Non-convex Optimization Program Research Articles

A modification piecewise convexification method with a classification strategy for box-constrained non-convex optimization programs

A modification piecewise convexification method with a classification strategy for box-constrained non-convex optimization programs

Global Optimality of Inverter Dynamic Voltage Support

This paper investigates the dynamic voltage support (DVS) from inverter-based resources (IBRs) under voltage sags to enhance the low-voltage ride-through performance. The DVS problem is formulated as a nonconvex optimization program that maximizes the positive-sequence voltage magnitude at the point of common coupling (PCC) subject to the current, active power, and synchronization stability constraints. Then, we perform the optimality analysis to explore the global optimum analytically. It is found that the unique global optimum has three scenarios/stages (S1–S3), which depends on the specific relationship among grid voltage, grid strength, and physical limits of IBRs. The explicit closed-form solutions in S1 and S3 are derived, and the optimality conditions (an implicit solution) of S2 are provided, which guarantees the global optimality and enables the compatibility with fast real-time control. Finally, we implement the optimum with a grid-connected single-stage photovoltaic (PV) power plant by integrating a DVS controller. Dynamic simulations are carried out under different scenarios to test our proposal. The robustness against model errors is also discussed.

Stochastic model predictive controller for wind farm frequency regulation in waked conditions

As renewable energy penetrations increase, wind farms may be required to provide ancillary services such as frequency regulation. For wind farms, two issues complicate frequency regulation: turbine-wake interactions alter the available wind power, and the future power reference signal is uncertain. This paper presents a stochastic model predictive controller (MPC) for wind farms to track an uncertain frequency regulation signal while accounting for wakes. The controller uses a simplified Park model with wake propagation delay to predict turbine-wake interactions. The controller operates without any knowledge of the future power regulation signal or wind speed. The controller solves a nonconvex stochastic optimization program using scenario optimization. Our stochastic MPC tracks with 7%–10% less error than our deterministic MPC controllers.

Beamforming Optimization for Information Enhancement Transmission in MISO SWIPT with NOMA System

This paper mainly considers a communication network with multiple-input single-output (MISO), in which nonorthogonal multiple access (NOMA) decoding technology is adopted, and wireless power-carrying technology is used to enhance the transmission quality of user information in the communication network. Through joint optimization of power splitting ratio and beamforming vector, we aim at minimizing transmission power (base station transmit power and secondary transmit power) while satisfying the quality-of-service requirement of all users and minimum power for secondary transmission. The problem is a nonconvex optimization program, and it is difficult to get the optimal value directly. First, in order to solve the problem for two-user case, the original problem is transformed into a semidefinite programming (SDP) problem, and then, the iterative updating algorithm is used to approximate the optimal value. When the number of users is greater than two, the original problem is transformed into a second-order cone problem, in which we deal with a sequence of second-order cone programming (SOCP). Results verify that the optimal value of the sequence of SOCPs is not increasing and converges to a local optimal value. Detailed simulation experiments indicate that the algorithm improves the performance of NOMA downlink beamforming under the condition of enhancing information and eliminating network interference.

Non-convex primal-dual algorithm for image reconstruction in spectral CT

The work seeks to develop an algorithm for image reconstruction by directly inverting the non-linear data model in spectral CT. Using the non-linear data model, we formulate the image-reconstruction problem as a non-convex optimization program, and develop a non-convex primal-dual (NCPD) algorithm to solve the program. We devise multiple convergence conditions and perform verification studies numerically to demonstrate that the NCPD algorithm can solve the non-convex optimization program and under appropriate data condition, can invert the non-linear data model. Using the NCPD algorithm, we then reconstruct monochromatic images from simulated and real data of numerical and physical phantoms acquired with a standard, full-scan dual-energy configuration. The result of the reconstruction studies shows that the NCPD algorithm can correct accurately for the non-linear beam-hardening effect. Furthermore, we apply the NCPD algorithm to simulated and real data of the numerical and physical phantoms collected with non-standard, short-scan dual-energy configurations, and obtain monochromatic images comparable to those of the standard, full-scan study, thus revealing the potential of the NCPD algorithm for enabling non-standard scanning configurations in spectral CT, where the existing indirect methods are limited.

Systems-Theoretic Approaches to Design Biological Networks with Desired Functionalities.

