A GLOBALLY CONVERGENT QP-FREE ALGORITHM FOR INEQUALITY CONSTRAINED MINIMAX OPTIMIZATION

Although QP-free algorithms have good theoretical convergence and are effective in practice,their applications to minimax optimization have not yet been investigated.In this article,on the basis of the stationary conditions,without the exponential smooth function or constrained smooth transformation,we propose a QP-free algorithm for the nonlinear minimax optimization with inequality constraints.By means of a new and much tighter working set,we develop a new technique for constructing the sub-matrix in the lower right corner of the coefficient matrix.At each iteration,to obtain the search direction,two reduced systems of linear equations with the same coefficient are solved.Under mild conditions,the proposed algorithm is globally convergent.Finally,some preliminary numerical experiments are reported,and these show that the algorithm is promising.

A NEWTON-TYPE GLOBALLY CONVERGENT INTERIOR-POINT METHOD TO SOLVE MULTI-OBJECTIVE OPTIMIZATION PROBLEMS

This paper proposes an interior-point technique for detecting the nondominated points of multi-objective optimization problems using the direction-based cone method.Cone method decomposes the multi-objective optimization problems into a set of single-objective optimization problems.For this set of problems,parametric perturbed KKT conditions are derived.Subsequently,an interior point technique is developed to solve the parametric perturbed KKT conditions.A differentiable merit function is also proposed whose stationary point satisfies the KKT conditions.Under some mild assumptions,the proposed algorithm is shown to be globally convergent.Numerical results of unconstrained and constrained multi-objective optimization test problems are presented.Also,three performance metrics(modified generational distance,hypervolume,inverted generational distance)are used on some test problems to investigate the efficiency of the proposed algorithm.We also compare the results of the proposed algorithm with the results of some other existing popular methods.

A GLOBALLY CONVERGENT ALGORITHM FOR THE EUCLIDEAN MULTIPLICITY LOCATION PROBLEM

The Euclidean single facility location problem (ESFL) and the Euclidean multiplicity lo-cation problem (EMFL) are two special nonsmooth convex programming problems which haveattracted a largr literature. For the ESFL problem. there are algorithms which converge bothglobally and quadratically For the EMFL problem, there are some quadratically convergentalgorithms. but for global convergencel they all need nontrivial assumptions on the problem.In this paper, we present an algorithm for EMFL. With no assumption on the problem, it isproved that from any initial point, this algorithm generates a sequence of points which convergesto the closed convex set of optimal solutions of EMFL.

A GLOBALLY AND SUPERLINEARLY CONVERGENT TRUST REGION METHOD FOR LC^1 OPTIMIZATION PROBLEMS

A new trust region algorithm for solving convex LC 1 optimization problem is presented.It is proved that the algorithm is globally convergent and the rate of convergence is superlinear under some reasonable assumptions.

A GLOBALLY CONVERGENT TRUST REGION ALGORITHM FOR OPTIMIZATION WITH GENERAL CONSTRAINTS AND SIMPLE BOUNDS

In this paper, we introduce a concept of substationary points and present a new trust region-based method for the optimization problems with general nonlinear equality constraints and simple bounds. Without the linear independent assumption on the gradients of the equalitiy constraints, we prove the global convergence results for the main algorithm and indicate that they extend the results on SQP and those on trust region methods for equality constrained optimizstion and for optimization with simple bounds. Moreover, since any nonlinear programming problem can be converted into the standard nonlinear programming by introducing slack variables, the trust region method preseated in this paper can be used for solving general nonlinear programming problems.

GLOBALLY CONVERGENT INEXACT GENERALIZED NEWTON METHODS WITH DECREASING NORM OF THE GRADIENT

Presents information on a study which proposed a type of globally convergent inexact generalized Newton methods to solve unconstrained optimization problems. Theorems on inexact generalized Newton algorithm with decreasing gradient norms; Discussion on the assumption given; Applications of algorithms and numerical tests.

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 Spectral Conjugate Gradient Method with Restart Strategy

As a generalization of the two-term conjugate gradient method(CGM),the spectral CGM is one of the effective methods for solving unconstrained optimization.In this paper,we enhance the JJSL conjugate parameter,initially proposed by Jiang et al.(Computational and Applied Mathematics,2021,40:174),through the utilization of a convex combination technique.And this improvement allows for an adaptive search direction by integrating a newly constructed spectral gradient-type restart strategy.Then,we develop a new spectral CGM by employing an inexact line search to determine the step size.With the application of the weak Wolfe line search,we establish the sufficient descent property of the proposed search direction.Moreover,under general assumptions,including the employment of the strong Wolfe line search for step size calculation,we demonstrate the global convergence of our new algorithm.Finally,the given unconstrained optimization test results show that the new algorithm is effective.

