fmincon

Parameters

f : +: a function, representing the objective function of the problem
x0 : +: a vector of doubles, containing the starting values of variables of size (1 X n) or (n X 1) where 'n' is the number of Variables
A : +: a matrix of doubles, containing the coefficients of linear inequality constraints of size (m X n) where 'm' is the number of linear inequality constraints
b : +: a vector of doubles, related to 'A' and containing the the Right hand side equation of the linear inequality constraints of size (m X 1)
Aeq : +: a matrix of doubles, containing the coefficients of linear equality constraints of size (m1 X n) where 'm1' is the number of linear equality constraints
beq : +: a vector of doubles, related to 'Aeq' and containing the the Right hand side equation of the linear equality constraints of size (m1 X 1)
lb : +: a vector of doubles, containing the lower bounds of the variables of size (1 X n) or (n X 1) where 'n' is the number of Variables
ub : +: a vector of doubles, containing the upper bounds of the variables of size (1 X n) or (n X 1) where 'n' is the number of Variables
nlc : +: a function, representing the Non-linear Constraints functions(both Equality and Inequality) of the problem. It is declared in such a way that non-linear inequality constraints are defined first as a single row vector (c), followed by non-linear equality constraints as another single row vector (ceq). Refer Example for definition of Constraint function.
options : +: a list, containing the option for user to specify. See below for details.
xopt : +: a vector of doubles, cointating the computed solution of the optimization problem
fopt : +: a scalar of double, containing the the function value at x
exitflag : +: a scalar of integer, containing the flag which denotes the reason for termination of algorithm. See below for details.
output : +: a structure, containing the information about the optimization. See below for details.
lambda : +: a structure, containing the Lagrange multipliers of lower bound, upper bound and constraints at the optimized point. See below for details.
gradient : +: a vector of doubles, containing the Objective's gradient of the solution.
hessian : +: a matrix of doubles, containing the Lagrangian's hessian of the solution.

Description

Search the minimum of a constrained optimization problem specified by : +Find the minimum of f(x) such that

The routine calls Ipopt for solving the Constrained Optimization problem, Ipopt is a library written in C++.

The options allows the user to set various parameters of the Optimization problem. +It should be defined as type "list" and contains the following fields. +

Syntax : options= list("MaxIter", [---], "CpuTime", [---], "GradObj", ---, "Hessian", ---, "GradCon", ---);
MaxIter : a Scalar, containing the Maximum Number of Iteration that the solver should take.
CpuTime : a Scalar, containing the Maximum amount of CPU Time that the solver should take.
GradObj : a function, representing the gradient function of the Objective in Vector Form.
Hessian : a function, representing the hessian function of the Lagrange in Symmetric Matrix Form with Input parameters x, Objective factor and Lambda. Refer Example for definition of Lagrangian Hessian function.
GradCon : a function, representing the gradient of the Non-Linear Constraints (both Equality and Inequality) of the problem. It is declared in such a way that gradient of non-linear inequality constraints are defined first as a separate Matrix (cg of size m2 X n or as an empty), followed by gradient of non-linear equality constraints as a separate Matrix (ceqg of size m2 X n or as an empty) where m2 & m3 are number of non-linear inequality and equality constraints respectively.
Default Values : options = list("MaxIter", [3000], "CpuTime", [600]);

The exitflag allows to know the status of the optimization which is given back by Ipopt. +

exitflag=0 : Optimal Solution Found
exitflag=1 : Maximum Number of Iterations Exceeded. Output may not be optimal.
exitflag=2 : Maximum amount of CPU Time exceeded. Output may not be optimal.
exitflag=3 : Stop at Tiny Step.
exitflag=4 : Solved To Acceptable Level.
exitflag=5 : Converged to a point of local infeasibility.

