hi,i have to find the maximum of a given constrained function but i have also a nonlinear constraint, something like a*b^2<1. Fmincon accepts just linear constraint so i don't know how can i add the non-linear inequality!any suggestion?

Hi, maybe there is something I don't see, but you can change variables by taking log. Tell us if it works.

Hi: I was struck!! I did remember using Fmincon in matlab with nonlinear constraints! A quick check with goggle told me I was right. fmincon Examples See Also Find the minimum of a constrained nonlinear multivariable function where x, b, beq, lb, and ub are vectors, A and Aeq are matrices, c(x) and ceq(x) are functions that return vectors, and f(x) is a function that returns a scalar. f(x), c(x), and ceq(x) can be nonlinear functions.Syntaxx = fmincon(fun,x0,A,b)x = fmincon(fun,x0,A,b,Aeq,beq)x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options,P1,P2, ...)[x,fval] = fmincon(...)[x,fval,exitflag] = fmincon(...)[x,fval,exitflag,output] = fmincon(...)[x,fval,exitflag,output,lambda] = fmincon(...)[x,fval,exitflag,output,lambda,grad] = fmincon(...)[x,fval,exitflag,output,lambda,grad,hessian] = fmincon(...)Descriptionfmincon finds the constrained minimum of a scalar function of several variables starting at an initial estimate. This is generally referred to as constrained nonlinear optimization or nonlinear programming.x = fmincon(fun,x0,A,b) starts at x0 and finds a minimum x to the function described in fun subject to the linear inequalities A*x <= b. x0 can be a scalar, vector, or matrix.x = fmincon(fun,x0,A,b,Aeq,beq) minimizes fun subject to the linear equalities Aeq*x = beq as well as A*x <= b. Set A=[] and b=[] if no inequalities exist.x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables, x, so that the solution is always in the range lb <= x <= ub. Set Aeq=[] and beq=[] if no equalities exist.x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon) subjects the minimization to the nonlinear inequalities c(x) or equalities ceq(x) defined in nonlcon. fmincon optimizes such that c(x) <= 0 and ceq(x) = 0. Set lb=[] and/or ub=[] if no bounds exist.x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options) minimizes with the optimization parameters specified in the structure options.x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options,P1,P2,...) passes the problem-dependent parameters P1, P2, etc., directly to the functions fun and nonlcon. Pass empty matrices as placeholders for A, b, Aeq, beq, lb, ub, nonlcon, and options if these arguments are not needed.[x,fval] = fmincon(...) returns the value of the objective function fun at the solution x.[x,fval,exitflag] = fmincon(...) returns a value exitflag that describes the exit condition of fmincon.[x,fval,exitflag,output] = fmincon(...) returns a structure output with information about the optimization.[x,fval,exitflag,output,lambda] = fmincon(...) returns a structure lambda whose fields contain the Lagrange multipliers at the solution x.[x,fval,exitflag,output,lambda,grad] = fmincon(...) returns the value of the gradient of fun at the solution x.[x,fval,exitflag,output,lambda,grad,hessian] = fmincon(...) returns the value of the Hessian of fun at the solution x.ArgumentsThe arguments passed into the function are described in Table 1-1. The arguments returned by the function are described in Table 1-2. Details relevant to fmincon are included below for fun, nonlcon, options, exitflag, lambda, and output. fun The function to be minimized. fun takes a vector x and returns a scalar value f of the objective function evaluated at x. You can specify fun to be an inline object. For example,fun = inline('sin(x''*x)');Alternatively, fun can be a string containing the name of a function (an M-file, a built-in function, or a MEX-file). If fun='myfun' then the M-file function myfun.m would have the formfunction f = myfun(x)f = ... % Compute function value at x If the gradient of fun can also be computed and options.GradObj is 'on', as set byoptions = optimset('GradObj','on')then the function fun must return, in the second output argument, the gradient value g, a vector, at x. Note that by checking the value of nargout the function can avoid computing g when fun is called with only one output argument (in the case where the optimization algorithm only needs the value of f but not g):function [f,g] = myfun(x)f = ... % compute the function value at xif nargout > 1 % fun called with two output arguments g = ... % compute the gradient evaluated at xend The gradient is the partial derivatives of f at the point x. That is, the ith component of g is the partial derivative of f with respect to the ith component of x. If the Hessian matrix can also be computed and options.Hessian is 'on', i.e., options = optimset('Hessian','on'), then the function fun must return the Hessian value H, a symmetric matrix, at x in a third output argument. Note that by checking the value of nargout we can avoid computing H when fun is called with only one or two output arguments (in the case where the optimization algorithm only needs the values of f and g but not H):function [f,g,H] = myfun(x)f = ... % Compute the objective function value at xif nargout > 1 % fun called with two output arguments g = ... % gradient of the function evaluated at x if nargout > 2 H = ... % Hessian evaluated at xendThe Hessian matrix is the second partial derivatives matrix of f at the point x. That is, the (ith,jth) component of H is the second partial derivative of f with respect to xi and xj, . The Hessian is by definition a symmetric matrix. nonlcon The function that computes the nonlinear inequality constraints c(x)<=0 and nonlinear equality constraints ceq(x)=0. nonlcon is a string containing the name of a