NUMERICAL RESOLVENT METHODS FOR CONSTRAINED PROBLEMS IN MECHANICS

Resolvent methods are presented for generating systematically iterative numerical algorithms for constrained problems in mechanics.The abstract framework corresponds to a general mixed finite element subdif-ferential model,with dual and primal evolution versions,which is shown to apply to problems of fluid dynamics,transport phenomena and solid mechanics,among others.In this manner,Uzawa's type methods and penalization-duality schemes,as well as macro-hybrid formulations,are generalized to non necessarily potential nanlinear mechanical problems.

Integral Global Minimization of Constrained Problems with Discontinuous Penalty Functions

A class of discontinuous penalty functions was proposed to solve constrained minimization problems with the integral approach to global optimization, m-mean value and v-variance optimality conditions of a constrained and penalized minimization problem were investigated. A nonsequential algorithm was proposed. Numerical examples were given to illustrate the effectiveness of the algorithm.

Modified Exact Jacobian Semidefinite Programming Relaxation for Celis-Dennis-Tapia Problem

A modified exact Jacobian semidefinite programming(SDP)relaxation method is proposed in this paper to solve the Celis-Dennis-Tapia(CDT)problem using the Jacobian matrix of objective and constraining polynomials.In the modified relaxation problem,the number of introduced constraints and the lowest relaxation order decreases significantly.At the same time,the finite convergence property is guaranteed.In addition,the proposed method can be applied to the quadratically constrained problem with two quadratic constraints.Moreover,the efficiency of the proposed method is verified by numerical experiments.

A New Strategy for Solving a Class of Constrained Nonlinear Optimization Problems Related to Weather and Climate Predictability

There are three common types of predictability problems in weather and climate, which each involve different constrained nonlinear optimization problems： the lower bound of maximum predictable time, the upper bound of maximum prediction error, and the lower bound of maximum allowable initial error and parameter error. Highly effcient algorithms have been developed to solve the second optimization problem. And this optimization problem can be used in realistic models for weather and climate to study the upper bound of the maximum prediction error. Although a filtering strategy has been adopted to solve the other two problems, direct solutions are very time-consuming even for a very simple model, which therefore limits the applicability of these two predictability problems in realistic models. In this paper, a new strategy is designed to solve these problems, involving the use of the existing highly effcient algorithms for the second predictability problem in particular. Furthermore, a series of comparisons between the older filtering strategy and the new method are performed. It is demonstrated that the new strategy not only outputs the same results as the old one, but is also more computationally effcient. This would suggest that it is possible to study the predictability problems associated with these two nonlinear optimization problems in realistic forecast models of weather or climate.

Improvement of Lagrangian relaxation performance for open pit mines constrained long-term production scheduling problem

Constrained long-term production scheduling problem(CLTPSP) of open pit mines has been extensively studied in the past few decades due to its wide application in mining projects and the computational challenges it poses become an NP-hard problem.This problem has major practical significance because the effectiveness of the schedules obtained has strong economical impact for any mining project.Despite of the rapid theoretical and technical advances in this field,heuristics is still the only viable approach for large scale industrial applications.This work presents an approach combining genetic algorithms(GAs) and Lagrangian relaxation(LR) to optimally determine the CLTPSP of open pit mines.GAs are stochastic,parallel search algorithms based on the natural selection and the process of evolution.LR method is known for handling large-scale separable problems; however,the convergence to the optimal solution can be slow.The proposed Lagrangian relaxation and genetic algorithms(LR-GAs) combines genetic algorithms into Lagrangian relaxation method to update the Lagrangian multipliers.This approach leads to improve the performance of Lagrangian relaxation method in solving CLTPSP.Numerical results demonstrate that the LR method using GAs to improve its performance speeding up the convergence.Subsequently,highly near-optimal solution to the CLTPSP can be achieved by the LR-GAs.

