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Near-Optimal Rapid MPC Using Neural Networks: A Primal-Dual Policy Learning Framework

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 2021年第5期29卷 2102-2114页

作者： Zhang, Xiaojing Bujarbaruah, Monimoy Borrelli, Francesco Univ Calif Berkeley Model Predict Control Lab Berkeley CA 94720 USA

In this article, we propose a novel framework for approximating the MPC policy for linear parameter-varying systems using supervised learning. Our learning scheme guarantees feasibility and near-optimality of the approximated MPC policy with high probability. Furthermore, in contrast to most existing approaches that only learn the MPC policy, we also learn the "dual policy," which enables us to keep a check on the approximated MPC's optimality online during the control process. If the check deems the control input from the approximated MPC policy safe and near-optimal, then it is applied to the plant;otherwise, a backup controller is invoked, thus filtering out (severely) suboptimal control inputs. The backup controller is only invoked with a bounded (low) probability, where the exact probability level can be chosen by the user. Since our framework does not require solving any optimization problem during the control process, it enables the deployment of MPC on resource-constrained systems. Specifically, we illustrate the utility of the proposed framework on a vehicle dynamics control problem. Compared with online optimization methods, we demonstrate a speedup of up to 62x on a desktop computer and 10x on an automotive-grade electronic control unit, while maintaining high control performance.

关键词： Deep neural networks (DNNs) explicit model predictive control model predictive control (MPC) policy learning randomized algorithms safe learning sample complexity

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Polynomial-time algorithms for probabilistic solutions of parameter-dependent linear matrix inequalities

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AUTOMATICA 2007年第3期43卷 538-545页

作者： Oishi, Yasuaki Univ Tokyo Grad Sch Informat Sci & Technol Dept Math Informat Bunkyo Ku Tokyo 1138656 Japan

A randomized approach is considered for a feasibility problem on a parameter-dependent linear matrix inequality (LMI). In particular, a gradient-based and an ellipsoid-based randomized algorithms are improved by introduction of a stopping rule. The improved algorithms stop after a bounded number of iterations and this bound is of polynomial order in the problem size. When the algorithms stop, either of the following two events occurs: (i) they find with high confidence a probabilistic solution, which satisfies the given LMI for most of the parameter values;(ii) they detect in an approximate sense the non-existence of a deterministic solution, which satisfies the given LMI for all the parameter values. These results are important because the original randomized algorithms have issues to be settled on detection of convergence, on the speed of convergence, and on the assumption of feasibility. The improved algorithms can be adapted for an optimization problem constrained by a parameter-dependent LMI. A numerical example shows the efficacy of the proposed algorithms. (C) 2007 Elsevier Ltd. All rights reserved.

关键词： randomized algorithms parameter-dependent linear matrix inequalities stopping rules probabilistic solutions computational complexity optimization

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A new approximation algorithm for the multilevel facility location problem

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DISCRETE APPLIED MATHEMATICS 2010年第5期158卷 453-460页

作者： Gabor, Adriana F. van Ommeren, Jan-Kees C. W. Erasmus Univ Inst Econometr NL-3000 DR Rotterdam Netherlands Univ Twente Fac Elect Engn Math & Comp Sci NL-7500 AE Enschede Netherlands

In this paper we propose a new integer programming formulation for the multilevel facility location problem and a novel 3-approximation algorithm based on LP-rounding. The linear program that we use has a polynomial number of variables and constraints, thus being more efficient than the one commonly used in the approximation algorithms for these types of problems. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Facility location Approximation algorithms randomized algorithms

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Attenuate Locally, Win Globally: Attenuation-Based Frameworks for Online Stochastic Matching with Timeouts

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ALGORITHMICA 2020年第1期82卷 64-87页

作者： Brubach, Brian Sankararaman, Karthik A. Srinivasan, Aravind Xu, Pan Univ Maryland College Pk MD 20742 USA New Jersey Inst Technol Dept Comp Sci Newark NJ 07102 USA

