Finite Convergence Research Articles

A hybrid discretization algorithm with guaranteed feasibility for the global solution of semi-infinite programs

A discretization-based algorithm for the global solution of semi-infinite programs (SIPs) is proposed, which is guaranteed to converge to a feasible, $$\varepsilon $$ź-optimal solution finitely under mild assumptions. The algorithm is based on the hybridization of two existing algorithms. The first algorithm (Mitsos in Optimization 60(10---11):1291---1308, 2011) is based on a restriction of the right-hand side of the constraints of a discretized SIP. The second algorithm (Tsoukalas and Rustem in Optim Lett 5(4):705---716, 2011) employs a discretized oracle problem and a binary search in the objective space. Hybridization of the approaches yields an algorithm, which leverages the strong convergence guarantees and the relatively tight upper bounding problem of the first approach while employing an oracle problem adapted from the second approach to generate cheap lower bounds and adaptive updates to the restriction of the first approach. These adaptive updates help in avoiding a dense population of the discretization. The hybrid algorithm is shown to be superior to its predecessors both theoretically and computationally. A proof of finite convergence is provided under weaker assumptions than the assumptions in the references. Numerical results from established SIP test cases are presented.

Solution to the quadratic assignment problem using semi-lagrangian relaxation

The semi-Lagrangian relaxation (SLR), a new exact method for combinatorial optimization problems with equality constraints, is applied to the quadratic assignment problem (QAP). A dual ascent algorithm with finite convergence is developed for solving the semi-Lagrangian dual problem associated to the QAP. We perform computational experiments on 30 moderately difficult QAP instances by using the mixed integer programming solvers, Cplex, and SLR+Cplex, respectively. The numerical results not only further illustrate that the SLR and the developed dual ascent algorithm can be used to solve the QAP reasonably, but also disclose an interesting fact: comparing with solving the unreduced problem, the reduced oracle problem cannot be always effectively solved by using Cplex in terms of the CPU time.

Finite-time synchronization of cyclic switched complex networks under feedback control

In this paper, the global finite-time synchronization between two cyclic switched complex dynamical networks is investigated via feedback control. Based on the cyclic dwell time approach, finite-time stability theorem and some inequality techniques, several sufficient criteria are proposed to ensure finite-time synchronization for a class of cyclic switched complex networks (CSCNs) with and without coupling delay. The obtained criteria not only provide a feasible approach to design state feedback controllers but also fully reveal the trade-off among the cyclic dwell time, the finite convergence time and the initial state. Finally, numerical simulations are given to illustrate the effectiveness of the proposed results.

Distributed finite-time leaderless consensus control for double-integrator multi-agent systems with external disturbances

We study the finite-time consensus control of double-integrator multi-agent systems with external bounded disturbances under the fixed leaderless network communication topology. Based on the relative position and the relative velocity information, a novel finite-time consensus protocol is constructed. By using the Lyapnuov finite-time theory, it is shown that the proposed protocol can coordinate the states of agents to guarantee the finite-time consensus under an undirected connected topology. And then we extend the results onto the strongly connected directed communication topology. Importantly, the finite convergence time of the leaderless systems is explicitly presented. Finally, the effectiveness of the results is illustrated by numerical simulation.

Multistage Generation and Network Expansion Planning in Distribution Systems Considering Uncertainty and Reliability

This paper describes the incorporation of uncertainty and reliability in the dynamic expansion planning of distribution network assets and distributed generation. Several alternatives for the installation of feeders, transformers, and distributed generation are considered. Thus, the optimal expansion plan identifies the best alternative, location, and installation time for the candidate assets under the uncertainty related to demand and renewable energy sources. To that end, an iterative algorithm is devised to yield a pool of high-quality candidate solutions in terms of total investment and operational costs. Each candidate solution results from a stochastic-programming-based model driven by the minimization of the expected investment and operational costs. The associated scenario-based deterministic equivalent is formulated as a mixed-integer linear program for which finite convergence to optimality is guaranteed and efficient off-the-shelf software is available. Standard metrics are subsequently applied to each candidate solution to characterize its reliability so that valuable information is provided to the distribution system planner. Numerical results illustrate the effective performance of the proposed approach.