For more details on exitflag see the ipopt documentation, go to http://www.coin-or.org/Ipopt/documentation/

The output data structure contains detailed informations about the optimization process. +It has type "struct" and contains the following fields. +

output.Iterations: The number of iterations performed during the search
output.Cpu_Time: The total cpu-time spend during the search
output.Objective_Evaluation: The number of Objective Evaluations performed during the search
output.Dual_Infeasibility: The Dual Infeasiblity of the final soution

The lambda data structure contains the Lagrange multipliers at the end +of optimization. In the current version the values are returned only when the the solution is optimal. +It has type "struct" and contains the following fields. +

lambda.lower: The Lagrange multipliers for the lower bound constraints.
lambda.upper: The Lagrange multipliers for the upper bound constraints.
lambda.eqlin: The Lagrange multipliers for the linear equality constraints.
lambda.ineqlin: The Lagrange multipliers for the linear inequality constraints.
lambda.eqnonlin: The Lagrange multipliers for the non-linear equality constraints.
lambda.ineqnonlin: The Lagrange multipliers for the non-linear inequality constraints.

Examples

//Find x in R^2 such that it minimizes:
+//f(x)= -x1 -x2/3
+//x0=[0,0]
+//constraint-1 (c1): x1 + x2 <= 2
+//constraint-2 (c2): x1 + x2/4 <= 1
+//constraint-3 (c3): x1 - x2 <= 2
+//constraint-4 (c4): -x1/4 - x2 <= 1
+//constraint-5 (c5): -x1 - x2 <= -1
+//constraint-6 (c6): -x1 + x2 <= 2
+//constraint-7 (c7): x1 + x2 = 2
+//Objective function to be minimised
+function y=f(x)
+y=-x(1)-x(2)/3;
+endfunction
+//Starting point, linear constraints and variable bounds
+x0=[0 , 0];
+A=[1,1 ; 1,1/4 ; 1,-1 ; -1/4,-1 ; -1,-1 ; -1,1];
+b=[2;1;2;1;-1;2];
+Aeq=[1,1];
+beq=[2];
+lb=[];
+ub=[];
+nlc=[];
+//Gradient of objective function
+function y=fGrad(x)
+y= [-1,-1/3];
+endfunction
+//Hessian of lagrangian
+function y=lHess(x, obj, lambda)
+y= obj*[0,0;0,0]
+endfunction
+//Options
+options=list("GradObj", fGrad, "Hessian", lHess);
+//Calling Ipopt
+[x,fval,exitflag,output,lambda,grad,hessian] =fmincon(f, x0,A,b,Aeq,beq,lb,ub,nlc,options)

Examples

//Find x in R^3 such that it minimizes:
+//f(x)= x1*x2 + x2*x3
+//x0=[0.1 , 0.1 , 0.1]
+//constraint-1 (c1): x1^2 - x2^2 + x3^2 <= 2
+//constraint-2 (c2): x1^2 + x2^2 + x3^2 <= 10
+//Objective function to be minimised
+function y=f(x)
+y=x(1)*x(2)+x(2)*x(3);
+endfunction
+//Starting point, linear constraints and variable bounds
+x0=[0.1 , 0.1 , 0.1];
+A=[];
+b=[];
+Aeq=[];
+beq=[];
+lb=[];
+ub=[];
+//Nonlinear constraints
+function [c, ceq]=nlc(x)
+c = [x(1)^2 - x(2)^2 + x(3)^2 - 2 , x(1)^2 + x(2)^2 + x(3)^2 - 10];
+ceq = [];
+endfunction
+//Gradient of objective function
+function y=fGrad(x)
+y= [x(2),x(1)+x(3),x(2)];
+endfunction
+//Hessian of the Lagrange Function
+function y=lHess(x, obj, lambda)
+y= obj*[0,1,0;1,0,1;0,1,0] + lambda(1)*[2,0,0;0,-2,0;0,0,2] + lambda(2)*[2,0,0;0,2,0;0,0,2]
+endfunction
+//Gradient of Non-Linear Constraints
+function [cg, ceqg]=cGrad(x)
+cg=[2*x(1) , -2*x(2) , 2*x(3) ; 2*x(1) , 2*x(2) , 2*x(3)];
+ceqg=[];
+endfunction
+//Options
+options=list("MaxIter", [1500], "CpuTime", [500], "GradObj", fGrad, "Hessian", lHess,"GradCon", cGrad);
+//Calling Ipopt
+[x,fval,exitflag,output] =fmincon(f, x0,A,b,Aeq,beq,lb,ub,nlc,options)