The deduction of design principles for complex biological functionalities has been a source of constant interest in the fields of systems and synthetic biology. A number of approaches have been adopted, to identify the space of network structures or topologies that can demonstrate a specific desired functionality, ranging from brute force to systems theory-based methodologies. The former approach involves performing a search among all possible combinations of network structures, as well as the parameters underlying the rate kinetics for a given form of network. In contrast to the search-oriented approach in brute force studies, the present chapter introduces a generic approach inspired by systems theory to deduce the network structures for a particular biological functionality. As a first step, depending on the functionality and the type of network in consideration, a measure of goodness of attainment is deduced by defining performance parameters. These parameters are computed for the most ideal case to obtain the necessary condition for the given functionality. The necessary conditions are then mapped as specific requirements on the parameters of the dynamical system underlying the network. Following this, admissible minimal structures are deduced. The proposed methodology does not assume any particular rate kinetics in this case for deducing the admissible network structures notwithstanding a minimum set of assumptions on the rate kinetics. The problem of computing the ideal set of parameter/s or rate constants, unlike the problem of topology identification, depends on the particular rate kinetics assumed for the given network. In this case, instead of a computationally exhaustive brute force search of the parameter space, a topology-functionality specific optimization problem can be solved. The objective function along with the feasible region bounded by the motif specific constraints amounts to solving a non-convex optimization program leading to non-unique parameter sets. To exemplify our approach, we adopt the functionality of adaptation, and demonstrate how network topologies that can achieve adaptation can be identified using such a systems-theoretic approach. The outcomes, in this case, i.e., minimum network structures for adaptation, are in agreement with the brute force results and other studies in literature.

Optimization-based algorithm for solving the discrete x-ray transform with nonlinear partial volume effect

Purpose: Inverting the discrete x-ray transform (DXT) with the nonlinear partial volume (NLPV) effect, which we refer to as the NLPV DXT, remains of theoretical and practical interest. We propose an optimization-based algorithm for accurately and directly inverting the NLPV DXT. Methods: Formulating the inversion of the NLPV DXT as a nonconvex optimization program, we propose an iterative algorithm, referred to as the nonconvex primal-dual (NCPD) algorithm, to solve the problem. We obtain the NCPD algorithm by modifying a first-order primal-dual algorithm to address the nonconvex optimization. Subsequently, we perform quantitative studies to verify and characterize the NCPD algorithm. Results: In addition to proposing the NCPD algorithm, we perform numerical studies to verify that the NCPD algorithm can reach the devised numerically necessary convergence conditions and, under the study conditions considered, invert the NLPV DXT by yielding numerically accurate image reconstruction. Conclusion: We have developed and verified with numerical studies the NCPD algorithm for accurate inversion of the NLPV DXT. The study and results may yield insights into the effective compensation for the NLPV artifacts in CT imaging and into the algorithm development for nonconvex optimization programs in CT and other tomographic imaging technologies.

Online Energy Management for Microgrids With CHP Co-Generation and Energy Storage

In this brief, the online energy management problem of a grid-connected microgrid is studied. The considered microgrid is a typical system that consists of renewable energy generations (RG), local co-generations with combined heat and power supply, the electricity and heat energy storage systems (ESS) (e.g., battery and thermal tank), and the centralized power grid (PG), as well as the external natural gas station. This brief aims to minimize the aggregate microgrid’s operation cost by formulating it as a stochastic nonconvex optimization programming, which is challenging to solve optimally due to the coupling feature of energy storage devices. First, by investigating the unique structure of the formulated optimization problem, we relax the constraints and transform the original nonconvex stochastic optimization problem into a convex optimization programming. Furthermore, we tackle the problem by developing a modified Lyapunov optimization approach and design an online energy management algorithm that does not require any statistic information of the random system inputs (e.g., the purchasing price from the PG, the harvesting electricity from the RG, and the charging levels of the ESS, and so on). Moreover, extensive empirical evaluations using real-world traces are performed to study the effectiveness of the proposed algorithm in practice. Our proposed algorithm can reduce 27.5% of the aggregate cost compared with the benchmark approach.

