Approximation Algorithms for Optimization of Combinatorial Dynamical Systems

Given a mixed 0–1 linear program in n binary variables, we studied in Chapter 2 the construction of an n + 1 level hierarchy of polyhedral approximations that ranges from the usual continuous relaxation at level 0 to an explicit algebraic representation of the convex hull of feasible solutions at level-n. While exponential in both the number of variables and constraints, level-n was shown in the previous chapter to serve as a useful tool for promoting valid inequalities and facets via a projection operation onto the original variable space. In this chapter, we once again invoke the convex hull representation at level-n, this time to provide a direct linkage between discrete sets and their polyhedral relaxations. Specifically, we study conditions under which a binary variable realizing a value of either 0 or 1 in an optimal solution to the linear programming relaxation of a mixed 0–1 linear program will persist in maintaining that same value in some discrete optimum.KeywordsDual SolutionContinuous RelaxationComplementary SlacknessOptimal Dual SolutionPersistency PropertyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper presents new linear programming techniques for the design of optimal 1D IIR digital filters. the first method is based on a linear minimax criterion and leads to a linear programming approximation problem whose optimal solution is attained by a new fast algorithm. Next, the linear minimax approximation problem is extended and formulated as two linear integer programming problems which permit the design of 1D IIR digital filters with coefficients of finite word length. the first of these methods is formulated as a 0‐1 integer linear programming problem and its optimal solution is attained rapidly by a new algorithm. This method appears to be suitable for the design of high‐order digital filters. the second method is more general and is formulated as a mixed integer linear programming problem. to solve the design problem efficiently, a new integer tree search algorithm is introduced. the feasibility of the proposed algorithms is illustrated with detailed solutions or numerical examples.

This thesis treats different aspects of the class of Mixed-Integer Optimal Control Problems (MIOCPs). These are optimization problems that combine the difficulties of underlying dynamic processes with combinatorial decisions. Typically, these combinatorial decisions are realized as switching decisions between the system’s different operations modes. During the last decades, direct methods emerged as the state-of-the-art solvers for MIOCPs. The formulation of a valid, tight and dependable integral relaxation, i.e., the formulation of a model for fractional values, plays an important role for these direct solution methods. We give detailed insight into several relaxation approaches for MIOCPs and compare them with regard to their respective structures. In particular, these are the typical solution’s structures and properties as convexity, problem size and numerical behavior. From these structural properties, we deduce some required specifications of a solver. Additionally, the modeling and subsequent limitation of the switching process directly tackle the class-specific typical issue of chattering solutions. One of the relaxation methods for MIOCPs is the outer convexification, where the binary variables only enter affinely. For the approximation of this relaxation’s solution, we took up on the control approximation problem in integral sense derived by Sager as part of a decomposition approach for MIOCPs with affine binary controls. This problem describes the optimal approximation of fractional controls with binary controls such that the corresponding dynamic process is changed as little as possible. For the multi-dimensional problem, we developed a new heuristic, which for the first time gives a bound that only depends on the control grid and not anymore on the number of the system’s controls. For the generalization of the control approximation problem with additional constraints, we derived a tailored branch-and-bound algorithm, which is based on the properties of the Lagrangian relaxation of the one-dimensional problem. This algorithm beats state-of-the-art commercial solvers for Mixed-Integer Linear Programs (MILPs) for this special approximation problem by several orders of magnitude. Overall, we present several, partially new modeling approaches for MIOCPs together with the accompanying structural properties. On this basis, we develop new theories for the approximation of certain relaxed solutions. We discuss the efficient implementation of the resulting structure exploiting algorithms. This leads to a deeper and better understanding of MIOCPs. We show the practicability of the theoretical observations with the help of four prototypical problems. The presented methods and algorithms allow on their basis the direct development of decision support and analysis tools in practice.

An effective linear approximation method for separable programming problems

Article Free Access Share on Exact sampling and approximate counting techniques Author: Mark Huber School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NYView Profile Authors Info & Claims STOC '98: Proceedings of the thirtieth annual ACM symposium on Theory of computingMay 1998 Pages 31–40https://doi.org/10.1145/276698.276709Published:23 May 1998Publication History 25citation666DownloadsMetricsTotal Citations25Total Downloads666Last 12 Months66Last 6 weeks15 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

— A gradient learning method to regulate the trajectories of some nonlinear chaotic systems is proposed. The method is motivated by the gradient descent learning algorithms for neural networks. It is based on two systems: dynamic optimization system and system for finding sensitivities. Numerical results of several examples are presented, which convincingly illustrate the efficiency of the method. Keywords — Chaos, Dynamical Systems, Learning, Neural Networks I. I NTRODUCTION H E study of controlling chaos in non-linear deterministic systems has been in focus during the last two decades. The ability to bring chaotic dynamical systems to regular motions is an important subject. The existing chaos control algorithms can be classified mainly into two categories: feed-back methods [1 - 4] and non-feedback methods [5, 6]. In [7, 8] the authors showed that adding noise to the excitation frequency of the Duffing system the initially chaotic motion becomes regular. In the present paper we propose a gradient learning method for adjusting the system parameters so that the trajectory of the system has certain specified properties. In the theory of neural networks the gradient descent learning requires that the performance of a dynamical system is assessable through certain error function which measures the discrepancy between the trajectories of the dynamical system and the desired behavior [9]. During gradient learning the interconnection weights between neurons are iteratively adjusted to reduce the error. In the present work we consider the system coefficients as adjustable parameters to obtain the desired behavior of the dynamical system. II. PROBLEMSTATEMENTConsider the dynamical system F u w I u t

Efficient Multi-Purpose Algorithm for Approximation and Alignment Problems in Coordinate Measurement Techniques