Online matching problems have garnered significant attention in recent years due to numerous applications in e-commerce, online advertisements, ride-sharing, etc. Many of them capture the uncertainty in the real world by including stochasticity in both the arrival and matching processes. The online stochastic matching with timeouts problem introduced by Bansal et al. (Algorithmica, 2012) models matching markets (e.g., E-Bay, Amazon). Buyers arrive from an independent and identically distributed (i.i.d.) known distribution on buyer profiles and can be shown a list of items one at a time. Each buyer has some probability of purchasing each item and a limit (timeout) on the number of items they can be shown. Bansal et al. (Algorithmica, 2012) gave a 0.12-competitive algorithm which was improved by Adamczyk et al. (ESA, 2015) to 0.24. We present several online attenuation frameworks that use an algorithm for offline stochastic matching as a black box. On the upper bound side, we show that one framework, combined with a black-box adapted from Bansal et al. (Algorithmica, 2012), yields an online algorithm which nearly doubles the ratio to 0.46. Additionally, our attenuation frameworks extend to the more general setting of fractional arrival rates for online vertices. On the lower bound side, we show that no algorithm can achieve a ratio better than 0.632 using the standard LP for this problem. This framework has a high potential for further improvements since new algorithms for offline stochastic matching can directly improve the ratio for the online problem. Our online frameworks also have the potential for a variety of extensions. For example, we introduce a natural generalization: online stochastic matching with two-sided timeouts in which both online and offline vertices have timeouts. Our frameworks provide the first algorithm for this problem achieving a ratio of 0.30. We once again use the algorithm of Bansal et al. (Algorithmica, 2012) as a black-box and plug it in

关键词： Online algorithms randomized algorithms Bipartite matching

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Quasi-fully dynamic algorithms for two-connectivity and cycle equivalence

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ALGORITHMICA 2002年第2期33卷 168-182页

作者： Korupolu, MR Ramachandran, V Univ Texas Dept Comp Sci Austin TX 78712 USA

We introduce a new class of dynamic graph algorithms called quasi-fully dynamic algorithms, which are much more general than backtracking algorithms and are much simpler than fully dynamic algorithms. These algorithms are especially suitable for applications in which a certain core connected portion of the graph remains fixed, and fully dynamic updates occur on the remaining edges in the graph, We present very simple quasi-fully dynamic algorithms with O(log n) worst-case time per operation for edge connectivity and O(log n) amortized time per operation for cycle equivalence, The former is deterministic while the latter is Monte-Carlo-type randomized. For 2-vertex connectivity, we give a deterministic quasi-fully dynamic algorithm with O(log(3)n) amortized time per operation. The quasi-fully dynamic algorithm we present for cycle equivalence (which has several applications in optimizing compilers) is of special interest since the algorithm is quite simple, and no special-purpose incremental or backtracking algorithm is known for this problem.

关键词： dynamic algorithms graph algorithms cycle equivalence connected graphs edge and vertex connectivity randomized algorithms compiler optimization

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A novel randomised particle swarm optimizer

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INTERNATIONAL JOURNAL OF MACHINE LEARNING AND CYBERNETICS 2021年第2期12卷 529-540页

作者： Liu, Weibo Wang, Zidong Zeng, Nianyin Yuan, Yuan Alsaadi, Fuad E. Liu, Xiaohui Shandong Univ Sci & Technol Coll Elect Engn & Automat Qingdao 266590 Peoples R China Brunel Univ London Dept Comp Sci Uxbridge UB8 3PH Middx England Xiamen Univ Dept Instrumental & Elect Engn Xiamen 361005 Fujian Peoples R China Northwestern Polytech Univ Sch Astronaut Xian 710072 Peoples R China King Abdulaziz Univ Fac Engn Dept Elect & Comp Engn Jeddah 21589 Saudi Arabia

The particle swarm optimization (PSO) algorithm is a popular evolutionary computation approach that has received an ever-increasing interest in the past decade owing to its wide application potential. Despite the many variants of the PSO algorithm with improved search ability by means of both the convergence rate and the population diversity, the local optima problem remains a major obstacle that hinders the global optima from being found. In this paper, a novel randomized particle swarm optimizer (RPSO) is proposed where the Gaussian white noise with adjustable intensity is utilized to randomly perturb the acceleration coefficients in order for the problem space to be explored more thoroughly. With this new strategy, the RPSO algorithm not only maintains the population diversity but also enhances the possibility of escaping the local optima trap. Experimental results demonstrate that the proposed RPSO algorithm outperforms some existing popular variants of PSO algorithms on a series of widely used optimization benchmark functions.

关键词： randomized algorithms Evolutionary computation Particle swarm optimization Gaussian white noise Acceleration coefficients

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Testing membership in languages that have small width branching programs

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SIAM JOURNAL ON COMPUTING 2002年第5期31卷 1557-1570页

作者： Newman, I Univ Haifa Dept Comp Sci IL-31905 Haifa Israel

Combinatorial property testing, initiated formally by Goldreich, Goldwasser, and Ron in [ J. ACM, 45 (1998), pp. 653-750] and inspired by Rubinfeld and Sudan [SIAM J. Comput., 25 (1996), pp. 252-271], deals with the following relaxation of decision problems: Given a fixed property and an input x, one wants to decide whether x has the property or is "far" from having the property. The main result here is that, if G = {g(n) : {0, 1}(n) --> {0, 1}} is a family of Boolean functions which have oblivious read-once branching programs of width w, then, for every n and epsilon > 0, there is a randomized algorithm that always accepts every x is an element of {0, 1}(n) if g(n) (x) = 1 and rejects it with high probability if at least epsilonn bits of x should be modified in order for it to be in g(n)(-1)(1). The algorithm makes (2(w)/epsilon)O-(w) queries. In particular, for constant epsilon and w, the query complexity is O(1). This generalizes the results of Alon et al. [Proceedings of the 40th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, 1999, pp. 645 655] asserting that regular languages are epsilon-testable for every epsilon > 0.

