Feasibility Problem Research Articles

Finite-Time Distributed Flow Balancing

We consider a flow network that is described by a digraph, each edge of which can admit a flow within a certain interval, with nonnegative end points that correspond to lower and upper flow limits. We propose and analyze a distributed iterative algorithm for solving, in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">finite time</i> , the so-called feasible circulation problem, which consists of computing flows that are <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">admissible</i> (i.e., within the given intervals at each edge) and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">balanced</i> (i.e., the total in-flow equals the total out-flow at each node). The algorithm assumes a communication topology that allows bidirectional message exchanges between pairs of nodes that are physically connected (i.e., nodes that share a directed edge in the physical topology) and is shown to converge to a feasible and balanced solution as long as the necessary and sufficient circulation conditions are satisfied with strict inequality. In case, the initial flows and flow limits are commensurable (i.e., they are integer multiples of a given constant), then the proposed algorithm reduces to a previously proposed finite-time balancing algorithm, for which we provide an explicit bound on the number of steps required for termination.

Optimal Resource Management for NOMA-Based Visible Light Communication Systems With Shot Noise

In this paper, the problem of power allocation is investigated for downlink visible light communication (VLC) systems with non-orthogonal multiple access. In the system, both input-dependent shot noise and input-independent Gaussian noise are considered due to the properties of practical VLC channels. We consider two scenarios for the transmitter with a single light emitting diode (LED) and multiple LEDs, respectively. For single-LED scenarios, the problem is posed as a joint optimization problem of alternating current and direct current power control, whose goal is to maximize the minimum signal-to-interference-plus-noise-ratio (SINR) of all users. Although the original problem is non-convex, a geometric programming based algorithm is proposed to acquire the global optimal solution. For multi-LED scenarios, the transmit and dimming beamforming vectors are jointly optimized to maximize the minimum SINR of all users. A joint optimization algorithm that alternately optimizes transmit and dimming beamforming vectors is proposed. Particularly, the transmit beamforming problem is converted into a second-order cone programming problem through the successive convex approximation method. The dimming beamforming problem is transformed into a feasibility problem utilizing a low-complexity bisection algorithm. Numerical results reveal the superior performance of the proposed algorithm compared with its orthogonal multiple access counterpart and conventional schemes without shot noise.

Constrained Reference Tracking via Structured Input–Output Finite-Time Stability

In this article, the control approach based on structured input–output finite-time stability (IO-FTS) is extended to tackle the reference tracking problem. IO-FTS was originally introduced to deal with the disturbance rejection problem. By applying the finite-time stability control to a properly augmented system, we show that it is possible to enforce a set of specific requirements on the response of the closed-loop system during the transients, taking also into account saturation constraints on the actuators. Adding IO-FTS constraints to classic-state-feedback control law leads to a feasibility problem with bilinear matrix inequality (BMI) constraints. Herein, we show how the original BMI problem can be relaxed to a linear matrix inequality (LMI) one, which comes at a price of more conservatism, but turns out to be computationally more efficient. To prove the effectiveness of the proposed approach, we consider the case of the longitudinal control of a missile.

Regularization methods for solving the split feasibility problem with multiple output sets in Hilbert spaces

We study the split feasibility problem with multiple output sets in Hilbert spaces. In order to solve this problem, we introduce several new iterative processes by using the Tikhonov regularization method.

Image restorations using a modified relaxed inertial technique for generalized split feasibility problems

In this article, we propose a new efficient self‐adaptive method and prove that it converges strongly to a minimum‐norm solution of a generalized split feasibility problem in real Hilbert spaces. The proposed method is derived from a definite discrete dynamical system in time, which combines both the relaxation and inertial techniques for the purpose of increasing the rate of convergence of the iterative scheme. Furthermore, the method requires the monotonicity and Lipschitz continuity condition of the underlying single‐valued cost operator , and it employs some simple self‐adaptive stepsizes that are generated at each iteration by some easy computations. As a by‐product, we obtain methods for solving other classes of generalized split feasibility problems in real Hilbert spaces. Two major merits of our scheme in solving image restoration problems over related schemes are the higher signal‐to‐noise ratio value and lower CPU time for generating recovered images. Finally, we analyze our methods with different related strong convergent methods in the literature.