Convergence analysis of a nonlinear Lagrange algorithm for general nonlinear constrained optimization problems

The convergence analysis of a nonlinear Lagrange algorithm for solving nonlinear constrained optimization problems with both inequality and equality constraints is explored in detail. The estimates for the derivatives of the multiplier mapping and the solution mapping of the proposed algorithm are discussed via the technique of the singular value decomposition of matrix. Based on the estimates, the local convergence results and the rate of convergence of the algorithm are presented when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions. Furthermore, the condition number of the Hessian of the nonlinear Lagrange function with respect to the decision variables is analyzed, which is closely related to efficiency of the algorithm. Finally, the preliminary numericM results for several typical test problems are reported.

A new evolutionary algorithm for constrained optimization problems

To solve single-objective constrained optimization problems,a new population-based evolutionary algorithm with elite strategy(PEAES) is proposed with the concept of single and multi-objective optimization.Constrained functions are combined to be an objective function.During the evolutionary process,the current optimal solution is found and treated as the reference point to divide the population into three sub-populations:one feasible and two infeasible ones.Different evolutionary operations of single or multi-objective optimization are respectively performed in each sub-population with elite strategy.Thirteen famous benchmark functions are selected to evaluate the performance of PEAES in comparison of other three optimization methods.The results show the proposed method is valid in efficiency,precision and probability for solving single-objective constrained optimization problems.

Discontinuous penalty approach with deviation integral for global constrained minimization

In this paper, we use the discontinuous exact penalty functions to solve the constrained minimization problems with an integral approach. We examine a general form of the constrained deviation integral and its analytical properties. The optimality conditions of the penalized minimization problems are proven. To implement the al- gorithm, the cross-entropy method and the importance sampling are used based on the Monte-Carlo technique. Numerical tests show the effectiveness of the proposed algorithm.

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 hybrid cuckoo search algorithm with feasibility-based rule for constrained structural optimization

Constrained optimization problems are very important as they are encountered in many science and engineering applications.As a novel evolutionary computation technique,cuckoo search(CS) algorithm has attracted much attention and wide applications,owing to its easy implementation and quick convergence.A hybrid cuckoo pattern search algorithm(HCPS) with feasibility-based rule is proposed for solving constrained numerical and engineering design optimization problems.This algorithm can combine the stochastic exploration of the cuckoo search algorithm and the exploitation capability of the pattern search method.Simulation and comparisons based on several well-known benchmark test functions and structural design optimization problems demonstrate the effectiveness,efficiency and robustness of the proposed HCPS algorithm.

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.

A New Algorithm for Resource Constraint Project Scheduling Problem Based on Multi-Agent Systems

The resource constrained project scheduling problem (RCPSP) and a decision-making model based on multi-agent systems (MAS) and general equilibrium marketing are proposed. An algorithm leading to the resource allocation decision involved in RCPSP has also been developed. And this algorithm can be used in the multi-project scheduling field as well.Finally, an illustration is given.

An Algorithm Based on Mobile Agent for Delay Constrained Least Cost Unicast Routing

The delay constrained least cost path problem with imprecise delay information is discussed, and a distributed heuristic algorithm without any assumption of imprecise state information is presented. The algorithm employs mobile agents to search feasible paths in parallel and requires limited network state information kept at each node. The simulations indicate that the presented solution provides better call acceptance probability and better fairness between short paths and long paths. And the algorithm can tolerate high degree of delay imprecision.

New smooth gap function for box constrained variational inequalities

A new smooth gap function for the box constrained variational inequality problem （VIP） is proposed based on an integral global optimality condition. The smooth gap function is simple and has some good differentiable properties. The box constrained VIP can be reformulated as a differentiable optimization problem by the proposed smooth gap function. The conditions, under which any stationary point of the optimization problem is the solution to the box constrained VIP, are discussed. A simple frictional contact problem is analyzed to show the applications of the smooth gap function. Finally, the numerical experiments confirm the good theoretical properties of the method.

