This paper develops a gradient descent (GD) method for solving a system of nonlinear equations with an explicit formulation. We theoretically prove that the GD method has linear convergence in ...general and, under certain conditions, is equivalent to Newton’s method locally with quadratic convergence. A stochastic version of the gradient descent is also proposed for solving large-scale systems of nonlinear equations. Finally, several benchmark numerical examples are used to demonstrate the feasibility and efficiency compared to Newton’s method.
•Develop a gradient descent method for solving nonlinear equations.•Prove the convergence of the gradient descent method.•The gradient descent method has quadratic convergence under certain conditions.
This paper introduces the potential-function based method for secondary (as well as tertiary) control of a microgrid, in both islanded and grid-connected modes. A potential function is defined for ...each controllable unit of the microgrid such that the minimum of the potential function corresponds to the control goal. The dynamic set points are updated, using communication within the microgrid. The proposed potential function method is applied for the secondary voltage control of two microgrids with single and multiple feeders. Both islanded and grid-connected modes are investigated. The studies are conducted in the time-domain, using the PSCAD/EMTDC software environment. The study results demonstrate feasibility of the proposed potential function method and viability of the secondary voltage control method for a microgrid.
Making optical networks more efficient and reliable requires further automation of the optical layer. In this context, we propose a closed control loop that automatically performs fine frequency ...adjustments of the subchannels of superchannels to maintain optimal performance despite time-dependent impairments, thus achieving three main goals. At design, our scheme reduces the need for margins and guard-bands dedicated to spectrum-related impairments such as filter or channel detunings. This allows for a more efficient network at deployment. Then, during operation, the quality of transmission (QoT) of each subchannel is maximized to make the superchannel more resilient to any kind of soft failure, thus improving network resilience. Finally, considering the elastic network paradigm, performance improvements can be converted into significant capacity upgrades. We demonstrate the ability of our solution to achieve these three goals with split-step Fourier simulations. For a four-subchannel superchannel, we demonstrate robustness to +/- 2 GHz frequency detunings and gains of up to 3.7 dB in quality of transmission.
Efficiently enhancing heat conduction through optimized distribution of a limited quantity of high thermal conductivity material is paramount in cooling electronic devices and numerous other ...applications. This paper introduces a target-driven all-at-once approach for PDE-constrained optimization and derives a splitting smoothed particle hydrodynamics (SPH) method for optimizing the distribution of thermal conductivity in heat conduction problems. In this method, the optimization iteration of the system is split into several easily addressed steps. A targeting step is employed to progressively enforce the direct target, which potentially leads to increased PDE residuals. Then, these residuals are recovered through an evolution step of the design variable. After this, a PDE solution step is carried out to further decrease the PDE residuals, and the system is ready for the next iteration. Unlike the simulation-based approaches, the present method does not rely on the adjoint state equation and converged state variable field in each iteration, and the optimization process is significantly simplified and accelerated. With the utilization of an implicit SPH splitting operator and a general numerical regularization formulation, the information propagation is further accelerated and the numerical stability is greatly enhanced. Typical examples of heat conduction optimization demonstrate that the current method yields optimal results comparable to previous methods and exhibits considerable computational efficiency. Moreover, the optimal results feature more moderate extreme values, which offers distinct advantages for the easier selection of appropriate material with high thermal conductivity.
•The optimization of the heat conduction is split into several easily addressed weakly coupled steps.•The target is directly imposed on the temperature, and fields only converge at the termination.•The splitting SPH method is employed to implicitly update temperature and thermal conductivity.•General regularization formulation is proposed to guarantee numerical stability.•Optimal results with moderate peak values of thermal conductivity are obtained with good efficiency.
With the continuous development of distributed energy, the energy storage system (ESS) is indispensable in improving power quality. Aiming at the application of large-capacity storage battery access ...to medium voltage dc power grid, a dc cascaded energy storage system based on the dc collector is proposed, and the characteristic, topology and control are presented in detail. In this scheme, the low-voltage storage batteries are accessed to medium voltage dc bus directly with dc collectors, which not only has higher efficiency, but also improves power density. Especially, the diversity of state of charge (SOC) is only reflected in the duty cycle of submodules (SMs), rather than the voltage deviation. Further, a variable angle carrier phase-shifted modulation with gradient descent method is discussed to eliminate the dominant current harmonics. By calculating the harmonic amplitude and solving the Jacobian matrix, the gradient that reduces harmonic current is obtained, and the optimal combination of the carrier phases is deduced by iterative calculation. Meanwhile, the method can be extended in the dc collector with any number of SMs to eliminate the dominant harmonics. Finally, the experimental prototype is built and the benefit of proposed solution is verified.
Least squares support vector machine (LSSVM) considerably simplifies problem solving, however, there are restrictions. The first is that it treats samples on both sides of the proximal hyperplane ...equally and does not differentiate them based on locations; the second is that it is sensitive to noise and outliers. To address the aforementioned issues, this paper proposes an asymmetric robust least squares support vector machine termed QTLS by combining Quadratic Type Squared Error Loss Function (QTSELF) with LSSVM, in which QTSELF is introduced into machine learning for the first time. On the one hand, QTLS can impose various penalties on samples according to their locations and place greater emphasis on samples that are susceptible to misclassification. On the other hand, it improves the model robustness by imposing a tiny penalty on noise or outliers located far from the proximal hyperplane. Using Rademacher complexity theory, we investigate the generalization capacity of QTLS. The RMSProp technique is utilized to solve both linear and nonlinear QTLS. Extensive experiments indicate the effectiveness of QTLS in addressing binary classification problems.
A novel stepsize for gradient descent method Hoai, Pham Thi; Vinh, Nguyen The; Chung, Nguyen Phung Hai
Operations research letters,
March 2024, 2024-03-00, Volume:
53
Journal Article
Peer reviewed
We propose a novel adaptive stepsize for the gradient descent scheme to solve unconstrained nonlinear optimization problems. With the convex and smooth objective satisfying locally Lipschitz gradient ...we obtain the complexity O(1k) of f(xk)−f⁎ at most. By using the idea of the new stepsize, we propose another new algorithm based on the projected gradient for solving a class of nonconvex optimization problems over a closed convex set. The computational experiments show the efficiency of the new method.
