I think you need to explain the problem correctly, and clearly, before anyone can be of any help. If English is a problem, try to receive some assistance in stating your question.

Ok, I will try to explain the problem clearly, Thanks

use the model MULTIMOORA. Many of your countrymen used it. See Google or World of Science.

NSGAII, MOEA/D,etc are all good choices, good luck

Use Bat algorithm, Cuckoo Search algorithm

• Alireza Soroudi, M. Ehsan, "Multi-Objective Planning Model for Integration of Distributed Generations in Deregulated Power Systems", Iranian Journal of Science and Technology, Transaction B. Engineering Vol 34. No B3. Pages: 307-324. File
• Alireza Soroudi; Afrasiab, M., "Binary PSO-based dynamic multi-objective model for distributed generation planning under uncertainty," Renewable Power Generation, IET , vol.6, no.2, pp.67,78, March 2012. File
• Alireza Soroudi; Ehsan, M.; Caire, R.; Hadjsaid, N.; , "Hybrid immune-genetic algorithm method for benefit maximisation of distribution network operators and distributed generation owners in a deregulated environment," Generation, Transmission & Distribution, IET , vol.5, no.9, pp.961-972, September 2011. File

yes, you can try Genetic Algorithm, it is very good for multi-objective functions. Also may be one of swarm optimization methods are good to converge multi-obj. fun's.

C. A. C. Coello, G. B. Lamont, and D. A. Van Veldhuizen, "Evolutionary Algorithms for Solving Multi-Objective Problems," New-York, USA: Springer-Verlag New-York Inc., 2007.

A problem of the general form :

Min J(x,u)

subject to x_dot = f(x,u)

is an optimal control problem, that should be solved either analytically (which is not your case) or by Dynamic Programming.

Moreover, to combine the two objectives there are different methods. A simple one is to simply sum up them after dividing each by their individual optima..

NSGA - II  is known to be the best.

NSGA

https://www.researchgate.net/publication/225141932_Application_of_a_Modified_NSGA_Method_for_Multi-Objective_Static_Distributed_Generation_Planning?ev=prf_pub 

PSO

https://www.researchgate.net/publication/258836033_Binary_PSO-based_dynamic_multi-objective_model_for_distributed_generation_planning_under_uncertainty?ev=prf_pub

Immune-GA

https://www.researchgate.net/publication/224258140_Hybrid_immune-genetic_algorithm_method_for_benefit_maximisation_of_distribution_network_operators_and_distributed_generation_owners_in_a_deregulated_environment?ev=prf_pub

Article Application of a Modified NSGA Method for Multi-Objective St...

Article Binary PSO-based dynamic multi-objective model for distribut...

Article Hybrid immune-genetic algorithm method for benefit maximisat...

For these types of problems you can use the constraint method, in which you first solve one objective function without constraints on the other o.f., and then solve the optimization problem for the second objective function without constraints on the first one. Select one of them to run the multiobjective problem and use the other one as constraints, bounded from the results of those two solutions. Run the optimization for values of the constraint within those bounds to prepare a set of optimal solutions, typically a Pareto front.

For the optimization, if the function is highly nonlinear and prone to not convergence with the use of gradient methods, use a meta-heuristic method, e.g. genetic algorithms, which work quite well due to its robust nature.

(For an application, see Medina-Herrera et al., IECR, 2014, 12023-12031)

One can also use a penalty method tied to GAs (see for instance Ponce-Ortega et al., CACE, 2008, 32, 2320-2332, and Applied Thermal Eng, 2009, 29, 203-209)

Genetic Algorithms can be used. 

Dear Zhang,

I do not understand your problem. Could you please formulate what are the objectives to be minimized, what are the parameters or function which represent the degrees of freedom. x seems to be a space variable, isn't it. If yes. of which dimension? The variable t seems to be the time.

However, what is then Df(x) = M(x, \tilde y(t)) f(x) with \tilde y(t) = y(tau(t)))? This looks like a linear  ODE system in space for the vector-valued (?) function f(x) with time as parameter. In this case D stands for the differentiation w.r.t. space. If this is true, the equation is an ODE vector constraint. Parameters are a and c. However, b, w, and r do not enter your formulation.

Sorry, the problem is not well formulated.

Kind regards,

Hans Josef

If you have the derivatives available, you could use a simple mono-objective gradient optimization on a linear combination of your different objectives.

You could use several starting point and several combination factor to ensure robustness of the solution found.

Regards,

Julien

Evolutionary Algorithm and Differential Evolutionary optimization algorithms are also very efficient in the MO field.

Dear Hans Josef,

  The problem is diffifercult to describe for me.But I will try again.The differential equation are as follows:

•          D2_phi=Tau+M(D_phi,D_P)*phi+lamda* D_phi, phi is  n(n>=3) dimensions vector.
•          D2_psi=qT+Tau+Q(D_phi, D_P)*phi+lamda* D_phi, psi is a scalar.
•          D2_P(t)=epsilong(phi, D_phi, D_psi)+Psi(phi,psi)*D_P(t), P(t) is a 2 dimension vector.

 tau is the input, which has the form:  tau=a*sin(w*t+c)+r,  t stand for time.  The two-objective functions are tau(minimum values) and velocity=norm(P(t)) (maximum value).

Parameters (a,b,c,w,r) are needed to optimization. D stand for differentiation.D2 stand for Differential again. lamda is an constant coefficient,  Others coefficients like M(D_phi,D_P), Q(D_phi, D_P)  and epsilong(phi, D_phi, D_psi) ,Psi(phi,psi)  are relate to variables.

Kind regards,

zhang

I think PSO can solve such kind of problem. 

Try MOORA or MULTIMOORA.

See therefore Google or the Web of Science

There are lots of different approches and software for ordinary (non-)linear optimzation problems. But as I understand your question the problem is optimizing the definition of a PDE or ODE though the problem is not completey defined in your question:

Is x one- or multidimensional, i.e. is the equation a PDE or an ODE?

For which values of t should the equation hold? If it should hold for all t in an interval - is the existence of such solution to the equation guaranteed?

In which sense should the functions f(x) and tau=a*sin(w*t+c)+r be objectives?

All in all your question does not clearly concern a multiobjective problem - at least as I understand this notion.

Because  the differential equation is  complex and mult-dimension. Genetic Algorithm ,NSGA_2 can solve this problem ,but the Program will run for a long time. each time I get a new objective function value,It will take 10 minutes.

So I hope to find  a Algorithm ,which has a higher  convergence efficiency and do not need the gradient information.

Modified NSGA II is a good choice

Without looking at your problem.

NSGA-II doesn't need any gradient information. If you want the solution to converge faster use GeDEA-II  

@Kondeti Lakshmi Vasudev ,Thanks for your advice.