Finite convergence analysis and weak sharp solutions for variational inequalities

In this paper, we study the weak sharpness of the solution set of variational inequality problem (in short, VIP) and the finite convergence property of the sequence generated by some algorithm for finding the solutions of VIP. In particular, we give some characterizations of weak sharpness of the solution set of VIP without considering the primal or dual gap function. We establish an abstract result on the finite convergence property for a sequence generated by some iterative methods. We then apply such abstract result to discuss the finite termination property of the sequence generated by proximal point method, exact proximal point method and gradient projection method. We also give an estimate on the number of iterates by which the sequence converges to a solution of the VIP.

Adaptive uniform finite-/fixed-time convergent second-order sliding-mode control

ABSTRACTThis paper presents an adaptive gain algorithm for second-order sliding-mode control (2-SMC), specifically a super-twisting (STW)-like controller, with uniform finite/fixed convergence time, that is robust to perturbations with unknown bounds. It is shown that a second-order sliding mode is established as exact finite-time convergence to the origin if the adaptive gain does not have the ability to get reduced and converge to a small vicinity of the origin if the adaptation algorithm does not overestimate the control gain. The estimate of fixed convergence time of the studied adaptive STW-like controller is derived based on the Lyapunov analysis. The efficacy of the proposed adaptive algorithm is illustrated in a tutorial example, where the adaptive STW-like controller with uniform finite/fixed convergence time is compared to the adaptive STW controller with non-uniform finite convergence time.

Physics Based Finite Element Interpolation Functions for Rotating Beams

Rotating beams are ubiquitous members of industrial structures such as wing turbine rotors, helicopter rotors, turbomachinery, robotic systems and aerial robots. Typically, finite element analysis is used to solve the vibration problem for these structures. We show that it is possible to significantly enhance the efficiency of the finite element methods for rotating beams by creating basis functions which more closely satisfy the governing differential equation of the structure. Since the rotating beam equation cannot be solved as an exact solution, different approximate strategies are explored to improve finite element convergence, especially at higher rotating speeds where the centrifugal stiffening terms become dominant.

Two fast algorithms for projecting a point onto the canonical simplex

Two fast orthogonal projection algorithms of a point onto the canonical simplex are analyzed. These algorithms are called the vector and scalar algorithms, respectively. The ideas underlying these algorithms are well known. Improved descriptions of both algorithms are given, their finite convergence is proved, and exact estimates of the number of arithmetic operations needed for their implementation are derived, and numerical results of the comparison of their computational complexity are presented. It is shown that on some examples the complexity of the scalar algorithm is maximal but the complexity of the vector algorithm is minimal and conversely. The orthogonal projection of a point onto the solid simplex is also considered.

Sliding Mode Controlfor Robust Regulationof Chemical Processes

Objective: This paper implies a comparison between sliding mode control and linear state feedback controller. When the linear systems are given bounded disturbance their robustness properties are studied. Method: Pole placement, classical sliding mode control using constant rate reaching law and constant plus proportional rate reaching law, super-twisting controller. Findings: Finite time asymptotic convergence is not achieved in linear feedback controller, Sliding Mode Control when used produced finite time asymptotic convergence. The measure of the control variable is obtained by taking norm of the control input. Applications: Process Control, Aerodynamics.

Immersed normalized B-spline finite elements – A convergence study for 2D problems

Sanches et al. (2011) [21] introduced a novel B-spline based immersed finite element method (FEM), called I-spline FEM, for the computation of geometrically and topologically complex problems. Away from the domain boundaries, the standard B-spline basis functions are used for the finite element interpolation, however, close to domain boundaries, the B-spline basis functions are modified, so that they locally interpolate the Dirichlet boundary conditions. Although the new technique is demonstrated to be robust and efficient, in the first tests it appeared to present very small convergence rates considering the B-spline order. In this work we propose a quadrature procedure for boundary cells based on triangles subdivision and linear mapping and make use of 2D elasticity and Poisson problems to study the I-spline finite element method convergence. Cubic B-splines are considered and all changes due to the B-spline modifications are studied regarding its influence in convergence rate. Finally, we show that good convergence rates, cubic for displacements and quadratic for stresses, may be achieved.

