GLOBAL CONVERGENCE OF A CAUTIOUS PROJECTION BFGS ALGORITHM FOR NONCONVEX PROBLEMS WITHOUT GRADIENT LIPSCHITZ CONTINUITY

A cautious projection BFGS method is proposed for solving nonconvex unconstrained optimization problems.The global convergence of this method as well as a stronger general convergence result can be proven without a gradient Lipschitz continuity assumption,which is more in line with the actual problems than the existing modified BFGS methods and the traditional BFGS method.Under some additional conditions,the method presented has a superlinear convergence rate,which can be regarded as an extension and supplement of BFGS-type methods with the projection technique.Finally,the effectiveness and application prospects of the proposed method are verified by numerical experiments.

An adaptive genetic algorithm with diversity-guided mutation and its global convergence property

An adaptive genetic algorithm with diversity-guided mutation, which combines adaptive probabilities of crossover and mutation was proposed. By means of homogeneous finite Markov chains, it is proved that adaptive genetic algorithm with diversity-guided mutation and genetic algorithm with diversity-guided mutation converge to the global optimum if they maintain the best solutions, and the convergence of adaptive genetic algorithms with adaptive probabilities of crossover and mutation was studied. The performances of the above algorithms in optimizing several unimodal and multimodal functions were compared. The results show that for multimodal functions the average convergence generation of the adaptive genetic algorithm with diversity-guided mutation is about 900 less than that of (adaptive) genetic algorithm with adaptive probabilities and genetic algorithm with diversity-guided mutation, and the adaptive genetic algorithm with diversity-guided mutation does not lead to premature convergence. It is also shown that the better balance between overcoming premature convergence and quickening convergence speed can be gotten.

Global Convergence Analysis of Non-Crossover Genetic Algorithm and Its Application to Optimization

Selection, crossover, and mutation are three main operators of the canonical genetic algorithm (CGA). This paper presents a new approach to the genetic algorithm. This new approach applies only to mutation and selection operators. The paper proves that the search process of the non-crossover genetic algorithm (NCGA) is an ergodic homogeneous Markov chain. The proof of its convergence to global optimum is presented. Some nonlinear multi-modal optimization problems are applied to test the efficacy of the NCGA. NP-hard traveling salesman problem (TSP) is cited here as the benchmark problem to test the efficiency of the algorithm. The simulation result shows that NCGA achieves much faster convergence speed than CGA in terms of CPU time. The convergence speed per epoch of NCGA is also faster than that of CGA.

Fast integrated guidance and control with global convergence

A global fast convergent integrated guidance and control design approach is proposed. A disturbance observer is utilized to estimate the uncertainties of integrated guidance and control model in finite time. According to the multiple sliding-mode surface control, the independent nonsingular terminal sliding functions are presented in each step, and all the sliding-mode surfaces run parallel. These presented sliding-mode surfaces keep zero value from a certain time, and the system states converge quickly in sliding phase. Therefore, the system response speed is increased. The proposed method offers the global convergent time analytically, which is useful to optimize the transient performance of system. Simulation results are used to verify the proposed method.

Uncertainty of the Numerical Solution of a Nonlinear System’s Long-term Behavior and Global Convergence of the Numerical Pattern

The computational uncertainty principle in nonlinear ordinary differential equations makes the numerical solution of the long-term behavior of nonlinear atmospheric equations have no meaning. The main reason is that, in the error analysis theory of present-day computational mathematics, the non-linear process between truncation error and rounding error is treated as a linear operation. In this paper, based on the operator equations of large-scale atmospheric movement, the above limitation is overcome by using the notion of cell mapping. Through studying the global asymptotic characteristics of the numerical pattern of the large-scale atmospheric equations, the definitions of the global convergence and an appropriate discrete algorithm of the numerical pattern are put forward. Three determinant theorems about the global convergence of the numerical pattern are presented, which provide the theoretical basis for constructing the globally convergent numerical pattern. Further, it is pointed out that only a globally convergent numerical pattern can improve the veracity of climatic prediction.

GLOBAL CONVERGENCE OF A TRUST REGION ALGORITHM USING INEXACT GRADIENT FOR EQUALITY-CONSTRAINED OPTIMIZATION

A trust-region algorithm is presented for a nonlinear optimization problem of equality-constraints. The characterization of the algorithm is using inexact gradient information. Global convergence results are demonstrated where the gradient values are obeyed a simple relative error condition.

Maneuvering Angle Rigid Formations With Global Convergence Guarantees

Angle rigid multi-agent formations can simultaneously undergo translational,rotational,and scaling maneuvering,therefore combining the maneuvering capabilities of both distance and bearing rigid formations.However,maneuvering angle rigid formations in 2D or 3D with global convergence guarantees is shown to be a challenging problem in the existing literature even when relative position measurements are available.Motivated by angle-induced linear equations in 2D triangles and 3D tetrahedra,this paper aims to solve this challenging problem in both 2D and3D under a leader-follower framework.For the 2D case where the leaders have constant velocities,by using local relative position and velocity measurements,a formation maneuvering law is designed for the followers governed by double-integrator dynamics.When the leaders have time-varying velocities,a sliding mode formation maneuvering law is proposed by using the same measurements.For the 3D case,to establish an angle-induced linear equation for each tetrahedron,we assume that all the followers'coordinate frames share a common Z direction.Then,a formation maneuvering law is proposed for the followers to globally maneuver Z-weakly angle rigid formations in 3D.The extension to Lagrangian agent dynamics and the construction of the desired rigid formations by using the minimum number of angle constraints are also discussed.Simulation examples are provided to validate the effectiveness of the proposed algorithms.

