archives@tulane.edu / Choosing which numerical optimization algorithm will perform best on a given problem is a task that researchers often face. Optimization benchmarking experiments allow researchers to compare the performance of different algorithms on various problems and thus provide insights into which algorithms should be used for a given problem. We benchmarked the prototypical iterative optimization algorithms, gradient descent, and the BFGS algorithm on a suite of test problems using the COCO benchmarking software. Our results indicate that the performance of gradient descent and BFGS varies by dimension, problem class, and solution accuracy. We provide recommendations for improving algorithm accuracy while reducing computational cost based on the implications of our results. / 1 / Elliot Hill

Nowadays, many optimization problems in real applications share a separable structure in the objective and it becomes more and more common to solve these problems by decomposition methods such as fast iterative shrinkage-thresholding algorithm (FISTA), generalized alternating direction method of multipliers (GADMM), and first-order primal-dual algorithm (PD), to name just a few. In this thesis, we focus on two optimization algorithms for solving convex programs: θ-scheme and a preconditioned primal-dual algorithm. For the θ-scheme, we first present an elaborative convergence analysis in a Hilbert space and propose a general convergent inexact θ-scheme. Second, for unconstrained problems, we prove the convergence of the θ-scheme and show a sublinear convergence rate in terms of the objective function. Furthermore, a practical inexact θ-scheme is derived to solve l_2-loss based problems and its convergence is proved. Third, for constrained problems, even though the convergence of the θ-scheme is available in the literature, yet its sublinear convergence rate is unknown until we provide one via a variational reformulation of the solution set. Besides, in order to relax the condition imposed on the θ-scheme, we propose a new variant and show its convergence. Finally, some preliminary numerical experiments demonstrate the efficiency of the θ-scheme and our proposed methods. For the preconditioned primal-dual algorithm, noticing that a practical step size cannot lie in the theoretical region, we show that the range of dual step size can be enlarged by 1/3 at most and at the same time, the convergence and a sublinear convergence rate can be ensured. Therefore, this practical step size can indeed guarantee the convergence. Furthermore, if more regularity conditions are imposed on objective functions, we can obtain a linear convergence rate. Finally, some connection with other methods is revealed. In future work, we focus on the acceleration of the θ-scheme. Some preliminary numerical experiments demonstrate the potential efficiency of our proposed accelerated θ-scheme. Therefore, it would be our priority of further study.

This study examines the impact of workforce level restriction on the aggregate production planning problem. Based on the results of this study, it is concluded that the solution to the aggregate production problem appears to be the same regardless of workforce level restriction. Additional research utilizing multi-industry large-scale data is needed to test the applicability of findings derived from the current study.

Massive MIMO has been recognized as a key technology for 5G systems due to its high spectral efficiency. The capacity and optimal signaling for a MIMO channel under the total power constraint (TPC) are well-known and can be obtained by the water-filling (WF) procedure. However, much less is known about optimal signaling under the per-antenna power constraint constraint (PAC) or under the joint power constraints (TPC+PAC). In this thesis, we consider a massive MIMO Gaussian channel under favorable propagation (FP) and obtain the optimal transmit covariance under the joint constraints. The effect of the joint constraints on the optimal power allocation (OPA) is shown. While it has some similarities to the standard WF, it also has number of notable differences. The numbers of active streams and active PACs are obtained, and a closed-form expression for the optimal dual variable is given. A capped water-filling interpretation of the OPA is given, which is similar to the standard WF, where a container has both floor and ceiling profiles. An iterative water-filling algorithm is proposed to find the OPA under the joint constraints, and its convergence to the OPA is proven. The robustness of optimal signaling under FP is demonstrated in which it becomes nearly optimal for a nearly favorable propagation channel. An upper bound of the sub-optimality gap is given which characterizes nearly (or eps)-favorable propagation. This upper bound quantifies how close the channel is to the FP. A bisection algorithm is developed to numerically compute the optimal dual variable. Newton-barrier and Monte-Carlo algorithms are developed to find the optimal signaling under the joint constraints for an arbitrary channel, not necessarily for a favorable propagation channel. When the diagonal entries of the channel Gram matrix are fixed, it is shown that a favorable propagation channel is not necessarily the best among all possible propagation scenarios capacity-wise. We further show that the main theorems in [1] on favorable propagation are not correct in general. To make their conclusions valid, some modifications as well as additional assumptions are needed, which are given here.

