Cutting-plane Research Articles

Synthesizing Robust Communication Networks for Unmanned Aerial Vehicles With Resource Constraints

In this article, we address the problem of synthesizing communication networks for unmanned aerial vehicles (UAVs) in the presence of resource constraints. UAVs can be deployed as backbone nodes in ad hoc networks that can be central to civilian and military applications. The cost of operation of the network depends on the resources that are used such as the total power consumption associated with the network and the number of communication links in the network. The objective of the problem is to synthesize a communication network that maximizes connectivity subject to the cost of operation being within the specified budget for the resources. It is known that algebraic connectivity is a measure of robust connectivity and hence, it is chosen as an objective for optimization. We pose the network synthesis problem as a mixed-integer semidefinite program (MISDP): (1) provide an algorithm for computing optimal solutions using cutting plane methods; (2) develop lower bounds by posing the problem as a binary semidefinite program; and (3) construct feasible solutions using heuristics and estimate their quality. The network synthesis problem is a nondeterministic polynomial--time (NP)-hard problem. We provide some computational results to corroborate the performance of the proposed algorithms.

Analytic centre stabilization of column generation algorithm for the capacitated vehicle routing problem

We investigate various stabilization strategies for improving the column generation algorithm frequently used in recent exact methods and propose a novel stabilization technique specialized for the capacitated vehicle routing problem (CVRP). We start with exploring the relationship between Lagrangian relaxation and Dantzig–Wolfe (DW) decomposition for CVRP. The proposed technique, called analytic centre stabilization, is based on the analytic centre cutting plane method. In particular, we generate a cutting plane in the Lagrangian dual space and use it as a new column within DW decomposition for CVRP. In computational experiments, we compare the performance of standard column generation based on the Simplex method with those of the analytic centre stabilization, the bundle method, and a hybrid of the two approaches. The results show that the proposed method is quite competitive and the hybrid method has the best convergence properties among others.

A cutting-plane method without inclusions of approximating sets for conditional minimization

Propose a cutting-plane method with partially embedding of a feasible set for solving a conditional minimization problem. The proposed method is characterized by possibility of periodically dropping of an arbitrary number of any planes constructed in the solution process. Prove convergence of the method, discuss its features, represent assessments of the solution’s accuracy.

Comparison of Some Cutting-Plane Methods for the Integer Linear Programming Problem

The article proposes a modification of the Gomory cyclic algorithm for the integer linear programming problem. The proposed algorithm is numerically compared with the loop algorithm, the all-integer Gomory algorithm, and Martin's algorithm with improved choice of the generating row. Comparisons are carried out by running time and by the number of added cutting planes.

Global blending optimization of laminated composites with discrete material candidate selection and thickness variation

A method capable of simultaneous topology and thickness optimization of laminated composites has previously been published by one of the authors. Mass constrained compliance minimization subject to certain manufacturing constraints was solved on basis of interpolation schemes with penalization. In order to obtain large patchwise material candidate continuity while also accommodating variable laminate thickness, a bi-linear stiffness parameterization was introduced, causing a non-convex problem. In this present work, we introduce an alternative problem formulation that holds identical capabilities but is, however, convex in the original mixed binary nested form. Convexity is the foremost important property of optimization problems, and the proposed method can guarantee the global or near-global optimal solution; unlike most topology optimization methods. The material selection is limited to a distinct choice among predefined numbers of candidates. The laminate thickness is variable but the number of plies must be integer. We solve the convex mixed binary non-linear programming problem by an outer approximation cutting-plane method augmented with a few heuristics to accelerate the convergence rate. The capabilities of the method and the effect of active versus inactive manufacturing constraints are demonstrated on several numerical examples of limited size, involving at most 320 binary variables. Most examples are solved to guaranteed global optimality and may constitute benchmark examples for popular topology optimization methods and heuristics based on solving sequences of non-convex problems. The results will among others demonstrate that the difficulty of the posed problem is highly dependent upon the composition of the constitutive properties of the material candidates.

Maximum Likelihood Passive and Active Sensing of Wideband Power Spectra From Few Bits

Wideband power spectrum sensing is essential for cognitive radio and many other applications. Aiming to crowdsource spectrum sensing operations, a novel frugal sensing framework was recently proposed, employing a network of low duty-cycle sensors (e.g., running in background mode on consumer devices) reporting randomly filtered broadband power measurement bits to a fusion center, which in turn estimates the ambient power spectrum. Frugal sensing is revisited here from a statistical estimation point of view. Taking into account fading and insufficient sample averaging considerations, maximum likelihood (ML) formulations are developed which outperform the original minimum power and interior point solutions when the soft power estimates prior to thresholding are noisy. Assuming availability of a downlink channel that the fusion center can use to send threshold information, active sensing strategies are developed that quickly narrow down and track the power spectrum estimate, using ideas borrowed from cutting plane methods to develop active ML solutions. Simulations show that satisfactory wideband power spectrum estimates can be obtained with passive ML sensing from few bits, and much better performance can be attained using active sensing. Various other aspects, such as known emitter spectral shapes and different types of non-negativity constraints, are also considered.

