Use Of Convex Optimization Research Articles

Optimal Mismatched Filter Design by Combining Convex Optimization With Circular Algorithm

Mismatched filters (MMF) have been intensively explored for linear-frequency modulation (LFM) waveform sidelobe reduction. Unlike other design strategies, the use of convex optimization (CO) techniques is able to find the optimal design. However, existing studies for CO-based MMF design have focused on minimizing peak sidelobe level (PSL), which might not take the mainlobe widening sacrifice into account. To address this issue, this paper proposes a weighted CO model targeting a flexible tradeoff between sidelobe suppression and mainlobe widening. Moreover, to better control the optimization problem via the presented CO model, the cyclic algorithm (CA) is employed to determine upper limits on mainlobe width. Consequently, the proposed tradeoff MMF design by combining CA with CO would be feasible to meet the requirements of different target detection scenarios. Comprehensive simulations have been carried out to demonstrate the effectiveness of our presented MMF design.

Decentralized learning of randomization-based neural networks with centralized equivalence

We consider a decentralized learning problem where training data samples are distributed over agents (processing nodes) of an underlying communication network topology without any central (master) node. Due to information privacy and security issues in a decentralized setup, nodes are not allowed to share their training data and only parameters of the neural network are allowed to be shared. This article investigates decentralized learning of randomization-based neural networks that provides centralized equivalent performance as if the full training data are available at a single node. We consider five randomization-based neural networks that use convex optimization for learning. Two of the five neural networks are shallow, and the others are deep. The use of convex optimization is the key to apply alternating-direction-method-of-multipliers with decentralized average consensus. This helps us to establish decentralized learning with centralized equivalence. For the underlying communication network topology, we use a doubly-stochastic network policy matrix and synchronous communications. Experiments with nine benchmark datasets show that the five neural networks provide good performance while requiring low computational and communication complexity for decentralized learning. The performance rankings of five neural networks using Friedman rank are also enclosed in the results, which are ELM < RVFL< dRVFL < edRVFL < SSFN.

Long-only equal risk contribution portfolios for CVaR under discrete distributions

Portfolios in which all assets contribute equally to the conditional value-at-risk (CVaR) represent an interesting variation of the popular risk parity investment strategy. This paper considers the use of convex optimization to find long-only equal risk contribution (ERC) portfolios for CVaR given a set of equally likely scenarios of asset returns. We provide second-order conic and non-linear formulations of the problem, which yields an ERC portfolio when CVaR is both positive and differentiable at the optimal solution. We identify sufficient conditions for differentiability and develop a heuristic that obtains an approximate ERC portfolio when the conditions are not satisfied. Computational tests show that the approach performs well compared to non-convex formulations that have been proposed in the literature.

Formulation of optimal surrogate descriptions of fuels considering sensitivities to experimental uncertainties

Transportation fuels consist of a large number of species that belong to different families of compounds. Surrogate fuel representations have been formulated to better understand their fundamental chemical composition and to emulate combustion properties. These descriptions are formulated using experiments or through computations, which has thus led to the existence of two different notions of surrogates. There is further distinction of concepts through the use of physical and chemical surrogates, which are designed to emulate those specific properties. Although several surrogate design methodologies have been proposed in literature, they do not incorporate information on experimental uncertainty. By addressing this issue, it is shown that this information is crucial for the reliable construction of surrogates through computations. To incorporate physical fuel properties, a consistent approach through the use of the recent ASTM D2887 distillation curve standard is discussed. Then, a formal computational procedure is presented that incorporates information of experimental uncertainties into the surrogate description. It is shown that surrogates then describe a feasible region and are hence not unique. Both physical and chemical properties are utilized as combustion property targets (CPTs) and consistency with experimental formulations is demonstrated for JP-8 and Jet-A (POSF 4658) surrogates. In addition, the use of convex optimization puts existing concepts for surrogate representation on a more rigorous basis and several conclusions are drawn, particularly on the importance of specific CPTs and weighting factors of regression-based approaches. Also, the effect of using simplified models for the evaluation of CPTs on the final surrogate composition is shown by considering the example of linear blending rules for ignition delay. Finally, the surrogate representation problem is connected to multi-parametric optimization and bounds on surrogate compositions are calculated as a function of the experimental uncertainty along with comparisons against experimental results.

