this link may be help you

https://www.mathworks.com/matlabcentral/answers/59915-finding-an-initial-feasible-point-x0-for-fmincon

a big class of Matlab's/Octave's optimization tools accept a @pointer to a function as an input. So you need to write your problem as a callable function. However... not all tools are well constrained, or able to avoid known solutions, so 10 years ago I was still inventing quasi-Newtonian scripts for solving wave equations with arbitrary BCs.

Whatever how you write your function, you need to identify the unknown vector x and code a function of x to be minimized. Then, if you have no constraint, you can use the fminunc function to solve your problem of fmincon in case you have constraints.

Thank you Abbas Miry. It was a helpful link. I'm reading the information on the page and the related links.

Thank you AndreyAndrey Porokhnyuk

I'll search using the keywords you mentioned and I'll ask again if there were more questions.

Richard Epenoy,

Thanks for your kind answer. I have written my unknown variables as a vector, but the point is that the ultimate function is highly dependent on the initial condition. In other words, an initial value must be defined for the unknown vector and then the objective function can be calculated. Afterward, the unknown vector needs to be amended so that it could reduce the objective function. It will take a long time to run the code and find the best solution. And the optimization methods, which I know, cannot solve this problem simply.

Dear Sahand,

I do not understand well your problem. In case the evaluation of the objective function f(x) depends on the unknown vector x but also on an additional "initial condition" I, you need to include I in the unknown vector. Best.

Obviously, if the value of the objective function depends on the parameter vectors x it will also depend on the initial values x0 you provide for its components. However, if the objective function is well behaved, which is probably not the case, dependent on the initial value you'll get various local minima. What you are trying to find is the global minimum of the objective function and the parameter set x that gives this global minimum. Using gradient methods will usually not get to the global minimum unless you are lucky with the initial value. Check out global optimization methods. There are genetic algorithms that might work, also there is a global optimizer called MIPT (also available as a toolbox) and some optimizer that uses an analogy to ant colonies (I dont no exactly how the method is called) that might work.

How mandy dimensions has the parameter vector?

MATLAB provides a wide range of solvers for nonlinear problems. The solvers are generally organized into Optimization toolbox and Global Optimization toolbox.

Optimization toolbox provides solvers for convex problems where the gradient descent approach guarantees finding the global optimum. LP, ILP, QP, and convex smooth NLP are examples of problems addressed by Optimization toolbox. Convex NLP solvers usually require a starting point x0 to be provided by the user, which can affect the final solution, in case of being used for non-convex problems.

The Global Optimization toolbox provides solvers for optimization problems that contain multiple local or global minima. This class is characterized by multiple local optima that can trap convex solvers. This toolbox contains signle-objective and multi-objective metaheuristics among other approaches. Global Optimization solvers seek global optimum and do not require an initial state for optimization process, although user can provide initial population.

MATLAB Optimization toolbox documentation: mathworks.com/help/optim

MATLAB Global optimization toolbox documentation: mathworks.com/help/gads

Have a look at :: https://au.mathworks.com/help/optim/ug/optim.problemdef.optimizationproblem.solve.html

https://au.mathworks.com/videos/part-4-problem-based-nonlinear-programming-1549458887351.html

https://au.mathworks.com/help/matlab/math/partial-differential-equations.html