Cognitive Transmit Beamforming From Binary CSIT

Transmit beamforming is used to steer radiated power towards a receiver of interest and to limit interference to unintended receivers, thereby facilitating coexistence. Transmit beamforming requires accurate channel state information (CSI) at the transmitter, which is often difficult to acquire, particularly in cognitive underlay settings, where the primary receiver cannot be expected to cooperate with the secondary system to enable it to learn the secondary to primary crosstalk channel. This paper considers cases where it is not realistic to assume channel reciprocity, or that the receivers are capable of accurate CSI estimation and feedback-because they are legacy systems, or have limited computation/energy resources. Transmit beamforming from binary and infrequent CSI is first considered for an isolated link. An online beamforming and learning algorithm is developed using the analytic center cutting plane method and is shown to asymptotically attain optimal performance. A robust maximum-likelihood formulation is next developed to handle feedback errors and correlation drift. The setup is then generalized to a cognitive underlay setting, also exploiting the standard acknowledgement/negative-acknowledgement feedback on the reverse primary link. This is the first solution to jointly tackle secondary signal-to-noise ratio maximization and primary interference mitigation from only rudimentary CSI, without assuming channel reciprocity.

Probabilistic robust linear parameter-varying control of a small helicopter using iterative scenario approach

In this paper, we present an iterative scenario approach (ISA) to design robust controllers for complex linear parameter-varying (LPV) systems with uncertainties. The robust controller synthesis problem is transformed to a scenario design problem, with the scenarios generated by identically extracting random samples on both uncertainty parameters and scheduling parameters. An iterative scheme based on the maximum volume ellipsoid cutting-plane method is used to solve the problem. Heuristic logic based on relevance ratio ranking is used to prune the redundant constraints, and thus, to improve the numerical stability of the algorithm. And further, a batching technique is presented to remarkably enhance the computational efficiency. The proposed method is applied to design an output-feedback controller for a small helicopter. Multiple uncertain physical parameters are considered, and simulation studies show that the closed-loop performance is quite good in both aspects of model tracking and dynamic decoupling. For robust LPV control problems, the proposed method is more computationally efficient than the popular stochastic ellipsoid methods.

Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection

This paper introduces a method for computing the maximum volume inscribed ellipsoid and k-ball of a projected polytope. It is known that deriving an explicit description of a projected polytope is NP-hard. By using adjustable robust optimization techniques, we construct a computationally tractable method that does not require an explicit description of the projection. The obtained centers of the projected polytope are considered as the robust solutions, e.g., for design centering problems. We perform numerical experiments on a simple polytope and a color tube design problem. The color tube design problem demonstrates that the obtained solutions are much more robust than the nominal approach with a slight compromise on the objective value. Some other potential applications include ellipsoidal approximations to polytopic sets, nominal scenario recovery, and cutting-plane method.

A new cut-and-solve and cutting plane combined approach for the capacitated lane reservation problem

This paper investigates a capacitated lane reservation problem with residual capacity. The focus of the problem is to design time-guaranteed paths for a set of transportation tasks via lane reservation with respect to the residual capacity issue. The lane reservation can ensure the time-guaranteed travel but has negative impact on the normal traffic. Thus, the objective of the problem is to minimize the impact of such lane reservation by optimally selecting lanes in the network to be reserved. For this NP-hard problem, an exact approach based on the cut-and-solve and cutting plane methods is developed. New separation algorithms are proposed to find appropriate valid inequalities according to the characteristic of the problem to accelerate the optimal convergence of the resolution approach. Numerical computational results show that the developed approach is efficient as compared with state-of-the-art algorithms in the literature and the IP solver CPLEX.

Reactive Power Optimization for Distribution Network with Distributed Generators Based on Semi-Definite Programming

Reactive power optimization for distribution network with distributed generators is a complicated nonconvex nonlinear mixed integer programming problem. This paper built a mathematical model of reactive power optimization for distribution network and a new method to solve this problem was proposed based on semi-definite programming. The original mathematical model was transformed and relaxed into a convex SDP model, to guarantee the global optimal solution within the polynomial times. Then the model was extended to a mixed integer semi-definite programming model with discrete variables when considering discrete compensation equipment such as capacitor banks. Global optimal solution of this model can be obtained by cutting plane method and branch and bound method. Numerical tests on the modified IEEE 33-bus system show this method is exact and can be solved efficiently.