A QP-FREE AND SUPERLINEARLY CONVERGENT ALGORITHM FOR INEQUALITY CONSTRAINED OPTIMIZATIONS

In this paper, a new mixed quasi-Newton method for inequality constrained optimization problems is proposed. The feature of the method is that only the systems of linear equations are solved in each iteration, other than the quadratic programming, which decrease the amount of computations and is also efficient for large scale problem. Under some mild assumptions without the strict complementary condition., the method is globally and superlinearly convergent.

NEW HMM ALGORITHM FOR TOPOLOGY OPTIMIZATION

A new hybrid MMA-MGCMMA （HMM） algorithm for solving topology optimization problems is presented. This algorithm combines the method of moving asymptotes （MMA） algorithm and the modified globally convergent version of the method of moving asymptotes （MGCMMA） algorithm in the optimization process. This algorithm preserves the advantages of both MMA and MGCMMA. The optimizer is switched from MMA to MGCMMA automatically, depending on the numerical oscillation value existing in the calculation. This algorithm can improve calculation efficiency and accelerate convergence compared with simplex MMA or MGCMMA algorithms, which is proven with an example.

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.

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.

A Coordinate Gradient Descent Method for Nonsmooth Nonseparable Minimization

This paper presents a coordinate gradient descent approach for minimizing the sum of a smooth function and a nonseparable convex function.We find a search direction by solving a subproblem obtained by a second-order approximation of the smooth function and adding a separable convex function.Under a local Lipschitzian error bound assumption,we show that the algorithm possesses global and local linear convergence properties.We also give some numerical tests(including image recovery examples) to illustrate the efficiency of the proposed method.

Nonmonotone Adaptive Trust Region Algorithms with Indefinite Dogleg Path for Unconstrained Minimization

In this paper, we combine the nonmonotone and adaptive techniques with trust region method for unconstrained minimization problems. We set a new ratio of the actual descent and predicted descent. Then, instead of the monotone sequence, the nonmonotone sequence of function values are employed. With the adaptive technique, the radius of trust region △k can be adjusted automatically to improve the efficiency of trust region methods. By means of the Bunch-Parlett factorization, we construct a method with indefinite dogleg path for solving the trust region subproblem which can handle the indefinite approximate Hessian Bk. The convergence properties of the algorithm are established. Finally, detailed numerical results are reported to show that our algorithm is efficient.

AN INFEASIBLE-INTERIOR-POINT PREDICTOR-CORRECTOR ALGORITHM FOR THE SECOND-ORDER CONE PROGRAM

A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming （SOCP） by using the Alizadeh- Haeberly-Overton （AHO） search direction. This algorithm does not require the feasibility of the initial points and iteration points. Under suitable assumptions, it is shown that the algorithm can find an -approximate solution of an SOCP in at most O（√n ln（ε0/ε）） iterations. The iteration-complexity bound of our algorithm is almost the same as the best known bound of feasible interior point algorithms for the SOCP.

GLOBAL COVERGENCE OF THE NON-QUASI-NEWTON METHOD FOR UNCONSTRAINED OPTIMIZATION PROBLEMS

In this paper, the non-quasi-Newton＇s family with inexact line search applied to unconstrained optimization problems is studied. A new update formula for non-quasi-Newton＇s family is proposed. It is proved that the constituted algorithm with either Wolfe-type or Armijotype line search converges globally and Q-superlinearly if the function to be minimized has Lipschitz continuous gradient.

Some new step-size rules for optimization problems

The step-size procedure is very important for solving optimization problems. The Armijo step-size rule, the Armijo-Goldstein step-size rule and the Wolfe-Powell step-size rule are three well-known line search methods. On the basis of the above three types of line search methods and the idea of the proximal point methods, a new class of step-size rules was proposed. Instead of a single objective function f, f ＋1/2（x - xk）^TBk（x-Xk） was used as the merit function in iteration k, where Sk is a given symmetric positive definite matrix. The existence of the steplength for the new rules was proved. Some convergence properties were also discussed.

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.

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.

Smoothing Newton-Like Method for the Solution of Nonlinear Systems of Equalities and Inequalities

In this paper,we present a smoothing Newton-like method for solving nonlinear systems of equalities and inequalities.By using the so-called max function,we transfer the inequalities into a system of semismooth equalities.Then a smoothing Newton-like method is proposed for solving the reformulated system,which only needs to solve one system of linear equations and to perform one line search at each iteration. The global and local quadratic convergence are studied under appropriate assumptions. Numerical examples show that the new approach is effective.