Before reading this article, I suggest reading it first This article , it talks about the principle of nonlinear optimization, that is, the concept of related terms, and then introduces the use method of NLopt, which is based on C language. This film introduces an example of NLopt in C + + language.

Before the example, first introduce the algorithms supported by NLopt and the precautions for using the algorithm

NLopt supported algorithms

NLopt contains many different optimization algorithms.

In the header file, the enumeration type of algorithm name is

  enum algorithm {
     GN_DIRECT = 0,
     GN_DIRECT_L,
     GN_DIRECT_L_RAND,
     GN_DIRECT_NOSCAL,
     GN_DIRECT_L_NOSCAL,
     GN_DIRECT_L_RAND_NOSCAL,
     GN_ORIG_DIRECT,
     GN_ORIG_DIRECT_L,
     GD_STOGO,
     GD_STOGO_RAND,
     LD_LBFGS_NOCEDAL,
     LD_LBFGS,
     LN_PRAXIS,
     LD_VAR1,
     LD_VAR2,
     LD_TNEWTON,
     LD_TNEWTON_RESTART,
     LD_TNEWTON_PRECOND,
     LD_TNEWTON_PRECOND_RESTART,
     GN_CRS2_LM,
     GN_MLSL,
     GD_MLSL,
     GN_MLSL_LDS,
     GD_MLSL_LDS,
     LD_MMA,
     LN_COBYLA,
     LN_NEWUOA,
     LN_NEWUOA_BOUND,
     LN_NELDERMEAD,
     LN_SBPLX,
     LN_AUGLAG,
     LD_AUGLAG,
     LN_AUGLAG_EQ,
     LD_AUGLAG_EQ,
     LN_BOBYQA,
     GN_ISRES,
     AUGLAG,
     AUGLAG_EQ,
     G_MLSL,
     G_MLSL_LDS,
     LD_SLSQP,
     LD_CCSAQ,
     GN_ESCH,
     NUM_ALGORITHMS /*Not an algorithm, just the number of algorithms*/
  };

Naming rules:

G/L stands for global or local optimization
N/D represents the optimization without derivative or derivative

For example, LN_COBYLA is the COBYLA algorithm used, and then the algorithm is used for local (L) optimization without derivative (N)

Algorithm selection

So what kind of algorithm do we use? Any choice? NO
For the optimization problem of a certain mathematical model, a better way is to compare several available algorithms and then determine which one to choose, because there is no optimal algorithm, and different optimization problems have different optimal solutions.

When comparing algorithms, it should also be noted that the function value tolerance and parameter tolerance adapted by different algorithms should also be different.

There is a better way to compare the advantages and disadvantages of two algorithms for a mathematical model:
First run an algorithm to get the minimum value, and then set it to converge to that value when running the second algorithm, and then compare the running time

Problems needing attention in selecting global optimization

At present, all global optimization algorithms require boundary constraints for all optimization parameters.
And it must be noted that
Among these algorithms, only ISRES, AGS, and ORIG_DIRECT supports nonlinear inequality constraints
Only ISRES supports nonlinear equality constraints

A good way is to find a best advantage through the global algorithm, and then take this best advantage as the starting point of local optimization to make the results more accurate

The global optimization algorithm takes much more energy to find the best advantage than the local optimization algorithm

Code

Mathematical model:
Find: max (lnx1+lnx2) objective function
Constraints: p1x1+p2x2=5 equality constraints
X1 < = x2 inequality constraint
x1>=0; X2 > = 0 boundary constraint

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#Include < nlopt. HPP > / / header file of nlopt

Reference header file
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//Declare a global variable whether to use gradients
bool grad_bool=1;

Declare a global variable whether to use gradients
In this way, it is convenient to change this variable when switching between gradient based algorithm and algorithm without derivative
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double myfunc(const std::vector<double>& x,std::vector<double>&grad,void* f_data )
{
    double result;//Declare results
    if (grad_bool)
    {
        grad[0] = 1 / x[0];
        grad[1] = 1 / x[1];
    }
    result = log(x[0])+log(x[1]);//Calculation results
    return result;//Return results
}

The objective function is max (lnx1+x2)
When defining the objective function, note that the form of the parameters of the function cannot be changed
Parameter x is the parameter vector to be optimized
Grad returns to the gradient of the optimization parameter at this time, which is the result of partial derivation of X in the mathematical model. If partial derivation is required, lnx1+lnx2 partial derivation of x1, and 1/x1, that is, grad[0] = 1 / x[0]; This place
f_data is the parameter to be passed in
The return value is the value of the objective function under an x vector
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double myconstraint(const std::vector<double>& x,std::vector<double>&grad,void* f_data)
{
    double *p = (double *) f_data;
    if (grad_bool)
    {
        grad[0] = p[0];
        grad[1] = p[1];
    }
    double result;//Declare results    
    result = x[0] * p[0] + x[1] * p[1] - 5;//Calculation results
    return result;//Return results
}