Predicting neurological targets for chemical neurotoxins using ToxCast in vitro data and read-across within QSAR Toolbox

Transportation fuels consist of a large number of species that belong to different families of compounds. Surrogate fuel representations have been formulated to better understand their fundamental chemical composition and to emulate combustion properties. These descriptions are formulated using experiments or through computations, which has thus led to the existence of two different notions of surrogates. There is further distinction of concepts through the use of physical and chemical surrogates, which are designed to emulate those specific properties. Although several surrogate design methodologies have been proposed in literature, they do not incorporate information on experimental uncertainty. By addressing this issue, it is shown that this information is crucial for the reliable construction of surrogates through computations. To incorporate physical fuel properties, a consistent approach through the use of the recent ASTM D2887 distillation curve standard is discussed. Then, a formal computational procedure is presented that incorporates information of experimental uncertainties into the surrogate description. It is shown that surrogates then describe a feasible region and are hence not unique. Both physical and chemical properties are utilized as combustion property targets (CPTs) and consistency with experimental formulations is demonstrated for JP-8 and Jet-A (POSF 4658) surrogates. In addition, the use of convex optimization puts existing concepts for surrogate representation on a more rigorous basis and several conclusions are drawn, particularly on the importance of specific CPTs and weighting factors of regression-based approaches. Also, the effect of using simplified models for the evaluation of CPTs on the final surrogate composition is shown by considering the example of linear blending rules for ignition delay. Finally, the surrogate representation problem is connected to multi-parametric optimization and bounds on surrogate compositions are calculated as a function of the experimental uncertainty along with comparisons against experimental results.

A formulation of a matrix sparsity approach for the quantum ordered search algorithm

One specific subset of quantum algorithms is Grovers Ordered Search Problem (OSP), the quantum counterpart of the classical binary search algorithm, which utilizes oracle functions to produce a specified value within an ordered database. Classically, the optimal algorithm is known to have a [Formula: see text] complexity; however, Grovers algorithm has been found to have an optimal complexity between the lower bound of [Formula: see text] and the upper bound of [Formula: see text]. We sought to lower the known upper bound of the OSP. With Farhi et al. MITCTP 2815 (1999), arXiv:quant-ph/9901059], we see that the OSP can be resolved into a translational invariant algorithm to create quantum query algorithm restraints. With these restraints, one can find Laurent polynomials for various [Formula: see text] — queries — and [Formula: see text] — database sizes — thus finding larger recursive sets to solve the OSP and effectively reducing the upper bound. These polynomials are found to be convex functions, allowing one to make use of convex optimization to find an improvement on the known bounds. According to Childs et al. [Phys. Rev. A 75 (2007) 032335], semidefinite programming, a subset of convex optimization, can solve the particular problem represented by the constraints. We were able to implement a program abiding to their formulation of a semidefinite program (SDP), leading us to find that it takes an immense amount of storage and time to compute. To combat this setback, we then formulated an approach to improve results of the SDP using matrix sparsity. Through the development of this approach, along with an implementation of a rudimentary solver, we demonstrate how matrix sparsity reduces the amount of time and storage required to compute the SDP — overall ensuring further improvements will likely be made to reach the theorized lower bound.

Interpreting evidential distances by connecting them to partial orders: Application to belief function approximation

Distances between mass functions are instrumental tools in evidence theory, yet it is not always clear in which situation a particular distance should be used. Indeed, while the mathematical properties of distances have been well studied, how to interpret them is still a largely open issue. As a step towards answering this question, we propose to interpret distances by looking at their compatibility with partial orders. We formalize this compatibility through some mathematical properties thereby allowing to combine the advantages of both partial orders (clear semantics) and distances (richer structure and access to numerical tools). We explore in particular the case of informational partial orders, and how distances compatible with such orders can be used to approximate initial belief functions by simpler ones through the use of convex optimization. We finish by discussing some perspectives of the current work. • Distances and partial orders in the theory of belief functions are formally related through a desirable property. • The compatibility of several L k norm based distances with some informational partial orders is proved. • These results are exploited to approximate initial belief functions by simpler ones through the use of convex optimization.

Post-Prognostics Decision for Optimizing the Commitment of Fuel Cell Systems

Abstract: In a post-prognostics decision context, this paper addresses the problem of maximizing the useful life of a platform composed of several parallel machines under service constraint. Application on multi-stack fuel cell systems is considered. In order to propose a solution to the insufficient durability of fuel cells, the purpose is to define a commitment strategy by determining at each time the contribution of each fuel cell stack to the global output so as to reach the demand as long as possible. Two algorithms making use of convex optimization are proposed to cope with the assignment problem. First one is based on the Mirror-prox for Saddle Points method and second one uses the Lasso (Least Absolute Shrinkage and Selection Operator) principle. Results based on computational experiments assess the efficiency of these two approaches in comparison with an intuitive resolution performing successive basic convex projections onto the sets of constraints associated to the optimization problem.

