Examples
-function f1=gattainObjfun(x) + |  |  |
function f1=gattainObjfun(x) + | | |
Solves minimax constraint problem
x = fminimax(fun,x0) +x = fminimax(fun,x0,A,b) +x = fminimax(fun,x0,A,b,Aeq,beq) +x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub) +x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlinfun) +x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlinfun,options) +[x, fval] = fmincon(.....) +[x, fval, maxfval]= fmincon(.....) +[x, fval, maxfval, exitflag]= fmincon(.....) +[x, fval, maxfval, exitflag, output]= fmincon(.....) +[x, fval, maxfval, exitflag, output, lambda]= fmincon(.....)
The function to be minimized. fun is a function that accepts a vector x and returns a vector F, the objective functions evaluated at x.
a nx1 or 1xn matrix of doubles, where n is the number of variables, the initial guess for the optimization algorithm
a nil x n matrix of doubles, where n is the number of variables and nil is the number of linear inequalities. If A==[] and b==[], it is assumed that there is no linear inequality constraints. If (A==[] & b<>[]), fminimax generates an error (the same happens if (A<>[] & b==[]))
a nil x 1 matrix of doubles, where nil is the number of linear inequalities
a nel x n matrix of doubles, where n is the number of variables and nel is the number of linear equalities. If Aeq==[] and beq==[], it is assumed that there is no linear equality constraints. If (Aeq==[] & beq<>[]), fminimax generates an error (the same happens if (Aeq<>[] & beq==[]))
a nel x 1 matrix of doubles, where nel is the number of linear equalities
a nx1 or 1xn matrix of doubles, where n is the number of variables. The lower bound for x. If lb==[], then the lower bound is automatically set to -inf
a nx1 or 1xn matrix of doubles, where n is the number of variables. The upper bound for x. If ub==[], then the upper bound is automatically set to +inf
function that computes the nonlinear inequality constraints c(x) <= 0 and nonlinear equality constraints ceq(x) = 0.
a nx1 matrix of doubles, the computed solution of the optimization problem
a vector of doubles, the value of fun at x
a 1x1 matrix of doubles, the maximum value in vector fval
a 1x1 matrix of floating point integers, the exit status
a struct, the details of the optimization process
a struct, the Lagrange multipliers at optimum
a list, containing the option for user to specify. See below for details.
fminimax minimizes the worst-case (largest) value of a set of multivariable functions, starting at an initial estimate. This is generally referred to as the minimax problem.
+
Currently, fminimax calls fmincon which uses the ip-opt algorithm.
+max-min problems can also be solved with fminimax, using the identity
+
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. +
The objective function must have header : +
F = fun(x) | | |
By default, the gradient options for fminimax are turned off and and fmincon does the gradient opproximation of minmaxObjfun. In case the GradObj option is off and GradConstr option is on, fminimax approximates minmaxObjfun gradient using numderivative toolbox.
+If we can provide exact gradients, we should do so since it improves the convergence speed of the optimization algorithm.
+Furthermore, we must enable the "GradObj" option with the statement : +
minimaxOptions = list("GradObj",fGrad); | | |
The constraint function must have header : +
[c, ceq] = confun(x) | | |
By default, the gradient options for fminimax are turned off and and fmincon does the gradient opproximation of confun. In case the GradObj option is on and GradCons option is off, fminimax approximates confun gradient using numderivative toolbox.
+If we can provide exact gradients, we should do so since it improves the convergence speed of the optimization algorithm.
+Furthermore, we must enable the "GradCon" option with the statement : +
minimaxOptions = list("GradCon",confunGrad); | | |
The constraint derivative function must have header : +
[dc,dceq] = confungrad(x) | | |
The exitflag allows to know the status of the optimization which is given back by Ipopt. +
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. +
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. +
// A basic case : +// we provide only the objective function and the nonlinear constraint +// function +function f=myfun(x) +f(1)= 2*x(1)^2 + x(2)^2 - 48*x(1) - 40*x(2) + 304; //Objectives +f(2)= -x(1)^2 - 3*x(2)^2; +f(3)= x(1) + 3*x(2) -18; +f(4)= -x(1) - x(2); +f(5)= x(1) + x(2) - 8; +endfunction +// The initial guess +x0 = [0.1,0.1]; +// The expected solution : only 4 digits are guaranteed +//xopt = [4 4] +//fopt = [0 -64 -2 -8 0] +maxfopt = 0 +// Run fminimax +[xopt,fopt,maxfval,exitflag,output,lambda] = fminimax(myfun, x0) +// Press ENTER to continue | | |
// A case where we provide the gradient of the objective +// functions and the Jacobian matrix of the constraints. +// The objective function and its gradient +function f=myfun(x) +f(1)= 2*x(1)^2 + x(2)^2 - 48*x(1) - 40*x(2) + 304; +f(2)= -x(1)^2 - 3*x(2)^2; +f(3)= x(1) + 3*x(2) -18; +f(4)= -x(1) - x(2); +f(5)= x(1) + x(2) - 8; +endfunction +// Defining gradient of myfun +function G=myfungrad(x) +G = [ 4*x(1) - 48, -2*x(1), 1, -1, 1; +2*x(2) - 40, -6*x(2), 3, -1, 1; ]' +endfunction +// The nonlinear constraints and the Jacobian +// matrix of the constraints +function [c, ceq]=confun(x) +// Inequality constraints +c = [1.5 + x(1)*x(2) - x(1) - x(2), -x(1)*x(2) - 10] +// No nonlinear equality constraints +ceq=[] +endfunction +// Defining gradient of confungrad +function [DC, DCeq]=cgrad(x) +// DC(:,i) = gradient of the i-th constraint +// DC = [ +// Dc1/Dx1 Dc1/Dx2 +// Dc2/Dx1 Dc2/Dx2 +// ] +DC= [ +x(2)-1, -x(2) +x(1)-1, -x(1) +]' +DCeq = []' +endfunction +// Test with both gradient of objective and gradient of constraints +minimaxOptions = list("GradObj",myfungrad,"GradCon",cgrad); +// The initial guess +x0 = [0,10]; +// The expected solution : only 4 digits are guaranteed +//xopt = [0.92791 7.93551] +//fopt = [6.73443 -189.778 6.73443 -8.86342 0.86342] +maxfopt = 6.73443 +// Run fminimax +[xopt,fopt,maxfval,exitflag,output] = fminimax(myfun,x0,[],[],[],[],[],[], confun, minimaxOptions) | | |