Multi-Mode Resource Constrained Project Scheduling Models for Progress and Equal Time Interval Payments

This study utilizes a time-precedence network technique to construct two models of multi-mode resource constrained project scheduling problem with discounted cash flows （MRCPSPDCF）, individually including the progress payment （PP） and the payment at an equal time interval （ETI）. The objective of each model is to maximize the net present value （NPV） for all cash flows in the project, subject to the related operational constraints. The models are characterized as NP-hard. A heuristic algorithm, coupled with two upper bound solutions, is proposed to efficiently solve the models and evaluate the heuristic algorithm performance which was not performed in past studies. The results show that the performance of proposed models and heuristic algorithm is good.

MATRIX EQUATION AXB=E WITH PX=sXP CONSTRAINT

The matrix equation AXB = E with the constraint PX = sXP is considered, where P is a given Hermitian matrix satisfying P^2 = I and s = ±1. By an eigenvalue decomposition of P, the constrained problem can be equivalently transformed to a well-known unconstrained problem of matrix equation whose coefficient matrices contain the corresponding eigenvector, and hence the constrained problem can be solved in terms of the eigenvectors of P. A simple and eigenvector-free formula of the general solutions to the constrained problem by generalized inverses of the coefficient matrices A and B is presented. Moreover, a similar problem of the matrix equation with generalized constraint is discussed.

AN SQP ALGORITHM WITH NONMONOTONE LINE SEARCHFOR GENERAL NONLINEAR CONSTRAINED OPTIMIZATION PROBLEM

In this paper, an SQP type algorithm with a new nonmonotone line search technique for general constrained optimization problems is presented. The new algorithm does not have to solve the second order correction subproblems for each iterations, but still can circumvent the so-called Maratos effect. The algorithm's global convergence and superlinear convergent rate have been proved. In addition, we can prove that, after a few iterations, correction subproblems need not be solved, so computation amount of the algorithm will be decreased much more. Numerical experiments show that the new algorithm is effective.

Algorithm Studies on How to Obtain a Conditional Nonlinear Optimal Perturbation (CNOP)

The conditional nonlinear optimal perturbation （CNOP）, which is a nonlinear generalization of the linear singular vector （LSV）, is applied in important problems of atmospheric and oceanic sciences, including ENSO predictability, targeted observations, and ensemble forecast. In this study, we investigate the computational cost of obtaining the CNOP by several methods. Differences and similarities, in terms of the computational error and cost in obtaining the CNOP, are compared among the sequential quadratic programming （SQP） algorithm, the limited memory Broyden-Fletcher-Goldfarb-Shanno （L-BFGS） algorithm, and the spectral projected gradients （SPG2） algorithm. A theoretical grassland ecosystem model and the classical Lorenz model are used as examples. Numerical results demonstrate that the computational error is acceptable with all three algorithms. The computational cost to obtain the CNOP is reduced by using the SQP algorithm. The experimental results also reveal that the L-BFGS algorithm is the most effective algorithm among the three optimization algorithms for obtaining the CNOP. The numerical results suggest a new approach and algorithm for obtaining the CNOP for a large-scale optimization problem.

EXISTENCE AND STABILITY OF STANDING WAVES FOR A COUPLED NONLINEAR SCHRDINGER SYSTEM

We study the existence and stability of the standing waves of two coupled SchrSdinger equations with potentials ｜x｜bi（bi ∈ R,i = 1, 2）. Under suitable conditions on the growth of the nonlinear terms, we first establish the existence of standing waves of the SchrSdinger system by solving a L2-normalized minimization problem, then prove that the set of all minimizers of this minimization problem is stable. Finally, we obtain the least energy solutions by the Nehari method and prove that the orbit sets of these least energy solutions are unstable, which generalizes the results of [11] where b1 = b2 = 2.

Distributed H_(∞)Consensus Problem for First-Order Multi-Agent Systems with Antagonistic Interactions and Nonconvex Constraints

This paper investigates the distributed H_(∞)consensus problem for a first-order multiagent system where both cooperative and antagonistic interactions coexist.In the presence of external disturbances,a distributed control algorithm using local information is addressed and a sufficient condition to get the H_(∞)control gain is obtained,which make the states of the agents in the same group converge to a common point while the inputs of each agent are constrained in the nonconvex sets.Finally,a numerical simulation is exhibited to illustrate the theory.