A test of asymmetric comovement for state-dependent stock returns

I propose a test of asymmetric stock return comovement across states. The test can be viewed as a variation of Kendall's τ conditional on the state and has an asymptotic χ2-distribution. A refined version of the test is derived based on the Markov chain theory of regenerative cycles which substantially improves finite sample size and power properties. I show that the test has power against local alternatives, which is nonetheless compromised due to a finite sample convergence bound put on the implied local alternative data generating process. I evaluate the new test against traditional correlation-based measures and demonstrate power attrition of a state-free tail dependence test as parameter values are varied. Broad market-based ETFs and international indices are studied and in most cases there is no compelling evidence for asymmetric comovement across states. A list of related tests is given as an extension at the end.

Necessary conditions for convergence of difference schemes for fractional-derivative two-point boundary value problems

A fractional-derivative two-point boundary value problem of the form \({\tilde{D}}^\delta u=f\) on (0, 1) with Dirichlet boundary conditions is studied. Here \({\tilde{D}}^\delta \) is a Caputo or Riemann–Liouville fractional derivative operator of order \(\delta \in (1,2)\). The discretisation of this problem by an arbitrary difference scheme is examined in detail when u or f is a polynomial. For any convergent difference scheme, it is proved rigorously that the entries of the associated matrix must satisfy certain identities. It is shown that some of these identities are not satisfied by certain well-known schemes from the research literature; this clarifies the type of problem to which these schemes can be applied successfully. The effects of the special boundary condition \(u(0)=0\) and the special right-hand-side condition \(f(0)=0\) are also investigated. This leads, under certain circumstances, to a sharpening of a recently-published finite difference scheme convergence result of two of the authors.

Data-Driven Optimization of Ambiguous Reward-Risk Ratio Measures

We investigate a class of distributionally robust optimization problems that have direct applications in finance. They are semi-infinite programming problems with ambiguous expectation constraints in which fractional functions represent reward-risk ratios. We develop a reformulation and algorithmic data-driven framework based on the Wasserstein metric to model ambiguity and to derive probabilistic guarantee that the ambiguity set contains the true probability distribution. The reformulation phase involves the derivation of the support function of the ambiguity set and the concave conjugate of the ratio function, and yields a mathematical programming problem in a finite dimensional constraint space. We design modular bisection algorithms with finite convergence property. We specify new ambiguous portfolio optimization models for the Sharpe, Sortino, Sortino-Satchel, and Omega ratios. The computational study shows the applicability and scalability of the framework to solve large, industry-relevant size problems.

A nonlinearly-activated neurodynamic model and its finite-time solution to equality-constrained quadratic optimization with nonstationary coefficients

The recently-proposed Zhang dynamics (ZD) has been proven to achieve success for solving the linear-equality constrained time-varying quadratic program ideally when time goes to infinity. The convergence performance is a significant improvement, as compared to the gradient-based dynamics (GD) that cannot make the error converge to zero even after infinitely long time. However, this ZD model with the suggested activation functions cannot reach the theoretical time-varying solution in finite time, which may limit its applications in real-time calculation. Therefore, a nonlinearly-activated neurodynamic model is proposed and studied in this paper for real-time solution of the equality-constrained quadratic optimization with nonstationary coefficients. Compared with existing neurodynamic models (specifically the GD model and the ZD model) for optimization, the proposed neurodynamic model possesses the much superior convergence performance (i.e., finite-time convergence). Furthermore, the upper bound of the finite convergence time is derived analytically according to Lyapunov theory. Both theoretical and simulative results verify the efficacy and superior of the nonlinearly-activated neurodynamic model, as compared to these of the GD and ZD models.

Convergence studies on block iterative algorithms for image reconstruction

The sequential block-iterative scheme based on Landweber’s method is a general iterative one for image reconstruction. In this paper, we give the finite termination convergence for this scheme, which provides an approach to choose relaxation coefficients. Furthermore, sufficient and necessary convergence conditions are also established for the sequential block-iterative scheme case.