Inside-Ellipsoid Outside-Sphere (IEOS) model for general bilinear feasibility problems: Feasibility analysis and solution algorithm

This paper deals with general bilinear feasibility problems. A nonlinear transformation is introduced that reformulates a general bilinear feasibility problem as a Linear Matrix Inequality (LMI) problem augmented with a single non-convex quadratic constraint. The single non-convex quadratic constraint has a regular concave constraint function. Due to the LMI part of this formulation, it is easier to analyze, and we prove that the solution space of this formulation is located inside several ellipsoids and outside a sphere. This leads to our proposed Inside-Ellipsoid and Outside-Sphere (IEOS) model for general bilinear feasibility problems. Then, the feasibility analysis of our proposed IEOS model is performed. The related necessary feasibility conditions and sufficient feasibility conditions are theoretically developed. Moreover, an iterative algorithm for solving our IEOS model is also proposed.Two applications including matrix-factorization problem in control systems and power-flow problem in power systems are considered to evaluate the practicality of our proposed approach. Both problems are formulated as IEOS models. It is shown that our proposed model can provide more accurate solutions to these problems as compared to previous competing approaches in the relevant literature.

The method for solving the extension of general of the split feasibility problem and fixed point problem of the cutter

This paper first introduces a new iterative method for weak and strong convergence theorem to demonstrate the estimation potential for a fixed point of the cutter and the finite general split feasibility problem. Consequently, the set of fixed points of a quasi-nonexpansive mapping and the finite general split feasibility problem, the constrained minimization problem, and the general constrained minimization problems are proved using our main results. Finally, we give two numerical examples to advocate our main results.

Finite-time stabilization of state dependent impulsive dynamical linear systems

The problem of finite-time stabilizing control design for state-dependent impulsive dynamical linear systems (SD-IDLS) is tackled in this paper. Such systems are characterized by continuous-time, linear, possibly time-varying, dynamics coupled with discrete-time, linear, possibly time-varying, dynamics. The continuous-time part determines the system evolution in any time interval between two consecutive resetting events, while the discrete-time part governs its instantaneous state jump whenever the system trajectory intersects a resetting set, i.e. a region of the state space assumed to be time-independent. By making use of a quadratic control Lyapunov function, the finite-time stabilization of SD-IDLS through a static output feedback control design is specifically discussed in this paper. A sufficient and constructive result is provided based on the conical hulls of the resetting set subregions and on some cone copositivity properties of the chosen control Lyapunov function. Such a result is based on the solution of a feasibility problem that involves a set of coupled Difference/Differential Linear Matrix Inequalities (D/DLMI), which is shown to be less conservative and more numerically amenable with respect to other results available in the literature. An example illustrates the effectiveness of the proposed approach.

Optimizable Control Barrier Functions to Improve Feasibility and Add Behavior Diversity while Ensuring Safety

Ensuring safety while retaining maximum performance is a basic requirement for automatic cyber-physical systems, especially for safety-critical applications. A quadratic programming optimization framework called MPC-CBF has recently been presented, which directly unifies model predictive control (MPC) with control barrier functions (CBFs) over the prediction time horizon. However, the conservative nature of CBFs can lead to feasibility problems in real applications. Based on the analysis of the role of the decay rate and the conservative accumulation phenomenon in standard CBF formulations, this paper proposes to directly optimize CBF constraints within the MPC framework. By regarding CBFs as a safety restriction level indicator and an optimizable constraint within the MPC framework, the trade-off between feasibility and safety can be adaptively optimized. The proposed Optimizable CBF (OCBF) model removes the hyper-parameters selection problem in standard CBFs and can adaptively adjust the safety restriction level and increase behavior diversity by adding the corresponding objects in the cost function in MPC. To eliminate the accumulation effects of actual values of the CBF constraints in previous time steps, this paper further proposes a General OCBF (GOCBF) formulation. Compared with existing formulations, the safety margin defined in our GOCBF has intuitive physical meanings and thus provides a more flexible and intuitive mechanism to compromise different objects in terms of ensuring safety while not undermining basic feasibility. Experimental results demonstrate that our algorithm provides a more flexible and intuitive mechanism to achieve this, thus improving feasibility and adding behavior diversity in the MPC-CBF framework.

Approximating Common Fixed Points of Nonexpansive Mappings on Hadamard Manifolds with Applications

The point of this research is to present a new iterative procedure for approximating common fixed points of nonexpansive mappings in Hadamard manifolds. The convergence theorem of the proposed method is discussed under certain conditions. For the sake of clarity, we provide some numerical examples to support our results. Furthermore, we apply the suggested approach to solve inclusion problems and convex feasibility problems.

