Quasi-convex Optimization Research Articles

HIQCO: A Hierarchical Optimization Method for Computation Offloading and Resource Optimization in Multi-Cell Mobile-Edge Computing Systems

Mobile-Edge Computing (MEC) addresses the shortcomings of mobile devices in terms of computing performance and energy efficiency by offloading computation tasks to the edge of the network. In this paper, a multi-cell MEC system is considered, where each small cell connects to a common MEC server, and the problem of minimizing system overhead in the MEC system is solved by jointly optimizing the computation offloading decisions, communication and computational resources. This problem is a mixed integer nonlinear programming problem (MINLP). Owning to the combined nature and complexity of this problem, we adopt a method to transform the original problem into two sub-problems: (i) joint optimization computation offloading decision and channel assignment problem (JO-CODCA) and (ii) joint optimization transmission power and computational resource allocation problem (JO-TPCRA). For the transformed sub-problems, we propose a hierarchical optimization method (HIQCO) that combines immune algorithm (IA), quasi-convex optimization and convex optimization techniques. Experimental results show that the proposed HIQCO give better performance comparing with the other compared algorithms in terms of system overhead.

Fixed point quasiconvex subgradient method

Constrained quasiconvex optimization problems appear in many fields, such as economics, engineering, and management science. In particular, fractional programming, which models ratio indicators such as the profit/cost ratio as fractional objective functions, is an important instance. Subgradient methods and their variants are useful ways for solving these problems efficiently. Many complicated constraint sets onto which it is hard to compute the metric projections in a realistic amount of time appear in these applications. This implies that the existing methods cannot be applied to quasiconvex optimization over a complicated set. Meanwhile, thanks to fixed point theory, we can construct a computable nonexpansive mapping whose fixed point set coincides with a complicated constraint set. This paper proposes an algorithm that uses a computable nonexpansive mapping for solving a constrained quasiconvex optimization problem. We provide convergence analyses for constant diminishing step-size rules. Numerical comparisons between the proposed algorithm and an existing algorithm show that the proposed algorithm runs stably and quickly even when the running time of the existing algorithm exceeds the time limit.

Incremental quasi-subgradient methods for minimizing the sum of quasi-convex functions

The sum of ratios problem has a variety of important applications in economics and management science, but it is difficult to globally solve this problem. In this paper, we consider the minimization problem of a sum of a number of nondifferentiable quasi-convex component functions over a closed and convex set, which includes the sum of ratios problem as a special case. The sum of quasi-convex component functions is not necessarily to be quasi-convex, and so, this study goes beyond quasi-convex optimization. Exploiting the structure of the sum-minimization problem, we propose a new incremental subgradient method for this problem and investigate its convergence properties to a global optimal solution when using the constant, diminishing or dynamic stepsize rules and under a homogeneous assumption and the H\"{o}lder condition of order $p$. To economize on the computation cost of subgradients of a large number of component functions, we further propose a randomized incremental subgradient method, in which only one component function is randomly selected to construct the subgradient direction at each iteration. The convergence properties are obtained in terms of function values and distances of iterates from the optimal solution set with probability 1. The proposed incremental subgradient methods are applied to solve the sum of ratios problem, as well as the multiple Cobb-Douglas productions efficiency problem, and the numerical results show that the proposed methods are efficient for solving the large sum of ratios problem.

On the Sample Complexity of the Linear Quadratic Regulator

This paper addresses the optimal control problem known as the linear quadratic regulator in the case when the dynamics are unknown. We propose a multistage procedure, called Coarse-ID control, that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate. Our technique uses contemporary tools from random matrix theory to bound the error in the estimation procedure. We also employ a recently developed approach to control synthesis called System Level Synthesis that enables robust control design by solving a quasi-convex optimization problem. We provide end-to-end bounds on the relative error in control cost that are optimal in the number of parameters and that highlight salient properties of the system to be controlled such as closed-loop sensitivity and optimal control magnitude. We show experimentally that the Coarse-ID approach enables efficient computation of a stabilizing controller in regimes where simple control schemes that do not take the model uncertainty into account fail to stabilize the true system.

Quasiconvex optimization problems and asymptotic analysis in Banach spaces

ABSTRACT We use asymptotic analysis for dealing with quasiconvex optimization problems in reflexive Banach spaces. We study generalized asymptotic (recession) cones for nonconvex and nonclosed sets and its respective generalized asymptotic functions. We prove that the generalized asymptotic functions defined in previous works directly through closed formulae can also be generated from the generalized asymptotic cones. We establish three characterizations results for the nonemptiness and compactness of the solution set for noncoercive quasiconvex minimization problems using different asymptotic functions. Finally, we present a sufficient condition for the nonemptiness and boundedness of the solution set for quasiconvex pseudomonotone equilibrium problems.