关键词： property testing randomized algorithms branching programs

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On the Nystrom method for approximating a gram matrix for improved kernel-based learning

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JOURNAL OF MACHINE LEARNING RESEARCH 2005年第12期6卷 2153-2175页

作者： Drineas, P Mahoney, MW Rensselaer Polytech Inst Dept Comp Sci Troy NY 12180 USA Yale Univ Dept Math New Haven CT 06520 USA

A problem for many kernel-based methods is that the amount of computation required to find the solution scales as O(n(3)), where n is the number of training examples. We develop and analyze an algorithm to compute an easily-interpretable low-rank approximation to an n x n Gram matrix G such that computations of interest may be performed more rapidly. The approximation is of the form (G) over tilde (k) = CWk+ C-T, where C is a matrix consisting of a small number c of columns of G and W-k is the best rank-k approximation to W, the matrix formed by the intersection between those c columns of G and the corresponding c rows of G. An important aspect of the algorithm is the probability distribution used to randomly sample the columns;we will use a judiciously-chosen and data-dependent nonuniform probability distribution. Let parallel to(.)parallel to(2) and parallel to(.)parallel to(F) denote the spectral norm and the Frobenius norm, respectively, of a matrix, and let G(k) be the best rank-k approximation to G. We prove that by choosing O(k/epsilon(4)) columns parallel to G-(CWk+CT)parallel to(xi) <=parallel to G-G(k)parallel to(xi)+epsilon Sigma(n)(i=1) G(ii)(2) both in expectation and with high probability, for both xi = 2, F, and for all k : 0 <= k <= rank(W). This approximation can be computed using O(n) additional space and time, after making two passes over the data from external storage. The relationships between this algorithm, other related matrix decompositions, and the Nystrom method from integral equation theory are discussed.

关键词： kernel methods randomized algorithms Gram matrix Nystrom method

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Special cases of the minimum spanning tree problem under explorable edge and vertex uncertainty

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NETWORKS 2024年第3期83卷 587-604页

作者： Mathwieser, Corinna Cela, Eranda Rhein Westfal TH Aachen Lehr & Forschungsgebiet Kombinator Optimierung D-52062 Aachen Germany Graz Univ Technol Dept Discrete Math Graz Austria

This article studies the Minimum Spanning Tree Problem under Explorable Uncertainty as well as a related vertex uncertainty version of the problem. We particularly consider special instance types, including cactus graphs, for which we provide randomized algorithms. We introduce the problem of finding a minimum weight spanning star under uncertainty for which we show that no algorithm can achieve constant competitive ratio.

关键词： cactus graphs competitive analysis explorable uncertainty online algorithms randomized algorithms spanning tree

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Implementation of the simultaneous perturbation algorithm for stochastic optimization

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IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS 1998年第3期34卷 817-823页

作者： Spall, JC Johns Hopkins Univ Appl Phys Lab Laurel MD 20723 USA

The need for solving multivariate optimization problems is pervasive in engineering and the physical and social sciences. The simultaneous perturbation stochastic approximation (SPSA) algorithm has recently attracted considerable attention for challenging optimization problems where it is difficult or impossible to directly obtain a gradient of the objective function with respect to the parameters being optimized. SPSA is based on an easily implemented and highly efficient gradient approximation that relies on measurements of the objective function, not on measurements of the gradient of the objective function. The gradient approximation is based on only two function measurements (regardless of the dimension of the gradient vector). This contrasts with standard finite-difference approaches, which require a number of function measurements proportional to the dimension of the gradient vector. This paper presents a simple step-by-step guide to implementation of SPSA in generic optimization problems and offers some practical suggestions for choosing certain algorithm coefficients.

关键词： approximation theory iterative methods mathematics computing optimisation perturbation techniques randomised algorithms stochastic processes MATLAB code algorithm coefficients efficient gradient approximation gain sequences generic optimization problems dimensions randomized algorithms Stochastic Processes Approximation Theory iterative methods Markov chain Objective function perturbation techniques Gradient vector

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