Stabilizing control design for state-dependent impulsive dynamical linear systems

The stabilizing control design problem for state-dependent impulsive dynamical linear systems (SD-IDLS) is tackled in this paper. This class of systems consists of a continuous-time, linear time-invariant system combined with discrete-time linear time-invariant dynamics in a prescribed region of the state space. The former regulates the evolution of the system between any two consecutive resetting events, while the latter governs the instantaneous state jumps occurring whenever the system trajectory intersects a resetting set, which is a portion of the state space assumed to be time-independent. Stabilization of SD-IDLS through a state feedback control design is specifically discussed in this paper, by making use of a candidate quadratic control Lyapunov function. By considering the conical hulls of the resetting set subregions and imposing some cone copositivity properties on the chosen control Lyapunov function, a sufficient and constructive condition for the global asymptotic stabilization of SD-IDLS is provided. Such a result, based on the solution of a feasibility problem that involves a set of Linear Matrix Inequalities (LMIs), is shown to be less conservative and more numerically amenable with respect to other results available in the literature. An example illustrates the effectiveness of the proposed approach.

Convergence Analysis under Consistent Error Bounds

We introduce the notion of consistent error bound functions which provides a unifying framework for error bounds for multiple convex sets. This framework goes beyond the classical Lipschitzian and Hölderian error bounds and includes logarithmic and entropic error bounds found in the exponential cone. It also includes the error bounds obtainable under the theory of amenable cones. Our main result is that the convergence rate of several projection algorithms for feasibility problems can be expressed explicitly in terms of the underlying consistent error bound function. Another feature is the usage of Karamata theory and functions of regular variations which allows us to reason about convergence rates while bypassing certain complicated expressions. Finally, applications to conic feasibility problems are given and we show that a number of algorithms have convergence rates depending explicitly on the singularity degree of the problem.

Quantifying multiqubit magic channels with completely stabilizer-preserving operations

In this paper, we extend the resource theory of magic to the channel case by considering completely stabilizer preserving operations (CSPOs) as free. We introduce and characterize the set of CSPO preserving and completely CSPO preserving superchannels. We quantify the magic of quantum channels by extending the generalized robustness and the min relative entropy of magic from the state to the channel domain and show that they bound the single-shot dynamical magic cost and distillation. We also provide analytical conditions for qubit interconversion under CSPOs and show that it is a linear programming feasibility problem and hence can be efficiently solved. Lastly, we give a classical simulation algorithm whose runtime is related to the generalized robustness of magic for channels. Our algorithm depends on some pre-defined precision, and if there is no bound on the desired precision then it achieves a constant runtime.

MICRO, SMALL AND MEDIUM ENTERPRISES ACCESS TO FINANCE CHALLENGES IN SUPPLY SIDE PERSPECTIVE IN ETHIOPIA

Empirical evidence shows that the contribution of Micro, Small and Medium Enterprises Empirical evidence shows that the contribution of Micro, Small and Medium Enterprises (MSMEs) to gross domestic product growth, job creation, and poverty alleviation is high. Because of limited access to finance and other factors, a significant number of MSMEs, particularly in developing countries, cannot realize their full potential. The study investigates the micro, small, and medium enterprises' access to finance challenges from the supply-side perspective in Ethiopia. The study employed descriptive analysis using survey data collected from purposely selected 11 banks and 9 micro-finance institutions (MFI). The results show that the lack of collateral, unavailability of financial records, poor pre-loan savings, low loan repayment rate, business feasibility problems, credit information asymmetry, attitudinal problems of the MSMEs, and diversion of the loan are demand-side constraints. From the financial institution’s perspective, the result shows that the liquidity problem, the influence of unfair competition from government banks, the lack of competent human resources, political leaders’ intervention, especially in MFI, corruption, the COVID-19 pandemic, political instability, and inadequate capital are major obstacles. Macroeconomic conditions related to challenges such as foreign exchange shortages, inflation, and trade imbalances are also mentioned as major obstacles. According to the findings of this study, designing and implementing business skill development in mindset, business planning, business bookkeeping, and financial reports can help MSMEs access loans. For financial institutions, the study suggests cash flow-based lending products, applying psychometric testing for credit scoring, improving the governance of the government-affiliated MFIs, and automating the MFIs.

Three-Dimensional Cooperative Positioning for Internet of Things Provenance

A large number of Internet of Things (IoT) devices have been interconnected for information collection and exchange. The data are only meaningful if it is captured at the expected location (i.e., the IoT devices or sensors are not removed accidentally or intentionally). This article presents a new algorithm, which cooperatively locates multiple IoT devices deployed in a 3-D space based on pairwise Euclidean distance measurements. When the distance measurement noises are negligible, a new feasibility problem of rank-3 variables is formulated. We solve the problem using the difference-of-convex (DC) programming to preserve the rank-3 constraints, rather than relaxing the constraints, using semidefinite relaxation (SDR). When the distance measurements are corrupted by additive noises and nonlight-of-sight (NLOS) propagation, a maximum-likelihood estimation (MLE) problem is formulated and transformed to a DC program solved with the rank-3 constraints preserved. Simulation results indicate that the proposed approach can achieve satisfactory accuracy results with a low complexity and strong robustness to the irregular topology, poor connectivity, and measurement errors, as compared to existing SDR-based alternatives.