Robust Beamforming and Power Allocation in CR MISO Networks with SWIPT to Maximize the Minimum Achievable Rate

In this paper, a new method is presented to maximize the minimum achievable rate of the users in a cognitive radio multiple input single output network. The secondary system provides simultaneous wireless information and power transfer (SWIPT) for energy harvesting receivers. Considering random Gaussian vector for the estimation error of the channel state information (CSI), we impose some outage constraints to guarantee the quality of service of the network. Outage constraints on the received power and the information leakage at the energy harvesting users are imposed to assure SWIPT. We introduce new convex inequalities to replace the original non-convex constraints and write the objective function as the minimization of the maximum of some fractional functions. This new objective function and convex inequalities result in a new quasi-convex general fractional programming problem. The new quasi-convex optimization problem is written in such a way that the computational costs to solve this problem are as low as possible. We develop an algorithm to obtain the optimal beamforming weights and artificial noise covariance matrix. All stochastic constraints are rewritten for the special case that the covariance matrices of error in the estimation of CSI are a factor of identity matrix. This reformulation results in less computation costs to obtain optimal beamforming weights and AN covariance matrix. Simulation results confirm that, all stochastic constraints are satisfied with certain probability. In comparison with the previous robust related work, the proposed method achieves higher rates which confirms superiority of our method.

Joint Task Offloading and Resource Allocation for Multi-Server Mobile-Edge Computing Networks

Mobile-edge computing (MEC) is an emerging paradigm that provides a capillary distribution of cloud computing capabilities to the edge of the wireless access network, enabling rich services and applications in close proximity to the end users. In this paper, an MEC enabled multi-cell wireless network is considered where each base station (BS) is equipped with a MEC server that assists mobile users in executing computation-intensive tasks via task offloading. The problem of joint task offloading and resource allocation is studied in order to maximize the users’ task offloading gains, which is measured by a weighted sum of reductions in task completion time and energy consumption. The considered problem is formulated as a mixed integer nonlinear program (MINLP) that involves jointly optimizing the task offloading decision, uplink transmission power of mobile users, and computing resource allocation at the MEC servers. Due to the combinatorial nature of this problem, solving for optimal solution is difficult and impractical for a large-scale network. To overcome this drawback, we propose to decompose the original problem into a resource allocation (RA) problem with fixed task offloading decision and a task offloading (TO) problem that optimizes the optimal-value function corresponding to the RA problem. We address the RA problem using convex and quasi-convex optimization techniques, and propose a novel heuristic algorithm to the TO problem that achieves a suboptimal solution in polynomial time. Simulation results show that our algorithm performs closely to the optimal solution and that it significantly improves the users’ offloading utility over traditional approaches.

拟凸优化问题严格解的最优性必要条件

拟凸优化问题严格解的最优性必要条件

D2D Communication Mode Selection and Resource Optimization Algorithm With Optimal Throughput in 5G Network

Aiming at the problem of device-to-device (D2D) communication mode selection and resource optimization under the joint resource allocation mode in the 5G communication network, a probabilistic integrated resource allocation strategy and a quasi-convex optimization algorithm based on channel probability statistical characteristics are proposed. This strategy and algorithm guarantee D2D. Communication maximizes total system throughput while maximizing access. The analysis results show that this algorithm can significantly optimize the total throughput of the system and reduce the communication interference between the users, which proves the rationality and efficiency of the communication model. The research results obtained in Muwen can provide a theoretical basis for analyzing more complex D2D communication systems and provide a numerical basis for designing heuristic algorithms.

High-Performance Beampattern Synthesis via Linear Fractional Semidefinite Relaxation and Quasi-Convex Optimization

This paper presents a new method to synthesize high-performance beampatterns, with the aid of linear fractional semidefinite relaxation (LFSDR) technique and a quasi-convex optimization approach. We consider two beampattern synthesis problems. The first one is how to determine the weight vector to maximize the array gain (or equivalently, to minimize the gain loss in mainlobe), under the condition that the amplitude response satisfies specific requirements. The second one is how to minimize the notch level at a given region, on the premise of a permissible gain loss in mainlobe. To these ends, two nonconvex optimization problems are first formulated and then relaxed to their quasi-convex forms, by using the LFSDR technique. To further solve the resultant problems, the bisection method, which is commonly used in quasi-convex optimization, is adopted. Suboptimal solutions to the original nonconvex problems are finally obtained through eigenvalue decomposition or randomization manipulations. The proposed method performs well in both the cases described earlier. Moreover, our method is not limited to the array configurations and/or noise environments. Representative simulations are presented to demonstrate the effectiveness of the proposed method in high-performance beampattern synthesis.

