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Yajing Liu

Publications and source records attributed to Yajing Liu.

31 records · Page 2Linked to original sources
arXiv2020

Model-Free State Estimation Using Low-Rank Canonical Polyadic Decomposition

As electric grids experience high penetration levels of renewable generation, fundamental changes are required to address real-time situational awareness. This paper uses unique traits of tensors to devise a model-free situational awareness and energy forecasting framework for distribution networks. This work formulates the state of the network at multiple time instants as a three-way tensor; hence, recovering full state information of the network is tantamount to estimating all the values of the tensor. Given measurements received from $μ$phasor measurement units and/or smart meters, the recovery of unobserved quantities is carried out using the low-rank canonical polyadic decomposition of the state tensor---that is, the state estimation task is posed as a tensor imputation problem utilizing observed patterns in measured quantities. Two structured sampling schemes are considered: slab sampling and fiber sampling. For both schemes, we present sufficient conditions on the number of sampled slabs and fibers that guarantee identifiability of the factors of the state tensor. Numerical results demonstrate the ability of the proposed framework to achieve high estimation accuracy in multiple sampling scenarios.

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arXiv2019

Decentralized Low-Rank State Estimation for Power Distribution Systems

This paper considers the low-observability state estimation problem in power distribution networks and develops a decentralized state estimation algorithm leveraging the matrix completion methodology. Matrix completion has been shown to be an effective technique in state estimation that exploits the low dimensionality of the power system measurements to recover missing information. This technique can utilize an approximate (linear) load flow model, or it can be used with no physical models in a network where no information about the topology or line admittance is available. The direct application of matrix completion algorithms requires solving a semi-definite programming (SDP) problem, which becomes infeasible for large networks. We therefore develop a decentralized algorithm that capitalizes on the popular proximal alternating direction method of multipliers (proximal ADMM). The method allows us to distribute the computation among different areas of the network, thus leading to a scalable algorithm. By doing all computations at individual control areas and only communicating with neighboring areas, the algorithm eliminates the need for data to be sent to a central processing unit and thus increases efficiency and contributes to the goal of autonomous control of distribution networks. We illustrate the advantages of the proposed algorithm numerically using standard IEEE test cases.

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arXiv2019

Submodular Optimization Problems and Greedy Strategies: A Survey

The greedy strategy is an approximation algorithm to solve optimization problems arising in decision making with multiple actions. How good is the greedy strategy compared to the optimal solution? In this survey, we mainly consider two classes of optimization problems where the objective function is submodular. The first is set submodular optimization, which is to choose a set of actions to optimize a set submodular objective function, and the second is string submodular optimization, which is to choose an ordered set of actions to optimize a string submodular function. Our emphasis here is on performance bounds for the greedy strategy in submodular optimization problems. Specifically, we review performance bounds for the greedy strategy, more general and improved bounds in terms of curvature, performance bounds for the batched greedy strategy, and performance bounds for Nash equilibria.

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arXiv2019

Matrix Completion for Low-Observability Voltage Estimation

With the rising penetration of distributed energy resources, distribution system control and enabling techniques such as state estimation have become essential to distribution system operation. However, traditional state estimation techniques have difficulty coping with the low-observability conditions often present on the distribution system due to the paucity of sensors and heterogeneity of measurements. To address these limitations, we propose a distribution system state estimation algorithm that employs matrix completion (a tool for estimating missing values in low-rank matrices) augmented with noise-resilient power flow constraints. This method operates under low-observability conditions where standard least-squares-based methods cannot operate, and flexibly incorporates any network quantities measured in the field. We empirically evaluate our method on the IEEE 33- and 123-bus test systems, and find that it provides near-perfect state estimation performance (within 1\% mean absolute percent error) across many low-observability data availability regimes.