A VARIATIONAL IMAGE SMOOTHING MODEL WITH GLOBAL CONVERGENCE

A new smoothing method is proposed. The smoothing process adapts to image characteristics and is good at preserving local image structures. More importantly, in the theory under the conditions weaker than those in the original Kaanov method an approximal sequence of solutions to the variational problems can be constructed and the global convergence can be proved. And the conditions in the papers of Schno¨rr(1994) and Heers, et al (2001) are discussed. Numerical solutions of the model are given.

GLOBAL CONVERGENCE OF TRUST REGION ALGORITHM FOR EQUALITY AND BOUND CONSTRAINED NONLINEAR OPTIMIZATION

This paper presents a trust region two phase model algorithm for solving the equality and bound constrained nonlinear optimization problem. A concept of substationary point is given. Under suitable assumptions,the global convergence of this algorithm is proved without assuming the linear independence of the gradient of active constraints. A numerical example is also presented.

A Note on Global Convergence Result for Conjugate Gradient Methods

We extend a results presented by Y.F. Hu and C.Storey (1991) [1] on the global convergence result for conjugate gradient methods with different choices for the parameter β k . In this note, the conditions given on β k are milder than that used by Y.F. Hu and C. Storey.

Global Convergence of Adaptive Generalized Predictive Controller Based on Least Squares Algorithm

Some papers on stochastic adaptive control schemes have established convergence algorithm using a least-squares parameters. With the popular application of GPC, global convergence has become a key problem in automatic control theory. However, now global convergence of GPC has not been established for algorithms in computing a least squares iteration. A generalized model of adaptive generalized predictive control is presented. The global convergebce is also given on the basis of estimating the parameters of GPC by least squares algorithm.

GLOBAL CONVERGENCE OF THE GENERAL THREE TERM CONJUGATE GRADIENT METHODS WITH THE RELAXED STRONG WOLFE LINE SEARCH

The global convergence of the general three term conjugate gradient methods with the relaxed strong Wolfe line search is proved.

The global convergence of the non-quasi-Newton methods with non-monotone line search

The non-quasi-Newton methods for unconstrained optimization was investigated. Non-monotone line search procedure is introduced, which is combined with the non-quasi-Newton family. Under the uniform convexity assumption on objective function, the global convergence of the non-quasi-Newton family was proved. Numerical experiments showed that the non-monotone line search was more effective.

A Descent Gradient Method and Its Global Convergence

Y Liu and C Storey(1992)proposed the famous LS conjugate gradient method which has good numerical results.However,the LS method has very weak convergence under the Wolfe-type line search.In this paper,we give a new descent gradient method based on the LS method.It can guarantee the sufficient descent property at each iteration and the global convergence under the strong Wolfe line search.Finally,we also present extensive preliminary numerical experiments to show the efficiency of the proposed method by comparing with the famous PRP^+method.

A CLASS OF TRUST REGION METHODS FOR LINEAR INEQUALITY CONSTRAINED OPTIMIZATION AND ITS THEORY ANALYSIS:I.ALGORITHM AND GLOBAL CONVERGENCEXIU NAIHUA

A class of trust region methods for solving linear inequality constrained problems is proposed in this paper. It is shown that the algorithm is of global convergence.The algorithm uses a version of the two-sided projection and the strategy of the unconstrained trust region methods. It keeps the good convergence properties of the unconstrained case and has the merits of the projection method. In some sense, our algorithm can be regarded as an extension and improvement of the projected type algorithm.

Global Convergence of Curve Search Methods for Unconstrained Optimization

In this paper we propose a new family of curve search methods for unconstrained optimization problems, which are based on searching a new iterate along a curve through the current iterate at each iteration, while line search methods are based on finding a new iterate on a line starting from the current iterate at each iteration. The global convergence and linear convergence rate of these curve search methods are investigated under some mild conditions. Numerical results show that some curve search methods are stable and effective in solving some large scale minimization problems.

GLOBAL CONVERGENCE OF NONMONOTONIC TRUST REGION ALGORITHM FOR NONLINEAR OPTIMIZATION

A trust region algorithm for equality constrained optimization is given in this paper.The algorithm does not enforce strict monotonicity of the merit function for every iteration.Global convergence of the algorithm is proved under the same conditions of usual trust region method.

GLOBAL CONVERGENCE OF A CLASS OF OPTIMALLY CONDITIONED SSVM METHODS

This paper explores the convergence of a class of optimally conditioned self scaling variable metric (OCSSVM) methods for unconstrained optimization. We show that this class of methods with Wolfe line search are globally convergent for general convex functions.

Global convergence of a solution to p-Ginzburg-Landau type equations

The author studies the global convergence of a solution of p-Ginzburg-Landau equations when the parameter tends to zero. The convergence is in C^α sense, which is derived by establishing a uniform gradient estimate for some solution of a regularized p-Ginzburg-Landau equations.

GLOBAL CONVERGENCE PROPERTIES OF THREE-TERM CONJUGATE GRADIENT METHOD WITH NEW-TYPE LINE SEARCH

In this paper, a new Wolfe-type line search and a new Armijo-type line searchare proposed, and some global convergence properties of a three-term conjugate gradient method withthe two line searches are proved.