A Quasiconvex Asymptotic Function with Applications in Optimization

We introduce a new asymptotic function, which is mainly adapted to quasiconvex functions. We establish several properties and calculus rules for this concept and compare it to previous notions of generalized asymptotic functions. Finally, we apply our new definition to quasiconvex optimization problems: we characterize the boundedness of the function, and the nonemptiness and compactness of the set of minimizers. We also provide a sufficient condition for the closedness of the image of a nonempty closed and convex set via a vector-valued function.

Maximizing the minimum achievable rates in Cognitive Radio networks subject to stochastic constraints

In this paper, considering errors in estimating the channel state information (CSI), we investigate the problem of robust beamforming in cognitive radio (CR) networks to maximize the minimum achievable rates for secondary users (SU). In addition to the constraints on the transmit power of the users, stochastic constraints on the signal to interference plus noise ratio (SINR) at SUs and interference power at the primary users (PU) are imposed to guarantee the quality of service (QoS) of the network. Bernstein inequalities and semi-definite relaxation are used to transform stochastic constraints to equivalent deterministic inequalities. By replacing new deterministic constraints in the optimization problem and defining new matrices, we write the problem of finding optimal beamforming weights in the form of quasiconvex optimization problem. Generalization of the Dinkelbach’s method is used to obtain optimal beamforming weights. Also, the problem of finding optimum beamforming weights is solved for the case that perfect CSI is available at the transmitters. Simulation results confirm that, the proposed method provides higher achievable rates in comparison with the previous works that minimize the total transmit power. The proposed method is robust because stochastic constraints are satisfied while the estimation of CSI includes some errors.

Battery Energy Storage System Control for Intermittency Smoothing Using an Optimized Two-Stage Filter

A new method for the control of a battery energy storage system and its implementation on a 25 kW solar system to compensate solar power intermittency and improve distribution grid power quality is presented in this paper. The novelty of the proposed method is to provide a systematic way to optimize the size of the battery capacity for the desired level of solar power smoothing. This goal is achieved by designing a two-stage filter solution. The first stage is a fast response digital finite impulse response (FIR) filter that makes a trade-off between smoothing of the solar output and battery capacity. This paper proposes an optimal design of a minimum-length, low-group-delay FIR filter by employing convex optimization, discrete signal processing, and polynomial stabilization techniques. The new strategy proposed in this paper formulates the design of a length- $N$ low-group-delay FIR filter as a convex second-order cone programming, which guarantees that all the filter zeros are inside the unit circle (minimum-phase). A quasi-convex optimization problem is formulated to minimize the length of the low-group-delay FIR filter. The second-stage filter is designed to level the battery charging load. The effectiveness and performance of the proposed approach is demonstrated by simulation results and also over a real-case implementation.

Abstract convergence theorem for quasi-convex optimization problems with applications

Quasi-convex optimization is fundamental to the modelling of many practical problems in various fields such as economics, finance and industrial organization. Subgradient methods are practical iterative algorithms for solving large-scale quasi-convex optimization problems. In the present paper, focusing on quasi-convex optimization, we develop an abstract convergence theorem for a class of sequences, which satisfy a general basic inequality, under some suitable assumptions on parameters. The convergence properties in both function values and distances of iterates from the optimal solution set are discussed. The abstract convergence theorem covers relevant results of many types of subgradient methods studied in the literature, for either convex or quasi-convex optimization. Furthermore, we propose a new subgradient method, in which a perturbation of the successive direction is employed at each iteration. As an application of the abstract convergence theorem, we obtain the convergence results of the proposed subgradient method under the assumption of the Hölder condition of order p and by using the constant, diminishing or dynamic stepsize rules, respectively. A preliminary numerical study shows that the proposed method outperforms the standard, stochastic and primal-dual subgradient methods in solving the Cobb–Douglas production efficiency problem.

Motion Estimation Based on Two Corresponding Points and Angular Deviation Optimization

Recently, there have been several studies on vision-based motion estimation under a supposition that planar motion follows a nonholonomic constraint. This allows reducing computational time. However, the vehicle motion in an outdoor environment does not accept this assumption. This paper presents a method for estimating the vision-based 3-D motion of a vehicle with several parts as follows. First, the Ackermann steering model is applied to reduce constraint parameters of the 3-D motion. In difference to the previous contribution, the proposed approach requires only two corresponding points of consecutive images to estimate the vehicle motion. Second, motion parameters are extracted based on a closed-form solution on geometric constraints. Third, the estimation approach applies the bundle adjustment-based quasiconvex optimization. This task aims to take into account advantage of omnidirectional vision-based features for reducing errors. The omnidirectional vision supports for landmarks tracking in long travel and large rotation, which is appropriate for a bundle adjustment technique. Evaluated results show that the proposed method is applicable in the practical condition of outdoor environments.