ML estimation of transition probabilities for an unknown maneuvering emitter tracking

We consider the problem of an unknown maneuvering emitter tracking by a wireless sensor network with time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements. Interacting multiple models combined with square-root cubature Kalman filter with correlated noises (IMM-SCKF-CN) is proposed to update the parameters in the maneuvering emitter tracking. Essential to this tracking framework is the Markov transition probability matrix (TPM) which governs the jumps between multiple dynamic motion models for the maneuvering target. However, in practice, the TPM is unknown and has to be estimated. In this paper, we consider the maximum likelihood (ML) estimation of the TPM and propose a recursive algorithm based on the improved weighted analytical center cutting plane method (ACCPM). Compared with some batch ML methods, the resulting recursive ML estimation method has a much lower per sample complexity. Simulation results show the efficacy of the proposed method with greatly improved tracking performance.

Warehouse location and two-echelon inventory management with concave operating cost

In this paper, we study a location–inventory network design problem which jointly optimises the warehouse location, the warehouse–retailer assignments, the warehouse–retailer echelon inventory replenishment and the safety stock-level decisions over an infinite planning horizon. The consideration of the facility operating cost, the safety stock cost and the two-echelon inventory cost results in an MIP model with several nonlinear terms. Due to the complex trade-offs among the various costs and multiple nonlinear terms in the model, traditional solution approaches no longer work for this problem. We outline a polymatroid cutting-plane approach based on the submodular property of the cost terms to address this problem. Computational results demonstrate that the cutting-plane method based on polymatroid inequalities can efficiently solve randomly generated instances with moderate sizes.

Provably Good Task Assignment for Two-Type Heterogeneous Multiprocessors Using Cutting Planes

Consider scheduling of real-time tasks on a multiprocessor where migration is forbidden. Specifically, consider the problem of determining a task-to-processor assignment for a given collection of implicit-deadline sporadic tasks upon a multiprocessor platform in which there are two distinct types of processors. For this problem, we propose a new algorithm, LPC (task assignment based on solving a Linear Program with Cutting planes). The algorithm offers the following guarantee: for a given task set and a platform, if there exists a feasible task-to-processor assignment, then LPC succeeds in finding such a feasible task-to-processor assignment as well but on a platform in which each processor is 1.5 × faster and has three additional processors. For systems with a large number of processors, LPC has a better approximation ratio than state-of-the-art algorithms. To the best of our knowledge, this is the first work that develops a provably good real-time task assignment algorithm using cutting planes.

Efficient algorithm for locating and sizing series compensation devices in large power transmission grids: II. Solutions and applications

In a companion manuscript (Frolov et al 2014 New J. Phys. 16 art. no.) , we developed a novel optimization method for the placement, sizing, and operation of flexible alternating current transmission system (FACTS) devices to relieve transmission network congestion. Specifically, we addressed FACTS that provide series compensation (SC) via modification of line inductance. In this sequel manuscript, this heuristic algorithm and its solutions are explored on a number of test cases: a 30-bus test network and a realistically-sized model of the Polish grid (∼2700 nodes and ∼3300 lines). The results from the 30-bus network are used to study the general properties of the solutions, including nonlocality and sparsity. The Polish grid is used to demonstrate the computational efficiency of the heuristics that leverage sequential linearization of power flow constraints, and cutting plane methods that take advantage of the sparse nature of the SC placement solutions. Using these approaches, we can use the algorithm to solve a Polish transmission grid in tens of seconds. We explore the utility of the algorithm by analyzing transmission networks congested by (i) uniform load growth, (ii) multiple overloaded configurations, and (iii) sequential generator retirements.

Efficient algorithm for locating and sizing series compensation devices in large power transmission grids: I. Model implementation

We explore optimization methods for planning the placement, sizing and operations of flexible alternating current transmission system (FACTS) devices installed to relieve transmission grid congestion. We limit our selection of FACTS devices to series compensation (SC) devices that can be represented by modification of the inductance of transmission lines. Our master optimization problem minimizes the l1 norm of the inductance modification subject to the usual line thermal-limit constraints. We develop heuristics that reduce this non-convex optimization to a succession of linear programs (LP) that are accelerated further using cutting plane methods. The algorithm solves an instance of the MatPower Polish Grid model (3299 lines and 2746 nodes) in 40 seconds per iteration on a standard laptop—a speed that allows the sizing and placement of a family of SC devices to correct a large set of anticipated congestions. We observe that our algorithm finds feasible solutions that are always sparse, i.e., SC devices are placed on only a few lines. In a companion manuscript, we demonstrate our approach on realistically sized networks that suffer congestion from a range of causes, including generator retirement. In this manuscript, we focus on the development of our approach, investigate its structure on a small test system subject to congestion from uniform load growth, and demonstrate computational efficiency on a realistically sized network.