Tensor Regression Using Low-Rank and Sparse Tucker Decompositions

This paper studies a tensor-structured linear regression model with a scalar response variable and tensor-structured predictors, such that the regression parameters form a tensor of order $d$ (i.e., a $d$-fold multiway array) in $\mathbb{R}^{n_1 \times n_2 \times \cdots \times n_d}$. It focuses on the task of estimating the regression tensor from $m$ realizations of the response variable and the predictors where $m\ll n = \prod \nolimits_{i} n_i$. Despite the seeming ill-posedness of this problem, it can still be solved if the parameter tensor belongs to the space of sparse, low Tucker-rank tensors. Accordingly, the estimation procedure is posed as a non-convex optimization program over the space of sparse, low Tucker-rank tensors, and a tensor variant of projected gradient descent is proposed to solve the resulting non-convex problem. In addition, mathematical guarantees are provided that establish the proposed method linearly converges to an appropriate solution under a certain set of conditions. Further, an upper bound on sample complexity of tensor parameter estimation for the model under consideration is characterized for the special case when the individual (scalar) predictors independently draw values from a sub-Gaussian distribution. The sample complexity bound is shown to have a polylogarithmic dependence on $\bar{n} = \max \big\{n_i: i\in \{1,2,\ldots,d \} \big\}$ and, orderwise, it matches the bound one can obtain from a heuristic parameter counting argument. Finally, numerical experiments demonstrate the efficacy of the proposed tensor model and estimation method on a synthetic dataset and a collection of neuroimaging datasets pertaining to attention deficit hyperactivity disorder. Specifically, the proposed method exhibits better sample complexities on both synthetic and real datasets, demonstrating the usefulness of the model and the method in settings where $n \gg m$.

Dimensionality Reduction of Sliced-Normal Distributions

Sliced-Normal (SN) distributions enable the characterization of complex multivariate data as both a vector of possibly dependent random variables and as a semi-algebraic, tightly enclosing set. SNs inject the physical space into a higher dimensional (so-called) feature space using a polynomial mapping. Optimization-based strategies for estimating SNs from data in both physical and feature space were recently developed. The formulations in physical space yield non-convex optimization programs whose solutions exhibit the best performance, whereas the formulations in feature space yield either an analytical solution or a convex program thereby facilitating their application to higher dimensional datasets. In both cases, however, the exponential dependency of the number of optimization variables on the dimension of feature space limits their application to moderately sized problems. Two strategies to mend for this deficiency are proposed herein. The first strategy identifies groups of highly interdependent parameters exhibiting a possibly nonlinear dependency using a distribution-free framework. This classification enables estimating a SN for any of such groups independently of the other groups thereby reducing the computational complexity of the estimation process. The second strategy reduces the dimension of feature space by only retaining the monomials of the polynomial mapping that significantly increase the likelihood of the data while leveraging lower dimensional SNs. A system identification example is used for illustration.

Efficient Resource Allocation for Relay-Assisted Computation Offloading in Mobile-Edge Computing

In this article, we consider the problem of relay assisted computation offloading (RACO), in which user A aims to share the results of computational tasks with another user B through wireless exchange over a relay platform equipped with mobile edge computing capabilities, referred to as a mobile edge relay server (MERS). To support the computation offloading, we propose a hybrid relaying (HR) approach employing two orthogonal frequency bands, where the amplify-and-forward scheme is used in one band to exchange computational results, while the decode-and-forward scheme is used in the other band to transfer the unprocessed tasks. The motivation behind the proposed HR scheme for RACO is to adapt the allocation of computing and communication resources both to dynamic user requirements and to diverse computational tasks. Within this framework, we seek to minimize the weighted sum of the execution delay and the energy consumption in the RACO system by jointly optimizing the computation offloading ratio, the bandwidth allocation, the processor speeds, as well as the transmit power levels of both user $A$ and the MERS, under practical constraints on the available computing and communication resources. The resultant problem is formulated as a non-differentiable and nonconvex optimization program with highly coupled constraints. By adopting a series of transformations and introducing auxiliary variables, we first convert this problem into a more tractable yet equivalent form. We then develop an efficient iterative algorithm for its solution based on the concave-convex procedure. By exploiting the special structure of this problem, we also propose a simplified algorithm based on the inexact block coordinate descent method, with reduced computational complexity. Finally, we present numerical results that illustrate the advantages of the proposed algorithms over state-of-the-art benchmark schemes.