A new self-adaptive inertial CQ-algorithm for solving convex feasibility and monotone inclusion problems

Abstract Using a dynamical step size technique, a new self-adaptive CQ-algorithm is proposed in the presence of an inertial term to find the solution of convex feasibility problem and monotone inclusion problem involving a finite number of maximal monotone set valued operators. To do this, in certain Banach spaces, we construct an algorithm which converges to the fixed point of right Bregman strongly nonexpansive mappings and coincidentally solves the convex feasibility and monotone inclusion problems. Strong convergence of the algorithm is achieved without computation of the associated operator norms. Interesting numerical examples which illustrate the implementation and efficiency of our scheme are also given. Results obtained via this work improve and extend on previous results of its kind, in the literature.

Error bounds, facial residual functions and applications to the exponential cone

We construct a general framework for deriving error bounds for conic feasibility problems. In particular, our approach allows one to work with cones that fail to be amenable or even to have computable projections, two previously challenging barriers. For the purpose, we first show how error bounds may be constructed using objects called one-step facial residual functions. Then, we develop several tools to compute these facial residual functions even in the absence of closed form expressions for the projections onto the cones. We demonstrate the use and power of our results by computing tight error bounds for the exponential cone feasibility problem. Interestingly, we discover a natural example for which the tightest error bound is related to the Boltzmann–Shannon entropy. We were also able to produce an example of sets for which a Hölderian error bound holds but the supremum of the set of admissible exponents is not itself an admissible exponent.

Implementation of a projection and rescaling algorithm for second-order conic feasibility problems

This paper documents a computational implementation of a projection and rescaling algorithm for solving one of the alternative feasibility problems where L is a linear subspace in , is its orthogonal complement, and is the interior of a direct product of second order cones. The gist of the projection and rescaling algorithm is to enhance a low-cost first-order method (a basic procedure) with an adaptive reconditioning transformation (a rescaling step). We give a full description of a Python implementation of this algorithm and present multiple sets of numerical experiments on synthetic problem instances with varied levels of conditioning. Our computational experiments provide promising evidence of the effectiveness of the projection and rescaling algorithm. Our Python code is publicly available. Furthermore, the simplicity of the algorithm makes a computational implementation in other environments completely straightforward.

On split feasibility problem for finite families of equilibrium and fixed point problems in Banach spaces

Abstract In this article, motivated by the works of Ali Akbar and Elahe Shahrosvand [Split equality common null point problem for Bregman quasi-nonexpansive mappings, Filomat 32 (2018), no. 11, 3917–3932], Eskandani et al. [A hybrid extragradient method for solving pseudomonotone equilibrium problem using Bregman distance, J. Fixed Point Theory Appl. 20 (2018), 132], B. Ali and M. H. Harbau [Convergence theorems for Bregman K-mappings and mixed equilibrium problems in reflexive Banach spaces, J. Funct. Spaces (2016) Article ID 5161682, 18 pages], and some other related results in the literature, we introduce a hybrid extragradient iterative algorithm that employs a Bregman distance approach for approximating a split feasibility problem for a finite family of equilibrium problems involving pseudomonotone bifunctions and fixed point problems for a finite family of Bregman quasi-asymptotically nonexpansive mappings using the concept of Bregman K-mapping in reflexive Banach spaces. Using our iterative algorithm, we state and prove a strong convergence result for approximating a common solution to the aforementioned problems. Furthermore, we give an application of our main result to variational inequalities and also report a numerical example to illustrate the convergence of our method. The result presented in this article extends and complements many related results in the literature.

Restricted Gröbner fans and re-embeddings of affine algebras

In this paper we continue the study of good re-embeddings of affine K-algebras started in Kreuzer et al. (J Algebra Appl, 2021. https://doi.org/10.1142/S0219498822501882 ). The idea is to use special linear projections to find isomorphisms between a given affine K-algebra K[X]/I, where $$X=(x_1,\dots ,x_n)$$ , and K-algebras having fewer generators. These projections are induced by particular tuples of indeterminates Z and by term orderings $$\sigma $$ which realize Z as leading terms of a tuple F of polynomials in I. In order to efficiently find such tuples, we provide two major new tools: an algorithm which reduces the check whether a given tuple F is Z-separating to an LP feasibility problem, and an isomorphism between the part of the Gröbner fan of I consisting of marked reduced Gröbner bases which contain a Z-separating tuple and the Gröbner fan of $$I\cap K[X\setminus Z]$$ . We also indicate a possible generalization to tuples Z which consist of terms. All results are illustrated by explicit examples.