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arXiv2018

Improved Bounds for the Greedy Strategy in Optimization Problems with Curvature

Consider the problem of choosing a set of actions to optimize an objective function that is a real-valued polymatroid function subject to matroid constraints. The greedy strategy provides an approximate solution to the optimization problem, and it is known to satisfy some performance bounds in terms of the total curvature. The total curvature depends on the value of objective function on sets outside the constraint matroid. If we are given a function defined only on the matroid, the problem still makes sense, but the existing bounds involving the total curvature do not apply. This is puzzling: If the optimization problem is perfectly well defined, why should the bounds no longer apply? This motivates an alternative formulation of such bounding techniques. The first question that comes to mind is whether it is possible to extend a polymatroid function defined on a matroid to one on the entire power set. This was recently shown to be negative in general. Here, we provide necessary and sufficient conditions for the existence of an \emph{incremental} extension of a polymatroid function defined on the uniform matroid of rank $k$ to one defined on the uniform matroid of rank $k+1$, together with an algorithm for constructing the extension. Whenever a polymatroid objective function defined on a matroid can be extended to the entire power set, the greedy approximation bounds involving the total curvature of the extension apply. However, these bounds still depend on sets outside the constraint matroid. Motivated by this, we define a new notion of curvature called \emph{partial curvature}, involving only sets in the matroid. We derive necessary and sufficient conditions for an extension of the function to have a total curvature that is equal to the partial curvature. Moreover, we prove that the bounds in terms of the partial curvature are in general improved over the previous ones......

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arXiv2017

Performance Bounds for Nash Equilibria in Submodular Utility Systems with User Groups

In this paper, we consider variations of the utility system considered by Vetta, in which users are grouped together. Our aim is to establish how grouping and cooperation among users affect performance bounds. We consider two types of grouping. The first type is from \cite{Zhang2014}, where each user belongs to a group of users having social ties with it. For this type of utility system, each user's strategy maximizes its social group utility function, giving rise to the notion of \emph{social-aware Nash equilibrium}. We prove that this social utility system yields to the bounding results of Vetta for non-cooperative system, thus establishing provable performance guarantees for the social-aware Nash equilibrium. For the second type of grouping, the set of users is partitioned into $l$ disjoint groups, where the users within a group cooperate to maximize their group utility function, giving rise to the notion of \emph{group Nash equilibrium}. In this case, each group can be viewed as a new user with vector-valued actions, and a 1/2 bound for the performance of group Nash equilibrium follows from the result of Vetta. But as we show tighter bounds involving curvature can be established. By defining the group curvature $c_{k_i}$ associated with group $i$ with $k_i$ users, we show that if the social utility function is nondecreasing and submodular, then any group Nash equilibrium achieves at least $1/(1+\max_{1\leq i\leq l}c_{k_i})$ of the optimal social utility, which is tighter than that for the case without grouping. As a special case, if each user has the same action space, then we have that any group Nash equilibrium achieves at least $1/(1+c_{k^*})$ of the optimal social utility, where $k^*$ is the least number of users among the $l$ groups. Finally, we present an example of a utility system for database assisted spectrum access to illustrate our results.

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arXiv2017

Performance Bounds with Curvature for Batched Greedy Optimization

The batched greedy strategy is an approximation algorithm to maximize a set function subject to a matroid constraint. Starting with the empty set, the batched greedy strategy iteratively adds to the current solution set a batch of elements that results in the largest gain in the objective function while satisfying the matroid constraints. In this paper, we develop bounds on the performance of the batched greedy strategy relative to the optimal strategy in terms of a parameter called the total batched curvature. We show that when the objective function is a polymatroid set function, the batched greedy strategy satisfies a harmonic bound for a general matroid constraint and an exponential bound for a uniform matroid constraint, both in terms of the total batched curvature. We also study the behavior of the bounds as functions of the batch size. Specifically, we prove that the harmonic bound for a general matroid is nondecreasing in the batch size and the exponential bound for a uniform matroid is nondecreasing in the batch size under the condition that the batch size divides the rank of the uniform matroid. Finally, we illustrate our results by considering a task scheduling problem and an adaptive sensing problem.