The distribution system restoration (DSR) problem is traditionally modeled as a mixed-integer linear programming (MILP) model. However, a significant number of integer variables are introduced to describe the DSR process, which introduce additional complexities in both time and space dimensions. Moreover, an enormous number of constraints are constructed to establish a rational DSR decision, while some of them may not be considered tight in practice. The enormous number of binary variables and inactive constraints could make the DSR problem very hard to solve and apply in real-time. The DSR computation burden would be reduced significantly if binary variables and binding constraints are pre-determined. This paper proposes an acceleration framework and solution algorithm based on the end-to-end optimization, which applies deep neural network (DNN) and gradient boosting decision tree (GBDT) methods to DSR. The DSR problem, which is solved in offline and online stages, will accordingly be reduced to a linear programming problem which can be solved more efficiently and reliably. Case studies are carried out on the modified IEEE 33-bus and 123-bus systems and a practical 1069-bus system. The proposed results indicate that the DSR problem with the proposed end-to-end acceleration framework is solved more than tenfold faster than those of traditional solvers.

Robust optimization of nonlinear dynamic systems with application to a jacketed tubular reactor

In this paper we introduce a new general framework for set covering problems, based on the combination of randomized rounding of the (near-)optimal solution of the linear programming (LP) relaxation, leading to a partial integer solution, and the application of a well-behaved approximation algorithm to complete this solution. If the value of the solution returned by the latter can be bounded in a suitable way, as is the case for the most relevant generalizations of bin packing, the method leads to improved approximation guarantees, along with a proof of tighter integrality gaps for the LP relaxation. Applying our general framework we obtain a polynomial-time randomized algorithm for d-dimensional vector packing with approximation guarantee arbitrarily close to ln d + 1. For d = 2, this value is 1.693 ..., i.e., we break the natural 2 "barrier" for this case. Moreover, for small values of d this is a notable improvement over the previously-known O(ln d) guarantee by Chekuri and Khanna (2004). For 2-dimensional bin packing with and without rotations, we construct algorithms with performance guarantee arbitrarily close to 1.525..., improving upon previous algorithms with performance guarantee of 2 + epsiv by Jansen and Zhang (2004) for the problem with rotations and1.691... by Caprara (2002) for the problem without rotations. The previously-unknown key property used in our proofs follows from a retrospective analysis of the implications of the landmark bin packing approximation scheme by Fernandez de la Vega and Lueker (1981). We prove that their approximation scheme is "subset oblivious", which leads to numerous applications. Another byproduct of our paper is an algorithm that solves a well-known configuration LP for 2-dimensional bin packing within a factor of (1 + epsiv) for any epsiv gt; 0. Interestingly, we do it without using an approximate separation oracle, which would correspond to a well-known geometric 2-dimensional knapsack. Although separation and optimization are equivalent (M. Grotschel et al, 1988) and the existence of an approximation scheme for the separation problem remains open, we are able to design an approximation scheme for the configuration LP since its objective function is unweighed

Hydropower producers must submit bids to electricity market auctions where they state their willingness to produce power. These bids may be determined using a mixed-integer linear stochastic program. However, for large interconnected river systems, this program may be too complex to be solved within the time limits set by current market rules. This paper investigates whether a linear approximation to start-ups can be used to reduce the computational burden without significantly degrading the solution quality. In order to investigate the trade-off of time versus solution quality, linear approximation is compared to a formulation that uses binary variables in a case study that simulates the operation of a reservoir system over time.

Solving linear programs is often a challenging task in distributed settings. While there are good algorithms for solving packing and covering linear programs in a distributed manner (Kuhn et al. 2006), this is essentially the only class of linear programs for which such an algorithm is known. In this work we provide a distributed algorithm for solving a different class of convex programs which we call “distance-bounded network design convex programs”. These can be thought of as relaxations of network design problems in which the connectivity requirement includes a distance constraint (most notably, graph spanners). Our algorithm runs in O((D/e) log n) rounds in the LOCAL model and with high probability finds a (1+e)-approximation to the optimal LP solution for any 0 < e ≤ 1, where D is the largest distance constraint. While solving linear programs in a distributed setting is interesting in its own right, this class of convex programs is particularly important because solving them is often a crucial step when designing approximation algorithms. Hence we almost immediately obtain new and improved distributed approximation algorithms for a variety of network design problems, including Basic 3- and 4-Spanner, Directed k-Spanner, Lowest Degree k-Spanner, and Shallow-Light Steiner Network Design with a spanning demand graph. Our algorithms do not require any “heavy” computation and essentially match the best-known centralized approximation algorithms, while previous approaches which do not use heavy computation give approximations which are worse than the best-known centralized bounds.

A local search approximation algorithm for k-means clustering

In k-means clustering we are given a set of n data points in d-dimensional space ℜd and an integer k, and the problem is to determine a set of k points in ℜd, called centers, to minimize the mean squared distance from each data point to its nearest center. No exact polynomial-time algorithms are known for this problem. Although asymptotically efficient approximation algorithms exist, these algorithms are not practical due to the extremely high constant factors involved. There are many heuristics that are used in practice, but we know of no bounds on their performance.We consider the question of whether there exists a simple and practical approximation algorithm for k-means clustering. We present a local improvement heuristic based on swapping centers in and out. We prove that this yields a (9+e)-approximation algorithm. We show that the approximation factor is almost tight, by giving an example for which the algorithm achieves an approximation factor of (9-e). To establish the practical value of the heuristic, we present an empirical study that shows that, when combined with Lloyd's algorithm, this heuristic performs quite well in practice.

A novel MILP formulation for short-term scheduling of multi-stage multi-product batch plants with sequence-dependent constraints