Indiana University-Purdue University Indianapolis (IUPUI) / Energy management has become a major concern in the past two decades with the increasing energy prices, overutilization of natural resources and increased carbon emissions. According to the department of Energy the industrial sector solely consumes 22.4% of the energy produced in the country [1]. This calls for an urgent need for the industries to design and implement energy efficient practices by analyzing the energy consumption, electricity data and making use of energy efficient equipment. Although, utility companies are providing incentives to consumer participating in Demand Response programs, there isn’t an active implementation of energy management principles from the consumer’s side. Technological advancements in controls, automation, optimization and big data can be harnessed to achieve this which in other words is referred to as “Smart Manufacturing”. In this research energy management techniques have been designed for two SEU (Significant Energy Use) equipment HVAC systems, Compressors and load shifting in manufacturing environments using control and optimization. The addressed energy management techniques associated with each of the SEUs are very generic in nature which make them applicable for most of the industries. Firstly, the loads or the energy consuming equipment has been categorized into flexible and non-flexible loads based on their priority level and flexibility in running schedule. For the flexible loads, an optimal load scheduler has been modelled using Mixed Integer Linear Programming (MILP) method that find carries out load shifting by using the predicted demand of the rest of the plant and scheduling the loads during the low demand periods. The cases of interruptible loads and non-interruptible have been solved to demonstrate load shifting. This essentially resulted in lowering the peak demand and hence cost savings for both “Time-of-Use” and Demand based price schemes. The compressor load sharing problem was next considered for optimal distribution of loads among VFD equipped compressors running in parallel to meet the demand. The model is based on MILP problem and case studies was carried out for heavy duty (>10HP) and light duty compressors (<=10HP). Using the compressor scheduler, there was about 16% energy and cost saving for the light duty compressors and 14.6% for the heavy duty compressors HVAC systems being one of the major energy consumer in manufacturing industries was modelled using the generic lumped parameter method. An Electroplating facility named Electro-Spec was modelled in Simulink and was validated using the real data that was collected from the facility. The Mean Absolute Error (MAE) was about 0.39 for the model which is suitable for implementing controllers for the purpose of energy management. MATLAB and Simulink were used to design and implement the state-of-the-art Model Predictive Control for the purpose of energy efficient control. The MPC was chosen due to its ability to easily handle Multi Input Multi Output Systems, system constraints and its optimal nature. The MPC resulted in a temperature response with a rise time of 10 minutes and a steady state error of less than 0.001. Also from the input response, it was observed that the MPC provided just enough input for the temperature to stay at the set point and as a result led to about 27.6% energy and cost savings. Thus this research has a potential of energy and cost savings and can be readily applied to most of the manufacturing industries that use HVAC, Compressors and machines as their primary energy consumer.

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Growing request for wideband communications requires innovation in power efficiency and signal processing. Without the use of any peak to average power ratio (PAPR) reduction technique, the efficiency of power consumption at the transmitter end becomes very poor. PAPR reduction in this work is accomplished based on using a unique class of functions, prolate spheroidal wave functions (PSWFs). The difficulty arises from the fact that these pulses do not belong to the Nyquist family. A zero forcing equalizer (ZFE) is designed to compensate intersymbol interference (ISI), and its performance is studied under the presence of AWGN. Considering PAPR and ISI as the constraints of communication systems, based on the properties of PSWF, a set of pulses with minimum ISI with respect to a specific amount of PAPR is achieved by defining an optimization problem. The desired level of PAPR is considered to be moved to the constraint set to convert the multi-objective problem into a single objective problem. The results of the numerical optimization of both ISI and PAPR are presented along with a couple of examples of comparison between the resultant pulse and the conventional square root raised cosine. It is shown that by achieving the same level of PAPR of the SRRC, the obtained pulse is a close approximation of SRRC. An implementation based on state variable filters is introduced to realize PSWF for high speed applications. An example based on this approach is presented to compare the finite pole approximation result with the original pulse.

