Inertial Step Research Articles

Projection-type methods with alternating inertial steps for solving multivalued variational inequalities beyond monotonicity

Projection-type methods with alternating inertial steps for solving multivalued variational inequalities beyond monotonicity

Inertial-Based Derivative-Free Method for System of Monotone Nonlinear Equations and Application

Iterative methods for solving nonlinear problems are of great importance due to their appearance in various areas of applications. In this paper, based on the inertial effect, we propose two projection derivative-free iterative methods for solving system of nonlinear equations. For the purpose of improving the numerical performance, the two methods incorporated the inertial step into the modified Barzilai and Borwein (BB) spectral parameters to generate the sequence of their respective search directions. The two spectral parameters are shown to be well-defined. For each method, the sequence of the search direction is bounded and satisfies the sufficient descent property. We establish the convergence analysis of the two methods based on the assumption that the underlying mapping is Lipschitzian and monotone. We demonstrate the efficiencies of the two methods on some collection of monotone system of nonlinear equations test problems. Finally, we apply the two methods to solve motion control problem involving a two planar robot.

Flexible Multi-Positional Microsensors for Cryoablation Temperature Monitoring

Cryoablation is a minimally invasive interventional therapy for the treatment of atrial fibrillation, but its current temperature measurement in ablation region is deficient during operation. In this study, a flexible device with multi-positional microsensors for temperature detection of ablation region is proposed, which can be attached to the surface of the medical cryoballoon to provide multi-positional temperature information. And the device makes up for the deficiency that current cryoablation catheter only detects the internal temperature of cryoballoon during surgery. In the temperature range from 295 K to 200 K, the static and dynamic characteristics of the device prove its high repeatability, linearity, and a response time of 0.3 s is obtained. In the meantime, the first-order inertial step response curve of the sensor is given. Finally, the ability of the flexible sensors to temperature detection is verified by simulation experiment based on pulmonary vein ostium agar model and ex vivo porcine heart cryoablation experiment.

A Gauss–Seidel type inertial proximal alternating linearized minimization for a class of nonconvex optimization problems

In this paper we study a broad class of nonconvex and nonsmooth minimization problems, whose objective function is the sum of a smooth function of the entire variables and two nonsmooth functions of each variable. We adopt the framework of the proximal alternating linearized minimization (PALM), together with the inertial strategy to accelerate the convergence. Since the inertial step is performed once the x-subproblem/y-subproblem is updated, the algorithm is a Gauss–Seidel type inertial proximal alternating linearized minimization (GiPALM) algorithm. Under the assumption that the underlying functions satisfy the Kurdyka–Łojasiewicz (KL) property and some suitable conditions on the parameters, we prove that each bounded sequence generated by GiPALM globally converges to a critical point. We apply the algorithm to signal recovery, image denoising and nonnegative matrix factorization models, and compare it with PALM and the inertial proximal alternating linearized minimization.

A gradient-type algorithm with backward inertial steps associated to a nonconvex minimization problem

We investigate an algorithm of gradient type with a backward inertial step in connection with the minimization of a nonconvex differentiable function. We show that the generated sequences converge to a critical point of the objective function, if a regularization of the objective function satisfies the Kurdyka-Łojasiewicz property. Further, we provide convergence rates for the generated sequences and the objective function values formulated in terms of the Łojasiewicz exponent. Finally, some numerical experiments are presented in order to compare our numerical scheme with some algorithms well known in the literature.

A note on the inertial proximal point method

The proximal point method (PPM) for solving maximal monotone operator inclusion problem is a highly powerful tool for algorithm design, analysis and interpretation. To accelerate convergence of the PPM, inertial PPM (iPPM) was proposed in the literature. In this note, we point out that some of the attractive properties of the PPM, e.g., the generated sequence is contractive with the set of solutions, do not hold anymore for iPPM. To partially inherit the advantages of the PPM and meanwhile incorporate inertial extrapolation steps, we propose an iPPM with alternating inertial steps. Our analyses show that the even subsequence generated by the proposed iPPM is contractive with the set of solutions. Moreover, we establish global convergence result under much relaxed conditions on the inertial extrapolation stepsizes, e.g., monotonicity is no longer needed and the stepsizes are significantly enlarged compared to existing methods. Furthermore, we establish certain $o(1/k)$ convergence rate results, where $k$ denotes the iteration counter. These features are new to inertial type PPMs.