StructBoost: Boosting Methods for Predicting Structured Output Variables.

Boosting is a method for learning a single accurate predictor by linearly combining a set of less accurate weak learners. Recently, structured learning has found many applications in computer vision. Inspired by structured support vector machines (SSVM), here we propose a new boosting algorithm for structured output prediction, which we refer to as StructBoost. StructBoost supports nonlinear structured learning by combining a set of weak structured learners. As SSVM generalizes SVM, our StructBoost generalizes standard boosting approaches such as AdaBoost, or LPBoost to structured learning. The resulting optimization problem of StructBoost is more challenging than SSVM in the sense that it may involve exponentially many variables and constraints. In contrast, for SSVM one usually has an exponential number of constraints and a cutting-plane method is used. In order to efficiently solve StructBoost, we formulate an equivalent 1-slack formulation and solve it using a combination of cutting planes and column generation. We show the versatility and usefulness of StructBoost on a range of problems such as optimizing the tree loss for hierarchical multi-class classification, optimizing the Pascal overlap criterion for robust visual tracking and learning conditional random field parameters for image segmentation.

Efficient Localization Methods for Passivity Enforcement of Linear Dynamical Models

This paper describes a novel approach for passivity enforcement of compact dynamical models of electrical interconnects. The proposed approach is based on a parameterization of general state-space scattering models with fixed poles. We formulate the passivity constraints as a unitary boundedness condition on the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm of the system transfer function. When this condition is not verified, we use it as an explicit constraint within an iterative perturbation loop of the system state-space matrices. Since the resulting optimization framework is convex but nonsmooth, we solve it via localization based algorithms, such as the ellipsoid and the cutting plane methods. The proposed technique solves two critical bottleneck issues of the existing approaches for passivity enforcement of linear macromodels. Compared to quasi-optimal schemes based on singular value or Hamiltonian eigenvalue perturbation, we are able to guarantee convergence to the optimal solution. Compared to convex formulations based on direct Bounded Real Lemma constraints, we are able to reduce both memory and time requirements by orders of magnitude. We demonstrate the effectiveness of our approach on a number of cases for which existing algorithms either fail or exhibit very slow convergence.

Energy Beamforming With One-Bit Feedback

Multi-antenna or multiple-input multiple-output (MIMO) techniques are appealing to enhance the transmission efficiency and range for radio frequency (RF) signal enabled wireless energy transfer (WET). In order to reap the energy beamforming gain in MIMO WET, acquiring the channel state information (CSI) at the energy transmitter (ET) is an essential task. This task is particularly challenging, since existing channel training and feedback methods used for communication receivers may not be implementable at the energy receiver (ER) due to its hardware limitation. To tackle this problem, we consider in this paper a multiuser MIMO WET system, and propose a new channel learning method that requires only one feedback bit from each ER to the ET per feedback interval. Specifically, each feedback bit indicates the increase or decrease of the harvested energy by each ER in the present as compared to the previous intervals, which can be measured without changing the existing structure of the ER. Based on such feedback information, the ET adjusts transmit beamforming in subsequent training intervals and at the same time obtains improved estimates of the MIMO channels to different ERs by applying an optimization technique called analytic center cutting plane method (ACCPM). For the proposed ACCPM based channel learning algorithm, we analyze its worst-case convergence, from which it is revealed that the algorithm is able to estimate multiuser MIMO channels simultaneously without reducing the analytic convergence speed. Also, we provide extensive simulations to show its performances in terms of both convergence speed and energy transfer efficiency.

A Lagrangian relaxation approach for expansion of a highway network

This paper deals with the problem of improving an existing road network in the context of strategic planning through the creation of new highway corridors. To address this problem we analyse three mixed integer programming models. The so-called [P1] is the classical capacitated multicommodity network design model. The model named [P2] imposes on [P1] the location of a single (main path) highway corridor in the road network and [P3] adds to [P2] a set of sub-tour breaking constraints. The stated goal is to minimize the total travel time for a known origin-destination demand matrix with a given budget. In this paper we propose an efficient method for [P3], based on a Lagrangian relaxation, to obtain easily-solved sub-problems. A cutting-plane method for solving the Lagrangian sub-problems is proposed. This method generates valid cuts until an optimal solution is found. The Lagrangian dual problem is solved using the sub-gradient optimization method. A case study has been carried out for the region of Castilla-La Mancha (Spain). Computational comparisons between the proposed method and a state-of-the-art mixed-integer code are presented. The Lagrangian relaxation approach is found to be capable of generating good feasible solutions to the case study within a reasonable computational time.