A Global Optimization Algorithm for Unconstrained Non-linear Optimization Problems Using Chaotic Neuron Map and Golden Number

This paper presents aninnovative global multi-variable optimization algorithm using one of the best chaotic sequences, the neuron map, a description of which is also provided in the paper. The algorithm uses neuron map in the first stage to move near the global minimum point, as well as in each iteration of the second stage of local search that is done using the N-dimensional golden section search algorithm. The generation and mapping of the neuron variables to the optimization variables along with the stagewise search for the global minimum is explained conscientiously in the work. Numerical results on some benchmark functions and the comparison with a latest state-of-the-art algorithm ispresented in order to demonstrate the efficiency of the proposed algorithm.

A Regularization Homotopy Strategy for the Constrained Parameter Inversion of Partial Differential Equations

The main difficulty posed by the parameter inversion of partial differential equations lies in the presence of numerous local minima in the cost function. Inversion fails to converge to the global minimum point unless the initial estimate is close to the exact solution. Constraints can improve the convergence of the method, but ordinary iterative methods will still become trapped in local minima if the initial guess is far away from the exact solution. In order to overcome this drawback fully, this paper designs a homotopy strategy that makes natural use of constraints. Furthermore, due to the ill-posedness of inverse problem, the standard Tikhonov regularization is incorporated. The efficiency of the method is illustrated by solving the coefficient inversion of the saturation equation in the two-phase porous media.

Generalized Geodesic Convexity on Riemannian Manifolds

We introduce log-preinvex and log-invex functions on a Riemannian manifold. Some properties and relationships of these functions are discussed. A characterization for the existence of a global minimum point of a mathematical programming problem is presented. Moreover, a mean value inequality under geodesic log-preinvexity is extended to Cartan-Hadamard manifolds.

Roughly geodesic B - r-preinvex functions on Cartan Hadamard manifolds

In this article, we introduce a new class of functions called roughly geodesic B????r????preinvex on a Hadamard manifold and establish some properties of roughly geodesic B - r-preinvex functions on Hadamard manifolds. It is observed that a local minimum point for a scalar optimization problem is also a global minimum point under roughly geodesic B-r- preinvexity on Hadamard manifolds. The results presented in this paper extend and generalize the results appeared in the literature.

UAV Formation Control Based on the Improved APF

Malfunction such as target non-reach ability, local minimum pole and oscillation happens when traditional APF is applied to route planning. This paper proposes an improved APF model, considering the relative distance between the UAV and the target, the relative distance between the lead craft and the wing craft and the safety. We ensure the point of target to be the global minimum point in the entire potential fieldtarget non-reach ability caused by the threat and target point being too close are solved by adopting the modified repulsive potential function; planning failure in which UAVs falling into local minimum point is solved by adopting random fluctuation method; considering the oscillation of traditional potential field, the obstacle link method is proposed. The results of the simulation indicate the single UAV simulation route after obstacles overall planning, track route of UAV formation and formation route with obstacles with turning angle constraints. Requirements of Formation control and formation obstacle avoidance have been well satisfied.

The Identification of Convex Function on Riemannian Manifold

The necessary and sufficient condition of convex function is significant in nonlinear convex programming. This paper presents the identification of convex function on Riemannian manifold by use of Penot generalized directional derivative and the Clarke generalized gradient. This paper also presents a method for judging whether a point is the global minimum point in the inequality constraints. Our objective here is to extend the content and proof the necessary and sufficient condition of convex function to Riemannian manifolds.

Parameter Estimation of a Single Wheel Station using Hybrid Differential Evolution

This paper investigates an application of a ‘discrete variable’ hybrid differential evolution (dvHDE) method to parameter estimation of a single wheel station. The parameters of the single wheel station represent a quarter of the suspension of a medium sized family car. The estimation method developed incorporated the dvHDE and use of a Kalman filter (KF). The KF provides estimates of the ‘unmeasured’ states of the system being studied. The dvHDE, which works as a function optimizer, provides a ‘best fit’ set of model parameters. The performance of the dvHDE method was examined and compared against the standard gradient-based (GB) method, downhill simplex (DS) method and original differential evolution (DE) method on simulated and experimentally obtained data. The normalized mean squared errors (MSEs) of the system outputs are considered as the fitting criterion in the optimization process. The identified model parameters gave an MSE of below 3.5 per cent. The dvHDE method performed better over the GB, DS and DE methods and has been shown to improve the convergence rate by approximately 19 per cent over the original DE method, without sacrificing ability to find the global minimum point.

A singularly perturbed elliptic problem involving supercritical Sobolev exponent

This paper deals with −Δu + εuq−1 = u2*−1, , where q > 2*, ε > 0. We first show that the minimiser of the associated minimisation problem blows up at the global minimum point of H(x, x), where H(y, x) is the regular part of the Green's function. We then prove that for each strictly local minimum point x0 of H(x, x), this problem has a solution concentrating at x0 as ε→0.