A Relaxed and Bound Algorithm Based on Auxiliary Variables for Quadratically Constrained Quadratic Programming Problem

Quadratically constrained quadratic programs (QCQP), which often appear in engineering practice and management science, and other fields, are investigated in this paper. By introducing appropriate auxiliary variables, QCQP can be transformed into its equivalent problem (EP) with non-linear equality constraints. After these equality constraints are relaxed, a series of linear relaxation subproblems with auxiliary variables and bound constraints are generated, which can determine the effective lower bound of the global optimal value of QCQP. To enhance the compactness of sub-rectangles and improve the ability to remove sub-rectangles, two rectangle-reduction strategies are employed. Besides, two ϵ-subproblem deletion rules are introduced to improve the convergence speed of the algorithm. Therefore, a relaxation and bound algorithm based on auxiliary variables are proposed to solve QCQP. Numerical experiments show that this algorithm is effective and feasible.

New Algorithm to Solve Mixed Integer Quadratically Constrained Quadratic Programming Problems Using Piecewise Linear Approximation

Techniques and methods of linear optimization underwent a significant improvement in the 20th century which led to the development of reliable mixed integer linear programming (MILP) solvers. It would be useful if these solvers could handle mixed integer nonlinear programming (MINLP) problems. Piecewise linear approximation (PLA) is one of most popular methods used to transform nonlinear problems into linear ones. This paper will introduce PLA with brief a background and literature review, followed by describing our contribution before presenting the results of computational experiments and our findings. The goals of this paper are (a) improving PLA models by using nonuniform domain partitioning, and (b) proposing an idea of applying PLA partially on MINLP problems, making them easier to handle. The computational experiments were done using quadratically constrained quadratic programming (QCQP) and MIQCQP and they showed that problems under PLA with nonuniform partition resulted in more accurate solutions and required less time compared to PLA with uniform partition.

Code Optimisation Framework for Multicode Sonar Systems

<p>The ocean is a temporally and spatially varying environment, the characteristics of which pose significant challenges to the development of effective underwater wireless communications and sensing systems. An underwater sensing system such as a sonar detects the presence of a known signal through correlation. It is advantageous to use multiple transducers to increase surveying area with reduced surveying costs and time. Each transducers is assigned a dedicated code. When using multiple codes, the sidelobes of auto- and crosscorrelations are restricted to theoretical limits known as bounds. Sets of codes must be optimised in order to achieve optimal correlation properties, and, achieve Sidelobe Level (SLL)s as low as possible. In this thesis, we present a novel code-optimisation method to optimise code-sets with any number of codes and up to any length of each code. We optimise code-sets for a matched filter for application in a multi-code sonar system. We first present our gradient-descent based algorithm to optimise sets of codes for flat and low crosscorrelations and autocorrelation sidelobes, including conformance of the magnitude of the samples of the codes to a target power profile. We incorporate the transducer frequency response and the channel effects into the optimisation algorithm. We compare the correlations of our optimised codes with the well-known Welch bound. We then present a method to widen the autocorrelation mainlobe and impose monotonicity. In many cases, we are able to achieve SLLs beyond the Welch bound. We study the Signal to Noise Ratio (SNR) improvement of the optimised codes for an Underwater Acoustic (UWA) channel. During its propagation, the acoustic wave suffers non-constant transmission loss which is compensated by the application of an appropriate Time Variable Gain (TVG). The effect of the TVG modifies the noise received with the signal. We show that in most cases, the matched filter is still the optimum filter. We also show that the accuracy in timing is very important in the application of the TVG to the received signal. We then incorporate Doppler tolerance into the existing optimisation algorithm. Our proposed method is able to optimise sets of codes for multiple Doppler scaling factors and non-integer delays in the arrival of the reflection, while still conforming to other constraints. We suggest designing mismatched filters to further reduce the SLLs, firstly using an existing Quadratically Constrained Qaudratic Program (QCQP) formulation and secondly, as a local optimisation problem, modifying our basic optimisation algorithm.</p>