Transient content caching and updating with modified harmony search for Internet of Things

Internet of Things (IoT) has emerged as one of the new use cases in the 5th Generation wireless networks. However, the transient nature of the data generated in IoT networks brings great challenges for content caching. In this paper, we study a joint content caching and updating strategy in IoT networks, taking both the energy consumption of the sensors and the freshness loss of the contents into account. In particular, we decide whether or not to cache the transient data and, if so, how often the servers should update their contents. We formulate this content caching and updating problem as a mixed 0–1 integer non-convex optimization programming, and devise a Harmony Search based content Caching and Updating (HSCU) algorithm, which is self-learning and derivative-free and hence stipulates no requirement on the relationship between the objective and variables. Finally, extensive simulation results verify the effectiveness of our proposed algorithm in terms of the achieved satisfaction ratio for content delivery, normalized energy consumption, and overall network utility, by comparing it with some baseline algorithms.

Solving Equations of Random Convex Functions via Anchored Regression

We consider the question of estimating a solution to a system of equations that involve convex nonlinearities, a problem that is common in machine learning and signal processing. Because of these nonlinearities, conventional estimators based on empirical risk minimization generally involve solving a non-convex optimization program. We propose anchored regression, a new approach based on convex programming that amounts to maximizing a linear functional (perhaps augmented by a regularizer) over a convex set. The proposed convex program is formulated in the natural space of the problem, and avoids the introduction of auxiliary variables, making it computationally favorable. Working in the native space also provides great flexibility as structural priors (e.g., sparsity) can be seamlessly incorporated. For our analysis, we model the equations as being drawn from a fixed set according to a probability law. Our main results provide guarantees on the accuracy of the estimator in terms of the number of equations we are solving, the amount of noise present, a measure of statistical complexity of the random equations, and the geometry of the regularizer at the true solution. We also provide recipes for constructing the anchor vector (that determines the linear functional to maximize) directly from the observed data.

Image reconstruction and scan configurations enabled by optimization-based algorithms in multispectral CT

Optimization-based algorithms for image reconstruction in multispectral (or photon-counting) computed tomography (MCT) remains a topic of active research. The challenge of optimization-based image reconstruction in MCT stems from the inherently non-linear data model that can lead to a non-convex optimization program for which no mathematically exact solver seems to exist for achieving globally optimal solutions. In this work, based upon a non-linear data model, we design a non-convex optimization program, derive its first-order-optimality conditions, and propose an algorithm to solve the program for image reconstruction in MCT. In addition to consideration of image reconstruction for the standard scan configuration, the emphasis is on investigating the algorithm’s potential for enabling non-standard scan configurations with no or minimum hardware modification to existing CT systems, which has potential practical implications for lowered hardware cost, enhanced scanning flexibility, and reduced imaging dose/time in MCT. Numerical studies are carried out for verification of the algorithm and its implementation, and for a preliminary demonstration and characterization of the algorithm in reconstructing images and in enabling non-standard configurations with varying scanning angular range and/or x-ray illumination coverage in MCT.

Optimal Energy Management for Microgrids with Combined Heat and Power (CHP) Generation, Energy Storages, and Renewable Energy Sources

This paper studies an energy management problem for a typical grid-connected microgrid system that consists of renewable energy sources, Combined Heat and Power (CHP) co-generation, and energy storages to satisfy electricity and heat demand simultaneously. We formulate this problem into a stochastic non-convex optimization programming to achieve the minimum microgrid’s operating cost, which is difficult to solve due to its non-convexity and coupling feature of constraints. Existing approaches such as dynamic programming (DP) assume that all the system dynamics are known, which results in a high computational complexity and thus are not feasible in practice. The focus of this paper is on the design of a real-time energy management strategy for the optimal operation of microgrids with low computational complexity. Specifically, derived from a modified Lyapunov optimization technique, an online algorithm with random inputs (e.g., the charging/discharging of energy storage devices, power from the CHP system, the electricity from external power grid, and the renewables generation, etc.), which requires no statistic system information, is proposed. We provide an implementation of the proposed energy management algorithm and prove its optimality theoretically. Based on real-world data traces, extensive empirical evaluations are presented to verify the performance of our algorithm.