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arXiv2016

Performance Bounds for the $k$-Batch Greedy Strategy in Optimization Problems with Curvature

The $k$-batch greedy strategy is an approximate algorithm to solve optimization problems where the optimal solution is hard to obtain. Starting with the empty set, the $k$-batch greedy strategy adds a batch of $k$ elements to the current solution set with the largest gain in the objective function while satisfying the constraints. In this paper, we bound the performance of the $k$-batch greedy strategy with respect to the optimal strategy by defining the total curvature $α_k$. We show that when the objective function is nondecreasing and submodular, the $k$-batch greedy strategy satisfies a harmonic bound $1/(1+α_k)$ for a general matroid constraint and an exponential bound $\left(1-(1-α_k/{t})^t\right)/α_k$ for a uniform matroid constraint, where $k$ divides the cardinality of the maximal set in the general matroid, $t=K/k$ is an integer, and $K$ is the rank of the uniform matroid. We also compare the performance of the $k$-batch greedy strategy with that of the $k_1$-batch greedy strategy when $k_1$ divides $k$. Specifically, we prove that when the objective function is nondecreasing and submodular, the $k$-batch greedy strategy has better harmonic and exponential bounds in terms of the total curvature. Finally, we illustrate our results by considering a task-assignment problem.

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arXiv2016

Bounding the Greedy Strategy in Finite-Horizon String Optimization

We consider an optimization problem where the decision variable is a string of bounded length. For some time there has been an interest in bounding the performance of the greedy strategy for this problem. Here, we provide weakened sufficient conditions for the greedy strategy to be bounded by a factor of $(1-(1-1/K)^K)$, where $K$ is the optimization horizon length. Specifically, we introduce the notions of $K$-submodularity and $K$-GO-concavity, which together are sufficient for this bound to hold. By introducing a notion of \emph{curvature} $η\in(0,1]$, we prove an even tighter bound with the factor $(1/η)(1-e^{-η})$. Finally, we illustrate the strength of our results by considering two example applications. We show that our results provide weaker conditions on parameter values in these applications than in previous results.

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arXiv2015

$L$-space surgeries on links

An $L$-space link is a link in $S^3$ on which all large surgeries are $L$-spaces. In this paper, we initiate a general study of the definitions, properties, and examples of $L$-space links. In particular, we find many hyperbolic $L$-space links, including some chain links and two-bridge links; from them, we obtain many hyperbolic $L$-spaces by integral surgeries, including the Weeks manifold. We give bounds on the ranks of the link Floer homology of $L$-space links and on the coefficients in the multi-variable Alexander polynomials. We also describe the Floer homology of surgeries on any $L$-space link using the link surgery formula of Ozsváth and Manolescu. As applications, we compute the graded Heegaard Floer homology of surgeries on 2-component $L$-space links in terms of only the Alexander polynomial and the surgery framing, and give a fast algorithm to classify $L$-space surgeries among them.

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arXiv2014

Heegaard Floer homology of surgeries on two-bridge links

We give an $O(p^{2})$ time algorithm to compute the generalized Heegaard Floer complexes $A_{s_{1},s_{2}}^{-}(\overrightarrow{L})$'s for a two-bridge link $\overrightarrow{L}=b(p,q)$ by using nice diagrams. Using the link surgery formula of Manolescu-Ozsváth, we also show that ${\bf HF}^{-}$ and their $d$-invariants of all integer surgeries on two-bridge links are determined by $A_{s_{1},s_{2}}^{-}(\overrightarrow{L})$'s. We obtain a polynomial time algorithm to compute ${\bf HF}^{-}$ of all the surgeries on two-bridge links, with $\mathbb{Z}/2\mathbb{Z}$ coefficients. In addition, we calculate some examples explicitly: ${\bf HF}^{-}$ and the $d$-invariants of all integer surgeries on a family of hyperbolic two-bridge links including the Whitehead link.

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arXiv2014

Guaranteed Bounds for General Approximate Dynamic Programming

In this paper, we will develop a systematic approach to deriving guaranteed bounds for approximate dynamic programming (ADP) schemes in optimal control problems. Our approach is inspired by our recent results on bounding the performance of greedy strategies in optimization of string-submodular functions over a finite horizon. The approach is to derive a string-submodular optimization problem, for which the optimal strategy is the optimal control solution and the greedy strategy is the ADP solution. Using this approach, we show that any ADP solution achieves a performance that is at least a factor of $β$ of the performance of the optimal control solution, which satisfies Bellman's optimality principle. The factor $β$ depends on the specific ADP scheme, as we will explicitly characterize. To illustrate the applicability of our bounding technique, we present examples of ADP schemes, including the popular rollout method.

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