Input-Constrained Controller Design of Linear Time-Varying Systems Based on Piecewise State Estimation

Finite-time control problem of linear time-varying systems with input constraints is considered in this paper. Successive ellipsoidal approximations are used to estimate the state evolution of linear time-varying systems during a certain finite-time interval. An algorithm to design a controller based on approximations of state evolution is proposed. According to the proposed algorithm, the speed of state approaching equilibrium is optimized piecewisely using admissible control. The controller gain can be obtained by solving several quasi-convex optimization problems, which makes the design process computationally tractable. Simulation results show that the proposed controller can quickly reduce state deviation without violating input constraints.

Performance analysis of superior selective reporting-based energy efficient cooperative spectrum sensing in cognitive radio networks

We study the energy efficiency of superior selective reporting-based schemes for spectrum sensing in cognitive radio networks. We first consider the superior selective reporting (SSR) scheme proposed earlier in the literature, and derive the achievable throughput, energy consumption and energy efficiency (EE) of the SSR scheme. We propose the maximization of EE for the SSR scheme as a multiple variable-based, non-convex optimization problem and provide approximations to reduce it to a quasi-convex optimization. We highlight that the errors due to these approximations are negligible. The SSR scheme is designed to optimize the energy consumption, which enhances the EE. Alternatively, EE can be improved by increasing the achievable throughput. Towards this end, we propose a novel variation on the SSR scheme called the opportunistic SSR (OSSR) scheme, and carry out its EE analysis. We study the tradeoff between performances of the SSR and OSSR schemes – the implicit tradeoff between achievable throughput and energy consumption, and discuss the regimes where OSSR is preferred over SSR and vice-versa, in terms of the EE. Also, through an extensive numerical study, we show that both schemes outperform the conventional schemes that employ the OR and AND fusion rules, in terms of energy efficiency.

Joint optimization of downlink and D2D transmissions for SVC streaming in cooperative cellular networks

We study the problem of broadcasting scalable video coded (SVC) streams in cellular networks, where all user equipments (UEs) require the same video content and cooperate with each other. To take the advantage of channel diversity gains, the base station (BS) uses network coding to generate linear combinations of video packets for broadcasting in the downlink. Once receiving sufficient number of the combinations, a UE can relay packets to others via device-to-device (D2D) connections. The optimal downlink and D2D transmission arrangement problem is formulated as a mixed integer non-linear programming (MINLP) problem, which becomes difficult to solve when the number of UEs increases. Then we convert the MINLP problem to a quasi-convex optimization problem using a continuous step function, and a primal-dual decomposition approach is used to solve it in a distributed way. In addition, we consider the influence of the transmission energy consumption to the optimal solution. Simulation results show that the proposed approach achieves near-optimal performance.

Optimal price-threshold control for battery operation with aging phenomenon: a quasiconvex optimization approach

This paper is concerned with grid-level battery storage operations, taking battery aging into consideration. Battery operations under price uncertainty are modeled as a Markov Decision Process with expected cumulated discounted rewards. The structure of the optimal policy is studied. An algorithm that takes advantage of the problem structure and works directly on the continuous state space is developed to maximize the objective over the life of the battery. The algorithm determines an optimal policy by solving a sequence of quasiconvex problems indexed by a battery-life state. Computational results are presented to compare the proposed approach to a standard dynamic programming method, and to evaluate the impact of refinements in the battery model. Error bounds for the proposed algorithm are established to demonstrate its accuracy.

Spectrum-Energy Efficiency Optimization for Downlink LTE-A for Heterogeneous Networks

Heterogeneous networks have been pointed out to be one of the key network architectures that help increase system capacity and reduce power consumption for efficient communications. Although conceivably, high operational efficiency brings a high profit for mobile service providers, it is noteworthy that the potential for maximizing the profit has not been explored for the heterogeneous environment. This paper investigates profitability for network operators with the spectrum-energy efficiency metric on the downlink of LTE Advanced communication systems. We pursue optimal policies by employing the techniques of cell size zooming, user migration, and sleep mode in the deployment of different base station types. The problem is formulated as a quasiconvex optimization problem and it is transformed into an equivalent form of the MILP problem; the former is solved with a bisection algorithm and the latter is approached by an off-the-shelf software package. Since the formulated optimization problem is NP hard, a sub-optimal approach with a lower computational complexity is also proposed. Numerical analysis through case studies are presented to evaluate the efficiency improvements, and demonstrate the performance of the near-optimal solution.