Stochastic Control With Uncertain Parameters via Chance Constrained Control

Chance constrained methods handle problem uncertainty by formulating the constraints probabilistically and only guaranteeing their satisfaction up to a given violation level. This paper extends existing chance constrained methods to handle both uncertainty in the system state as well as in the constraint parameters. Due to the imperfect knowledge of the system state caused by motion, sensor and environment uncertainty, the system constraints cannot be guaranteed to be satisfied and consequently must be considered probabilistically. Prior work has primarily only dealt with uncertainty in the system state or constraint parameters but not both simultaneously, however, in many motivating applications both frequently occur. To handle this case, previous approaches either approximate the distribution leading to a nonconvex optimization program, or use sampling alone to represent the uncertainty which requires a large number of samples to accurately represent the distribution. To address these limitations, a novel hybrid method is proposed that uses both analytical functions and sampling to represent the uncertainty. It is shown that under certain conditions, the resulting optimization program using this hybrid representation is convex. To check the convexity, an efficient a priori, sufficient condition is developed. Furthermore, by using this hybrid representation, this method drastically reduces the computational complexity over previous methods, which is demonstrated through two examples. Consequently, this method has the ability to enable real-time stochastic control for the motivating applications.

Influence of Interval Uncertainty on the Behavior of Geometrically Nonlinear Elastoplastic Structures

AbstractThis paper proposes an interval analysis scheme to map out the complete bound spectrum of the most maximum and most minimum responses of geometrically nonlinear elastoplastic structures subjected to both interval applied loads and interval inelastic material properties. The proposed heuristic method uses a finite-step holonomic formulation under pseudodisplacement control. Geometric nonlinearity is modeled using a conventional second-order approximation. The analysis thus determines directly the most maximum and most minimum bound solutions by processing a pair of optimization problems, known as interval mathematical programs with equilibrium constraints or interval MPECs. The simultaneous presence of complementarity constraints and interval data is the main cause of difficulties (associated with nonconvex and/or nonsmooth optimization programs) underpinning the interval MPECs considered. The simple solution approach proposed reformulates the interval MPECs into their noninterval nonlinear program...

Rate-optimal fair power allocation in complex field network coded relay communications

In this paper, we consider the complex field network coded relay assisted communication (CFNC-RAC) channel. Although CFNC-RAC is spectrally efficient, its bit error rate performance is degraded by multi-access interference, which can be improved by appropriately allocating the user and relay powers. Since the fairness is an important factor for a practical multi-user communication system, we have proposed a rate-optimal fair power adaptation (ROFPA) technique in this work. The proposed ROFPA policy not only aims to maximize the average achievable sum-rate of CFNC-RAC under the use of the decode and forward relaying but also intends to satisfy the average rate-fairness restriction while taking the total power constraint and the network topology into account. We formulate the ROFPA as a non-convex optimization program and then derive an analytical solution for it. Extensive performance evaluation and numerical simulations validate that ROFPA method can provide significant sum-rate with considerable user fairness when compared to symbol-error-rate optimized (SER-OPT) policy proposed by Eritmen et al. (Wirel Netw, 2015. doi:10.1007/s11276-015-0924-1).

Optimal deleveraging with nonlinear temporary price impact

Abstract In this paper, we first propose a portfolio management model where the objective is to balance equity and liability. The asset price dynamics includes both permanent and temporary price impact, where the permanent impact is a linear function of the cumulative trading amount and the temporary impact is a k th (between 0 and 1) order power function of the instantaneous trading rate. We construct efficient frontiers to visualize the tradeoff between equity and liability and obtain analytical properties regarding the optimal trading strategies. In the second part, we further consider an optimal deleveraging problem with leverage constraints. It reduces to a non-convex polynomial optimization program with polynomial and box constraints. A Lagrangian method for solving the problem is presented and the quality of the solution is studied.

Design of a flight control architecture using a non-convex bundle method

We design a feedback control architecture for longitudinal flight of an aircraft. The multi-level architecture includes the flight control loop to govern the short-term dynamics of the aircraft, and the autopilot to control the long-term modes. Using \(H_\infty \) performance and robustness criteria, the problem is cast as a non-convex and non-smooth optimization program. We present a non-convex bundle method, prove its convergence, and show that it is apt to solve the longitudinal flight control problem.