subjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC); Probability (math.PR)
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