Code Optimisation Framework for Multicode Sonar Systems

<p>The ocean is a temporally and spatially varying environment, the characteristics of which pose significant challenges to the development of effective underwater wireless communications and sensing systems. An underwater sensing system such as a sonar detects the presence of a known signal through correlation. It is advantageous to use multiple transducers to increase surveying area with reduced surveying costs and time. Each transducers is assigned a dedicated code. When using multiple codes, the sidelobes of auto- and crosscorrelations are restricted to theoretical limits known as bounds. Sets of codes must be optimised in order to achieve optimal correlation properties, and, achieve Sidelobe Level (SLL)s as low as possible. In this thesis, we present a novel code-optimisation method to optimise code-sets with any number of codes and up to any length of each code. We optimise code-sets for a matched filter for application in a multi-code sonar system. We first present our gradient-descent based algorithm to optimise sets of codes for flat and low crosscorrelations and autocorrelation sidelobes, including conformance of the magnitude of the samples of the codes to a target power profile. We incorporate the transducer frequency response and the channel effects into the optimisation algorithm. We compare the correlations of our optimised codes with the well-known Welch bound. We then present a method to widen the autocorrelation mainlobe and impose monotonicity. In many cases, we are able to achieve SLLs beyond the Welch bound. We study the Signal to Noise Ratio (SNR) improvement of the optimised codes for an Underwater Acoustic (UWA) channel. During its propagation, the acoustic wave suffers non-constant transmission loss which is compensated by the application of an appropriate Time Variable Gain (TVG). The effect of the TVG modifies the noise received with the signal. We show that in most cases, the matched filter is still the optimum filter. We also show that the accuracy in timing is very important in the application of the TVG to the received signal. We then incorporate Doppler tolerance into the existing optimisation algorithm. Our proposed method is able to optimise sets of codes for multiple Doppler scaling factors and non-integer delays in the arrival of the reflection, while still conforming to other constraints. We suggest designing mismatched filters to further reduce the SLLs, firstly using an existing Quadratically Constrained Qaudratic Program (QCQP) formulation and secondly, as a local optimisation problem, modifying our basic optimisation algorithm.</p>

Scalable Optimization Methods for Incorporating Spatiotemporal Fractionation into Intensity-Modulated Radiotherapy Planning

It has been recently shown that an additional therapeutic gain may be achieved if a radiotherapy plan is altered over the treatment course using a new treatment paradigm referred to in the literature as spatiotemporal fractionation. Because of the nonconvex and large-scale nature of the corresponding treatment plan optimization problem, the extent of the potential therapeutic gain that may be achieved from spatiotemporal fractionation has been investigated using stylized cancer cases to circumvent the arising computational challenges. This research aims at developing scalable optimization methods to obtain high-quality spatiotemporally fractionated plans with optimality bounds for clinical cancer cases. In particular, the treatment-planning problem is formulated as a quadratically constrained quadratic program and is solved to local optimality using a constraint-generation approach, in which each subproblem is solved using sequential linear/quadratic programming methods. To obtain optimality bounds, cutting-plane and column-generation methods are combined to solve the Lagrangian relaxation of the formulation. The performance of the developed methods are tested on deidentified clinical liver and prostate cancer cases. Results show that the proposed method is capable of achieving local-optimal spatiotemporally fractionated plans with an optimality gap of around 10%–12% for cancer cases tested in this study. Summary of Contribution: The design of spatiotemporally fractionated radiotherapy plans for clinical cancer cases gives rise to a class of nonconvex and large-scale quadratically constrained quadratic programming (QCQP) problems, the solution of which requires the development of efficient models and solution methods. To address the computational challenges posed by the large-scale and nonconvex nature of the problem, we employ large-scale optimization techniques to develop scalable solution methods that find local-optimal solutions along with optimality bounds. We test the performance of the proposed methods on deidentified clinical cancer cases. The proposed methods in this study can, in principle, be applied to solve other QCQP formulations, which commonly arise in several application domains, including graph theory, power systems, and signal processing.

Reconfigurable Intelligent Surface-assisted Edge Computing to Minimize Delay in Task Offloading

The advantage of computational resources in edge computing near the data source has kindled growing interest in delay-sensitive Internet of Things (IoT) applications. However, the benefit of the edge server is limited by the uploading and downloading links between end-users and edge servers when these end-users seek computational resources from edge servers. The scenario becomes more severe when the user-end's devices are in the shaded region resulting in low uplink/downlink quality. In this paper, we consider a reconfigurable intelligent surface (RIS)-assisted edge computing system, where the benefits of RIS are exploited to improve the uploading transmission rate. We further aim to minimize the delay of worst-case in the network when the end-users either compute task data in their local CPU or offload task data to the edge server. Next, we optimize the uploading bandwidth allocation for every end-user's task data to minimize the maximum delay in the network. The above optimization problem is formulated as quadratically constrained quadratic programming. Afterward, we solve this problem by semidefinite relaxation. Finally, the simulation results demonstrate that the proposed strategy is scalable under various network settings.

Reconfigurable Intelligent Surface-assisted Edge Computing to Minimize Delay in Task Offloading

The advantage of computational resources in edge computing near the data source has kindled growing interest in delay-sensitive Internet of Things (IoT) applications. However, the benefit of the edge server is limited by the uploading and downloading links between end-users and edge servers when these end-users seek computational resources from edge servers. The scenario becomes more severe when the user-end's devices are in the shaded region resulting in low uplink/downlink quality. In this paper, we consider a reconfigurable intelligent surface (RIS)-assisted edge computing system, where the benefits of RIS are exploited to improve the uploading transmission rate. We further aim to minimize the delay of worst-case in the network when the end-users either compute task data in their local CPU or offload task data to the edge server. Next, we optimize the uploading bandwidth allocation for every end-user's task data to minimize the maximum delay in the network. The above optimization problem is formulated as quadratically constrained quadratic programming. Afterward, we solve this problem by semidefinite relaxation. Finally, the simulation results demonstrate that the proposed strategy is scalable under various network settings.