Examples

//The below problem is an unbounded problem:
+//Find x in R^3 such that it minimizes:
+//f(x)= -(x1^2 + x2^2 + x3^2)
+//x0=[0.1 , 0.1 , 0.1]
+//  x1 <= 0
+//  x2 <= 0
+//  x3 <= 0
+//Objective function to be minimised
+function y=f(x)
+y=-(x(1)^2+x(2)^2+x(3)^2);
+endfunction
+//Starting point, linear constraints and variable bounds
+x0=[0.1 , 0.1 , 0.1];
+A=[];
+b=[];
+Aeq=[];
+beq=[];
+lb=[];
+ub=[0,0,0];
+//Options
+options=list("MaxIter", [1500], "CpuTime", [500]);
+//Calling Ipopt
+[x,fval,exitflag,output,lambda,grad,hessian] =fmincon(f, x0,A,b,Aeq,beq,lb,ub,[],options)

Examples

//The below problem is an infeasible problem:
+//Find x in R^3 such that in minimizes:
+//f(x)=x1*x2 + x2*x3
+//x0=[1,1,1]
+//constraint-1 (c1): x1^2 <= 1
+//constraint-2 (c2): x1^2 + x2^2 <= 1
+//constraint-3 (c3): x3^2 <= 1
+//constraint-4 (c4): x1^3 = 0.5
+//constraint-5 (c5): x2^2 + x3^2 = 0.75
+// 0 <= x1 <=0.6
+// 0.2 <= x2 <= inf
+// -inf <= x3 <= 1
+//Objective function to be minimised
+function y=f(x)
+y=x(1)*x(2)+x(2)*x(3);
+endfunction
+//Starting point, linear constraints and variable bounds
+x0=[1,1,1];
+A=[];
+b=[];
+Aeq=[];
+beq=[];
+lb=[0 0.2,-%inf];
+ub=[0.6 %inf,1];
+//Nonlinear constraints
+function [c, ceq]=nlc(x)
+c=[x(1)^2-1,x(1)^2+x(2)^2-1,x(3)^2-1];
+ceq=[x(1)^3-0.5,x(2)^2+x(3)^2-0.75];
+endfunction
+//Gradient of objective function
+function y=fGrad(x)
+y= [x(2),x(1)+x(3),x(2)];
+endfunction
+//Hessian of the Lagrange Function
+function y=lHess(x, obj, lambda)
+y= obj*[0,1,0;1,0,1;0,1,0] + lambda(1)*[2,0,0;0,0,0;0,0,0] + lambda(2)*[2,0,0;0,2,0;0,0,0] +lambda(3)*[0,0,0;0,0,0;0,0,2] + lambda(4)*[6*x(1    ),0,0;0,0,0;0,0,0] + lambda(5)*[0,0,0;0,2,0;0,0,2];
+endfunction
+//Gradient of Non-Linear Constraints
+function [cg, ceqg]=cGrad(x)
+cg = [2*x(1),0,0;2*x(1),2*x(2),0;0,0,2*x(3)];
+ceqg = [3*x(1)^2,0,0;0,2*x(2),2*x(3)];
+endfunction
+//Options
+options=list("MaxIter", [1500], "CpuTime", [500], "GradObj", fGrad, "Hessian", lHess,"GradCon", cGrad);
+//Calling Ipopt
+[x,fval,exitflag,output,lambda,grad,hessian] =fmincon(f, x0,A,b,Aeq,beq,lb,ub,nlc,options)

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fmincon

Calling Sequence

Parameters

Description

Examples

Examples

Examples

Examples

Authors