Transmit Subaperturing for Range and Angle Estimation in Frequency Diverse Array Radar

Different from conventional phased-array, which provides only angle-dependent beampattern, frequency diverse array (FDA) employs a small frequency increment across the antenna elements and results in a range-angle-dependent beampattern. This beampattern offers a potential to localize the targets in two dimensions in terms of slant ranges and azimuth angles. However, it is difficult to obtain the target location information from a standard FDA radar due to the couplings in range and angle responses. In this paper, we propose a transmit subaperturing scheme on the FDA radar for range and angle estimation of targets. The essence is to divide the FDA elements into multiple subarrays and optimize the transmit beamspace matrix with the use of convex optimization. We also discuss several practical issues for designing the FDA radar system parameters. Since the subarrays offer decoupled range and angle responses, the targets can be located using the beamspace-based multiple signal classification algorithm. The range and angle estimation performance is evaluated by comparing with the Cramer-Rao lower bound.

Optimal power control and coverage management in two-tier femtocell networks

In this article, we jointly consider the problem of efficient power control and coverage (PCC) management over an integrated two-tier macrocell/femtocell network towards maximizing the expected throughput of the system subject to appropriate power constraints, under the existence of both co-tier and cross-tier interferences. Optimal network design amounts to joint optimization of users’ allocated power levels and cell’s maximum aggregated downlink transmitted power, i.e., coverage area management. This problem is inherently difficult because it is in fact a non-convex optimization problem. A novel approach to address the latter is performed that entails a suitable transformation, which allows the use of convex optimization and also forms the basis for the design of a distributed PCC algorithm via performing two-level primal-dual decomposition. PCC algorithm’s convergence to optimality is established. We demonstrate that for realistic macrocell/femtocell deployment scenarios, overall system throughput increase up to approximately 50% can be achieved while guaranteeing 70% of power savings.

Information-Theoretic Lower Bounds on the Oracle Complexity of Stochastic Convex Optimization

Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardn4516420ess of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining an understanding of these complexity-theoretic issues is important. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We introduce a new notion of discrepancy between functions, and use it to reduce problems of stochastic convex optimization to statistical parameter estimation, which can be lower bounded using information-theoretic methods. Using this approach, we improve upon known results and obtain tight minimax complexity estimates for various function classes.

Sparse Optimization with Least-Squares Constraints

The use of convex optimization for the recovery of sparse signals from incomplete or compressed data is now common practice. Motivated by the success of basis pursuit in recovering sparse vectors, new formulations have been proposed that take advantage of different types of sparsity. In this paper we propose an efficient algorithm for solving a general class of sparsifying formulations. For several common types of sparsity we provide applications, along with details on how to apply the algorithm, and experimental results.

Convex Optimization in Signal Processing [From the Guest Editors

In recent years, we have witnessed technical breakthroughs in a wide variety of topics where the key to success is the use of convex optimization. In fact, convex optimization has now emerged as a major signal processing tool that has made a significant impact on numerous problems previously considered intractable. Considering the foundational nature and potential impact of convex optimization in signal processing, we have put together this special issue that aims to provide tutorials of convex optimization techniques (including available software) and various successful signal processing applications. Our goal is not only to contribute to the diffusion of recent developments in this research area within the signal processing community, but also to spur further advances in and applications of convex optimization for signal processing.

A covariance fitting approach for correlated acoustic source mapping

Microphone arrays are commonly used for noise source localization and power estimation in aeroacoustic measurements. The delay-and-sum (DAS) beamformer, which is the most widely used beamforming algorithm in practice, suffers from low resolution and high sidelobe level problems. Therefore, deconvolution approaches, such as the deconvolution approach for the mapping of acoustic sources (DAMAS), are often used for extracting the actual source powers from the contaminated DAS results. However, most deconvolution approaches assume that the sources are uncorrelated. Although deconvolution algorithms that can deal with correlated sources, such as DAMAS for correlated sources, do exist, these algorithms are computationally impractical even for small scanning grid sizes. This paper presents a covariance fitting approach for the mapping of acoustic correlated sources (MACS), which can work with uncorrelated, partially correlated or even coherent sources with a reasonably low computational complexity. MACS minimizes a quadratic cost function in a cyclic manner by making use of convex optimization and sparsity, and is guaranteed to converge at least locally. Simulations and experimental data acquired at the University of Florida Aeroacoustic Flow Facility with a 63-element logarithmic spiral microphone array in the absence of flow are used to demonstrate the performance of MACS.