Define equality constraint function
The equation of the mathematical model is: p1x1+p2x2=5
p1 and p2 are passed in as external parameters

The best way is to change the writing method, so that any parameter format can be applied

typedef struct{
    double p1,p2;
}my_constraint_data;

Declare the structure to pass in the parameter

double myconstraint(const std::vector<double>& x,std::vector<double>&grad,void* f_data)
{
    my_constraint_data *d = (my_constraint_data*)f_data;
    //double *p = (double *) f_data;
    double p1 = d->p1;
    double p2 = d->p2;
    if (grad_bool)
    {
        grad[0] = p1;
        grad[1] = p2;
    }
    double result;//Declare results    
    result = x[0] *  p1  + x[1] * p2 - 5;//Calculation results
    return result;//Return results
}

Then when taking parameters, you can

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double myinconstraint(const  std::vector<double>& x,std::vector<double>&grad,void* f_data)
{
    if (grad_bool)
    {
        grad[0] = 1;
	    grad[1] = -1;
    }

    double result;//Declare results    
    result = x[0]  - x[1] ;//Calculation results
    return result;//Return results
}

Constraint function of inequality
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int main(int argc, char** argv) 
{
    //Initialize node
    ros::init(argc, argv, "lidar_align");

    //Declare two handles   
    ros::NodeHandle nh;

Then there is the main function. I did it under ros, so I added a node initialization
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    //Parameters in mathematical model
    //double p[2]={1,2};
    my_constraint_data data = {1,2};

    //Parameters to be optimized
    std::vector<double> x{1,1};

Declare the parameters of equality constraints in the mathematical model_ constraint_ The data structure is defined above
Declaring the parameter 1,1 to be optimized is equivalent to initializing
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    //Declare an optimizer
    nlopt::opt opter;

Declare an optimizer for NLopt

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    /*Local optimization algorithm*/
    //opter = nlopt::opt(nlopt::LD_SLSQP,2);  // Derivative
    //opter = nlopt::opt(nlopt::LN_COBYLA,2); // derivative-free 
    /*Global optimization algorithm*/
    opter = nlopt::opt(nlopt::GN_ISRES,2);    //No derivative is possible X1 = 1.66667 x2 = 1.66667 Fmax = 1.02165

Set what optimization algorithm to use for the optimizer, and set the number of factors, that is, the number of x
There are optimization algorithms in nlopt.hpp, from which you can choose what you want to use

But note that not all algorithms can be calculated. For example, in the global optimization algorithm with inequality constraints and equality constraints in this mathematical model, only ISRES can be used

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    /*Boundary constraints of optimization parameters*/
    std::vector<double> lb {1,1};//Note that the number of parameters should correspond to
    std::vector<double> ub {10000,10000};//Note that the number of parameters should correspond to
        //Set parameter boundary
    opter.set_lower_bounds(lb);//Set parameter lower limit
    opter.set_upper_bounds(ub);//Set parameter upper limit

Boundary constraints of optimization parameters

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    /*Set optimization accuracy*/
	double tol = 1e-8;//tolerance
    opter.set_xtol_rel(tol);
    opter.set_force_stop(tol);

Set optimization accuracy
When does control optimization stop
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    /*Set the objective function, where the objective function is self-defined */
    // myfunc is the function name defined by itself. The second parameter is data that can be passed in. If there is no data, it is NULL
    opter.set_max_objective(myfunc,NULL);

Set objective function

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    /*Set equality constraints and inequality constraints of mathematical model   */
    //myconstraint is the name of the self-defined function, and the second parameter is the data that can be passed in,
    opter.add_equality_constraint(myconstraint,&data,tol);
    opter.add_inequality_constraint(myinconstraint, NULL,tol);

Set equality constraints and inequality constraints of mathematical model
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    /* The maximum value of the obtained result is stored in F_ The corresponding vector in Max is stored in x*/
	double f_max = -10000;
    nlopt::result res = opter.optimize(x,f_max);

Perform optimization calculations
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 std::cout<<"x1 = " <<x[0]<<"    x2= "<<x[1]<<"  fmax = "<<f_max<<std::endl;

Print calculation results

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Result