Slotted Waveguide Antenna Stiffened Structures (SWASS) offer a new way to integrate the antennas used by many aircraft systems in modern aircraft. Looking at the weather radars used by current aircraft and using the loading estimates of the X-47B from Northrop Grumman, the designs went through several stages in the optimization procedure. The first stage centered around accounting for the stress concentrations present at the corners of the slots. These points led to local failure around the slots prior to the buckling of the overall structure, but the development of a concentration factor curve fit accounted for these in the optimization procedure and filled in a gap in the current literature. The models are then optimized, exposing a weakness in that these stress concentrations would lead to failure well before buckling in most designs with a loaded copper insert. To avoid this and shift most of the load to the supporting material, an initial gap is implemented in the eigenvalue buckling analysis, thus allowing for the simple 1-D models to be rapidly optimized without the need for contact modelling upon the gap's closure. The waveguide designs are then analyzed to ensure that the optimization of the individual waveguides is not prioritizing the structural performance to the detriment of the electromagnetic performance. Multiple points along the optimized Pareto front are tested and showed that their electromagnetic performance was consistent across the various regions of the front, and that the desired frequency of 10 GHz used by weather radars was within the optimal operational range for the various designs. Continuing from the individual waveguides now to larger panels, high fidelity models were used to develop another curve fit that relates the buckling of a panel simply supported on all four sides to the buckling of a single constituent waveguide simply supported on both ends. This curve fit is then used to validate the larger panel's performance against anticipated flight loads, without the need to model entire panels during the optimization procedure. / Doctor of Philosophy / Modern aircraft utilize antennas for a variety of purpose, ranging from the weather radars in the nose of passenger airlines, to the communications antennas mounted on the exterior of military aircraft, and even the targeting radars used by weapons systems in modern military craft. However, these systems often require large empty spaces within the aircraft or interfere with the profile of the aircraft if mounted externally. Slotted Waveguide Antenna Stiffened Structures (SWASS) aims to eliminate these issues by integrating these antennas into the skin of the aircraft but uses the antennas themselves to help strengthen the structures, thereby eliminating the need to reroute the loads around them and making the aircraft lighter. These designs consist of a slotted metallic waveguide enclosed within supporting composite materials, which are substituted in place of the standard aircraft skin so as to fit seamlessly into the designs. Multiple issues can arise when attempting to do this, which this thesis tackles. To develop optimized, multifunctional designs the thesis balances the structural needs to integrate the designs into existing aircraft against the electromagnetic needs of the antenna systems it replaces. Gaps in the existing literature are addressed through the development of a curve fit to properly account for issues caused by the slots cut into the upper surface of the waveguides. New methods are also employed to simplify the optimization procedure. The first is reducing the load on the metallic waveguide through an initial gap by deriving a simplified model and eliminating the need for the complex models previously required. The next step is the creation of a new curve fit to relate the buckling of a single, less complex single waveguide model, to the buckling of the larger, more complex panel models. Throughout all of this, constraints and model validations are used to ensure that the designs meet their requirements, both as an antenna as well as a load bearing part of the aircraft's skin, specifically that of the X-47B.

京都大学 / 新制・課程博士 / 博士(工学) / 甲第25013号 / 工博第5190号 / 新制||工||1991(附属図書館) / 京都大学大学院工学研究科機械理工学専攻 / (主査)教授 泉井 一浩, 教授 小森 雅晴, 教授 井上 康博 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

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