On the Use of Convex Optimization in Sensor Network Localization and Synchronization

In this paper we report some new results obtained in the field of multi-agent systems that are based on convex optimization. First, we provide review of a set of polynomial function optimization tools including sum of squares (SOS) and semidefinite programming (SDP). Then we present several applications of these tools in various multiagent system localization and synchronization tasks. As the first application, we propose a method based on SOS relaxation for agent localization using noisy measurements and describe the solution through SDP. Later, we apply this method to address the problems of cooperative target localization in the presence of noise and robot pose determination based on range measurements. Then we introduce the problem of anchor selection for minimizing the effect of noise in sensor networks via SDP. We use the same machinery to propose a method based on SDP to enhance synchronizability in networks. We do so by proposing a distributed algorithm for adding new edges to the network to enhance synchronizability. Finally, we present a method to identify the node in a network loss of which inflicts the most damage on the synchronizability of the network. Conclusions are presented in the last section.

Real-time Embedded Convex Optimization

This talk concerns the use of convex optimization, embedded as part of a larger system that executes automatically with newly arriving data or changing conditions, in areas such as automatic control, signal processing, real-time estimation, real-time resource allocation and decision making, and fast automated trading. Such systems are already in use in applications such as model predictive control or supply chain optimization, with sample times measured in minutes (or longer); our focus is on systems with much faster dynamics, with execution times measured in milliseconds or microseconds for small and medium size problems. We describe a preliminary implementation of an automatic code generation system, which scans a description of the problem family and performs much of the analysis and optimization of the algorithm, such as choosing variable orderings used with sparse factorizations, at code generation time; compiling the generated source code yields an extremely efficient custom solver for the problem family.

Convexity, Differential Equations, and Games

Theoretical and experimental studies of noncooperative games increasingly recognize Nash equilibrium as a limiting outcome of players' repeated interaction. This note, while sharing that view, illustrates and advocates combined use of convex optimization and differential equations, the purpose being to render equilibrium both plausible and stable. Play, Differential Equations, Stability

Stability of uncertain systems with hysteresis nonlinearities

Stability of systems with hysteresis nonlinearities, parametric uncertainty and finite dimensional unmodelled dynamics is considered. Conditions for exponential decay of the signals in the system to an equilibrium position are given. The equilibrium is generally not unique. The stability condition is given in terms of a frequency domain inequality, which involves the transfer function of the nominal system and a multiplier, The search for a suitable multiplier can be performed by use of convex optimization in terms of linear matrix inequalities. A simple example is given that illustrates our result. © 1998 John Wiley & Sons, Ltd.

Simultaneous Optimization Problem of Passive Mechanical/Structural Elements and Active Control. Formulation of H.INF. Optimal Design Based on Descriptor Form.

A simultaneous optimization problem of mechanical/structural elements and controller parameters has been studied. In this paper, an H∞ simultaneous optimization problem is formulated, which is similar to the conventional H∞ optimization. The descriptor forms are used for a controlled object and a generalized plant because the structural parameters appear naturally in these forms. Static state feedback is considered as a control law. The objective function is the H∞ performance, which is the H∞ norm of the closed-loop transfer function from exogenous input to controlled output. In order to obtain the solvability conditions of the formulated problem, the descriptor form representation is transformed into a state space one. The conditions can be expressed in terms of linear matrix inequalities (LMI). An iterative procedure is proposed for solving LMI by use of convex optimization.

Robust control of constrained systems via convex optimization

AbstractA successful controller design paradigm must take into account both model uncertainty and design specifications. Model uncertainty can be successfully addressed using ℋ︁∞ robust control theory. However, this framework cannot directly accommodate the realistic case where in addition to robustness considerations the system is subject to both time‐ and frequency‐domain specifications, such as bounds on the control action. In this paper we propose a design procedure, based upon the use of convex optimization, that takes explicitly into account both time‐ and frequency‐domain specifications. The main result of the paper is a new framework to address problems having both control and output constraints and model uncertainty. Additionally, the paper serves as a brief tutorial on the issues involved in addressing design problems with multiple design specifications via convex optimization.