Quasi-convex Optimization Problem Research Articles

Karush-Kuhn-Tucker optimality conditions for non-smooth geodesic quasi-convex optimization on Riemannian manifolds

This paper is committed to studying Karush-Kuhn-Tucker (in short, KKT) type necessary and sufficient optimality conditions for non-smooth quasi-convex (geodesic sense) optimization problems on Riemannian manifolds. Recently, Ansari et al. [Ansari QH, Babu F, Zeeshan M. Incremental quasi-subgradient method for minimizing geodesic quasi-convex function on Riemannian manifolds with applications. Numer Funct Anal Optim. 2022;42(13):1492–1521. doi: 10.1080/01630563.2021.2001823] defined the quasi-subdifferential on Riemannian manifolds and established the existence results of the quasi-subdifferential. We provide several auxiliary results for the quasi-subdifferential in the current study. We offer the KKT optimality conditions for the quasi-convex optimization problems on Riemannian manifolds with or without the Slater-constraint qualifications. To verify the suggested outcomes, we formulate numerical examples. In addition, we also provide our results in the Euclidean spaces, which are original and distinct from earlier findings in the Euclidean spaces.

Inexact proximal point algorithm for quasiconvex optimization problems on Hadamard manifolds

ABSTRACT In this paper, by using the inexact scalarization proximal methods, we solve quasiconvex multiobjective optimization problems on Hadamard manifolds. Under some assumptions on the multifunction of problem and vector fields, our methods are proved to be convergent to a Pareto critical point of the problem. For the convex case, the sequences generated by the methods converge to a weak Pareto solution.

Robust Structural Stability, Stability Margins, and Maximum Uncertainty Amplification for 2D Uncertain Systems via Structured Lyapunov Functions and Matrix Annihilators

This paper considers 2D uncertain systems with homogeneous or mixed dynamics, and addresses three problems: establishing robust structural stability, computing robust structural stability margins, and determining the maximum uncertainty amplification that preserves robust structural stability. Two sufficient linear matrix inequality (LMI) conditions are proposed for establishing robust structural stability obtained by introducing an equivalent closed-loop complex system and by searching for real or complex structured Lyapunov functions and matrix annihilators parameterized by the uncertainties and auxiliary quantities. Moreover, it is shown that lower bounds of the introduced robust structural stability margins and maximum uncertainty amplification can be obtained by solving quasi-convex optimization problems under some restrictions on the sought Lyapunov functions or on the set of uncertainties. Lastly, the nonconservatism of the proposed results is analyzed, showing that the proposed LMI condition based on the use of complex Lyapunov functions is not only sufficient but also necessary under some assumptions.

Interior quasi-subgradient method with non-Euclidean distances for constrained quasi-convex optimization problems in hilbert spaces

Interior quasi-subgradient method with non-Euclidean distances for constrained quasi-convex optimization problems in hilbert spaces

Multi-sensor Kalman filtering over packet-dropping networks subject to Round-Robin protocol scheduling

This paper considers the filtering problem for a class of linear cyber-physical systems (CPSs) subject to the Round-Robin protocol (RRP) scheduling, where the RRP is adopted to efficiently avoid data collisions in multi-sensor application scenarios. Unlike most of the existing results concerning the scheduling effects of the RRP under reliable communication channels, the filtering problem over packet-dropping networks is investigated. In such a framework, an optimal Kalman-type recursive filter is derived in the minimum mean square error (MMSE) sense, which is different from the suboptimal filters with bounded error covariances proposed in the previous results. Due to the protocol-induced behaviors and the unreliability of the channels, the estimator may be unstable. Thus, the stability problem of the filter is mainly discussed. It can be proved that the filter is stable when the arrival rate of the measurements exceeds a certain threshold, where the threshold can be obtained by solving a quasi-convex optimization problem. Furthermore, a sufficient condition for the existence of the steady-state error covariance is presented and can be transferred into the feasibility of a certain linear matrix inequality (LMI). Finally, a simulation example is provided to demonstrate the developed results.

Time-Optimal Point Stabilization Control for WIP Vehicles Using Quasi-Convex Optimization and B-Spline Adaptive Interpolation Techniques

This paper addresses a time-optimal point stabilization control for inherent unstable wheeled inverted pendulum (WIP) vehicles using quasi-convex optimization and B-spline adaptive interpolation techniques. First, to handle the difficulty caused by the inherent unstable characteristic, a state feedback control law is introduced to the state-space model deduced with the kinematic coupling relationship between the longitudinal motion and tilt angle of vehicle body. Then, by system discretization, a standard quasi-convex optimization problem is formulated to plan a time-optimal trajectory with various constraints being taken into account, and the optimization problem can be solved by a bisection method combined with solving a series of convex feasibility problems. Next, to obtain the analytic expression of the discrete optimal trajectory, a B-spline interpolation algorithm with adaptive curve refinement is presented based on feature points recognition. Furthermore, as the time-optimal trajectories with analytic expressions, including displacement, velocity, and acceleration trajectories, have been obtained, a proportional-integral-derivative (PID) tracking controller is applied to control the vehicle to track the optimal trajectories and, thus, the time-optimal point stabilization control for the WIP vehicle can be realized. Finally, simulation results of a numerical example is presented to validate the feasibility and effectiveness of the proposed method.

Convergence rates of subgradient methods for quasi-convex optimization problems

Quasi-convex optimization acts a pivotal part in many fields including economics and finance; the subgradient method is an effective iterative algorithm for solving large-scale quasi-convex optimization problems. In this paper, we investigate the quantitative convergence theory, including the iteration complexity and convergence rates, of various subgradient methods for solving quasi-convex optimization problems in a unified framework. In particular, we consider a sequence satisfying a general (inexact) basic inequality, and investigate the global convergence theorem and the iteration complexity when using the constant, diminishing or dynamic stepsize rules. More importantly, we establish the linear (or sublinear) convergence rates of the sequence under an additional assumption of weak sharp minima of Holderian order and upper bounded noise. These convergence theorems are applied to establish the iteration complexity and convergence rates of several subgradient methods, including the standard/inexact/conditional subgradient methods, for solving quasi-convex optimization problems under the assumptions of the Holder condition and/or the weak sharp minima of Holderian order.

Convex Optimization Over Risk-Neutral Probabilities

We consider a collection of derivatives that depend on the price of an underlying asset at expiration or maturity. The absence of arbitrage is equivalent to the existence of a risk-neutral probability distribution on the price; in particular, any risk neutral distribution can be interpreted as a certificate establishing that no arbitrage exists. We are interested in the case when there are multiple risk-neutral probabilities. We describe a number of convex optimization problems over the convex set of risk neutral price probabilities. These include computation of bounds on the cumulative distribution, VaR, CVaR, and other quantities, over the set of risk-neutral probabilities. After discretizing the underlying price, these problems become finite dimensional convex or quasiconvex optimization problems, and therefore are tractable. We illustrate our approach using real options and futures pricing data for the S&P 500 index and Bitcoin.

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.

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.

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

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.

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.

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.

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.