Robustness analysis of a hybrid of recursive neural dynamics for online matrix inversion

Encouraged by superior convergence performance achieved by a recently-proposed hybrid of recursive neural dynamics for online matrix inversion, we investigate its robustness properties in this paper when there exists large model implementation errors. Theoretical analysis shows that the perturbed dynamic system is still global stable with the tight steady-state bound of solution error estimated. Moreover, this paper analyses global exponential convergence rate and finite convergence time of such a hybrid dynamical model to a relatively loose solution error bound. Computer simulation results substantiate our analysis on the perturbed hybrid neural dynamics for online matrix inversion when having large implementation errors.

Fault-tolerant finite time consensus for multiple uncertain nonlinear mechanical systems under single-way directed communication interactions and actuation failures

This paper investigates the problem of finite time consensus for a group of uncertain nonlinear mechanical systems under single-way directed communication topology and actuation failures. Due to the existence of the unknown inherent nonlinear dynamics and the undetectable actuation faults, the resultant control gain of the system becomes unknown and time-varying, making the control impact on the system uncertain and the finite time control synthesis nontrivial. The underlying problem becomes further complex as the communication among the agents is not only local but also one-way directed. In this work, three major steps are employed to circumvent the aforementioned difficulties, leading to a robust adaptive fault-tolerant finite time consensus solution. Firstly, by deriving a useful property on the newly constructed Laplacian matrix, the technical difficulty in finite time control design and stability analysis is circumvented; Secondly, to deal with the time-varying and uncertain control gain, the concept of virtual parameter estimation error is introduced and incorporated into a skillfully chosen Lyapunov function; Thirdly, to facilitate the global stability analysis of the proposed adaptive fault-tolerant finite time consensus scheme for multiple nonlinear systems, an important lemma (Lemma 7) containing a useful inequality is derived. In addition, the finite convergence time for each agent to reach the required consensus configuration is explicitly established and recipes for control parameter selection to make the residual errors as small as desired are provided. The effectiveness of the proposed control scheme is confirmed by numerical simulation.

Particle swarm optimization tuned fuzzy terminal sliding mode control for UPS inverters

This paper proposes a particle swarm optimization algorithm tuned fuzzy terminal sliding mode control for the appli- cation of UPS inverters. Though classic sliding mode control (SMC) is insensitive to system uncertainties, it possesses an infinite system-state convergence time. For high-accuracy tracking control, a terminal sliding mode control (TSMC) is developed to provide a finite system-state convergence time. However, difficult estimation occurs in TSMC, and incurs high UPS inverter voltage harmonics and slow dynamic response. To obtain high-quality UPS inverter output voltage, a fuzzy logic (FL) with a computationally simple and practically easy estimator is integrated into TSMC to resolve system uncertainties. Simultaneously, the particle swarm optimization (PSO) algorithm is applied to optimally tune the control gains of the TSMC with a fuzzy estimator. Results indicate that the presented combination of PSO, FL and TSMC yields a closed-loop UPS inverter with good performance under various loading conditions. Simulation and experimental results indicate that the proposed control can achieve low total harmonic distortion (THD) under nonlinear loading conditions and fast dynamic response under transient loading conditions.

On Slater’s condition and finite convergence of the Douglas–Rachford algorithm for solving convex feasibility problems in Euclidean spaces

The Douglas-Rachford algorithm is a classical and very successful method for solving optimization and feasibility problems. In this paper, we provide novel conditions sufficient for finite convergence in the context of convex feasibility problems. Our analysis builds upon, and considerably extends, pioneering work by Spingarn. Specifically, we obtain finite convergence in the presence of Slater's condition in the affine-polyhedral and in a hyperplanar-epigraphical case. Various examples illustrate our results. Numerical experiments demonstrate the competitiveness of the Douglas-Rachford algorithm for solving linear equations with a positivity constraint when compared to the method of alternating projections and the method of reflection-projection.