function (an M-file, a built-in, or a MEX-file). nonlcon takes a vector x and returns two arguments, a vector c of the nonlinear inequalities evaluated at x and a vector ceq of the nonlinear equalities evaluated at x. For example, if nonlcon='mycon' then the M-file mycon.m would have the formfunction [c,ceq] = mycon(x)c = ... % Compute nonlinear inequalities at xceq = ... % Compute the nonlinear equalities at x If the gradients of the constraints can also be computed and options.GradConstr is 'on', as set byoptions = optimset('GradConstr','on')then the function nonlcon must also return, in the third and fourth output arguments, GC, the gradient of c(x), and GCeq, the gradient of ceq(x). Note that by checking the value of nargout the function can avoid computing GC and GCeq when nonlcon is called with only two output arguments (in the case where the optimization algorithm only needs the values of c and ceq but not GC and GCeq): function [c,ceq,GC,GCeq] = mycon(x)c = ... % nonlinear inequalities at xceq = ... % nonlinear equalities at xif nargout > 2 % nonlcon called with 4 outputs GC = ... % gradients of the inequalities GCeq = ... % gradients of the equalitiesend If nonlcon returns a vector c of m components and x has length n, then the gradient GC of c(x) is an n-by-m matrix, where GC(i,j) is the partial derivative of c(j) with respect to x(i) (i.e., the jth column of GC is the gradient of the jth inequality constraint c(j)). Likewise, if ceq has p components, the gradient GCeq of ceq(x) is an n-by-p matrix, where GCeq(i,j) is the partial derivative of ceq(j) with respect to x(i) (i.e., the jth column of GCeq is the gradient of the jth equality constraint ceq(j)). options Optimization parameter options. You can set or change the values of these parameters using the optimset function. Some parameters apply to all algorithms, some are only relevant when using the large-scale algorithm, and others are only relevant when using the medium-scale algorithm. We start by describing the LargeScale option since it states a preference for which algorithm to use. It is only a preference since certain conditions must be met to use the large-scale algorithm. For fmincon, the gradient must be provided (see the description of fun above to see how) or else the medium-scale algorithm will be used.LargeScale - Use large-scale algorithm if possible when set to 'on'. Use medium-scale algorithm when set to 'off'. Parameters used by both the large-scale and medium-scale algorithms:Diagnostics - Print diagnostic information about the function to be minimized. Display - Level of display. 'off' displays no output; 'iter' displays output at each iteration; 'final' displays just the final output. GradObj - Gradient for the objective function defined by user. See the description of fun under the Arguments section above to see how to define the gradient in fun. The gradient must be provided to use the large-scale method. It is optional for the medium-scale method. MaxFunEvals - Maximum number of function evaluations allowed. MaxIter - Maximum number of iterations allowed. TolFun - Termination tolerance on the function value. TolCon - Termination tolerance on the constraint violation. TolX - Termination tolerance on x. Parameters used by the large-scale algorithm only:Hessian - Hessian for the objective function defined by user. See the description of fun under the Arguments section above to see how to define the Hessian in fun. HessPattern - Sparsity pattern of the Hessian for finite-differencing. If it is not convenient to compute the sparse Hessian matrix H in fun, the large-scale method in fmincon can approximate H via sparse finite-differences (of the gradient) provided the sparsity structure of H -- i.e., locations of the nonzeros -- is supplied as the value for HessPattern. In the worst case, if the structure is unknown, you can set HessPattern to be a dense matrix and a full finite-difference approximation will be computed at each iteration (this is the default). This can be very expensive for large problems so it is usually worth the effort to determine the sparsity structure. MaxPCGIter - Maximum number of PCG (preconditioned conjugate gradient) iterations (see the Algorithm section below). PrecondBandWidth - Upper bandwidth of preconditioner for PCG. By default, diagonal preconditioning is used (upper bandwidth of 0). For some problems, increasing the bandwidth reduces the number of PCG iterations. TolPCG - Termination tolerance on the PCG iteration. TypicalX - Typical x values. Parameters used by the medium-scale algorithm only:DerivativeCheck - Compare user-supplied derivatives (gradients of the objective and constraints) to finite-differencing derivatives. DiffMaxChange - Maximum change in variables for finite-difference gradients. DiffMinChange - Minimum change in variables for finite-difference gradients. LineSearchType - Line search algorithm choice. exitflag Describes the exit condition:> 0 indicates that the function converged to a solution x. 0 indicates that the maximum number of function evaluations or iterations was reached. < 0 indicates that the function did not converge to a solution. lambda A structure containing the Lagrange multipliers at the solution x (separated by constraint type):lambda.lower for the lower bounds lb. lambda.upper for the upper bounds ub. lambda.ineqlin for the linear inequalities. lambda.eqlin for the linear equalities. lambda.ineqnonlin for the nonlinear inequalities. lambda.eqnonlin for the nonlinear equalities. output A structure whose fields contain information about the optimization:output.iterations - The number of iterations taken. output.funcCount - The number of function evaluations. output.algorithm - The algorithm used. output.cgiterations - The number of PCG iterations (large-scale algorithm only). output.stepsize - The final step size taken (medium-scale algorithm only). output.firstorderopt - A measure of first-order optimality (large-scale algorithm only). ExamplesFind values of x that minimize , starting at the point x = [10; 10; 10] and subject to the constraints First, write an M-file that returns a scalar value f of the function evaluated at x:function f = myfun(x)f = -x(1) * x(2) * x(3);Then rewrite the constraints as both less than or equal to a constant, Since both constraints are linear, formulate them as the matrix inequality where Next, supply a starting point and invoke an optimization routine:x0 = [10; 10; 10]; % Starting guess at the solution[x,fval] = fmincon('myfun',x0,A,b)After 66 function evaluations, the solution isx = 24.0000 12.0000 12.0000where the function value is fval = -3.4560e+03and linear inequality constraints evaluate to be <= 0A*x-b= -72 0NotesLarge-scale optimization. To use the large-scale method, the gradient must be provided in fun (and options.GradObj set to 'on'). A warning is given if no gradient is provided and options.LargeScale is not 'off'. fmincon permits g(x) to be an approximate gradient but this option is not recommended: the numerical behavior of most optimization codes is considerably more robust when the true gradient is used. The large-scale method in fmincon is most effective when the matrix of second derivatives, i.e., the Hessian matrix H(x), is also computed. However, evaluation of the true Hessian matrix is not required. For example, if you can supply the Hessian sparsity structure (using the HessPattern parameter in options), then fmincon will compute a sparse finite-difference approximation to H(x). If x0 is not strictly feasible, fmincon chooses a new strictly feasible (centered) starting point.If components of x have no upper (or lower) bounds, then fmincon prefers that the corresponding components of ub (or lb) be set to Inf (or -Inf for lb) as opposed to an arbitrary but very large positive (or negative in the case of lower bounds) number.Several aspects of linearly constrained minimization should be noted:A dense (or fairly dense) column of matrix Aeq can result in considerable fill and computational cost. fmincon removes (numerically) linearly dependent rows in Aeq; however, this process involves repeated matrix factorizations and therefore can be costly if there are many dependencies. Each iteration involves a sparse least-squares solve with matrix where RT is the Cholesky factor of the preconditioner. Therefore, there is a potential conflict between choosing an effective preconditioner and minimizing fill in .Medium-scale optimization. Better numerical results are likely if you specify equalities explicitly using Aeq and beq, instead of implicitly using lb and ub.If equality constraints are present and dependent equalities are detected and removed in the quadratic subproblem, 'dependent' is printed under the Procedures heading (when output is asked for using options.Display = 'iter'). The dependent equalities are only removed when the equalities are consistent. If the system of equalities is not consistent, the subproblem is infeasible and 'infeasible' is printed under the Procedures heading.AlgorithmLarge-scale optimization. By default fmincon will choose the large-scale algorithm if the user supplies the gradient in fun (and GradObj is 'on' in options) and if only upper and lower bounds exists or only linear equality constraints exist. This algorithm is a subspace trust region method and is based on the interior-reflective Newton method described in [5],[6]. Each iteration involves the approximate solution of a large linear system using the method of preconditioned conjugate gradients (PCG). See the trust-region and preconditioned conjugate gradient method descriptions in the Large-Scale Algorithms chapter.Medium-scale optimization. fmincon uses a Sequential Quadratic Programming (SQP) method. In this method, a Quadratic Programming (QP) subproblem is solved at each iteration. An estimate of the Hessian of the Lagrangian is updated at each iteration using the BFGS formula (see fminunc, references [3, 6]).A line search is performed using a merit function similar to that proposed by [1] and [2, 3]. The QP subproblem is solved using an active set strategy similar to that described in [4]. A full description of this algorithm is found in the "Constrained Optimization" section of the Introduction to Algorithms chapter of the toolbox manual.See also the SQP implementation section in the Introduction to Algorithms chapter for more details on the algorithm used.DiagnosticsLarge-scale optimization. The large-scale code will not allow equal upper and lower bounds. For example if lb(2)==ub(2), then fmincon gives the error:Equal upper and lower bounds not permitted in this large-scale method.Use equality constraints and the medium-scale method instead.If you only have equality constraints you can still use the large-scale method. But if you have both equalities and bounds, you must use the medium-scale method.LimitationsThe function to be minimized and the constraints must both be continuous. fmincon may only give local solutions.When the problem is infeasible, fmincon attempts to minimize the maximum constraint value.The objective function and constraint function must be real-valued, that is they cannot return complex values.Large-scale optimization. To use the large-scale algorithm, the user must supply the gradient in fun (and GradObj must be set 'on' in options) , and only upper and lower bounds constraints may be specified, or only linear equality constraints must exist and Aeq cannot have more rows than columns. Aeq is typically sparse. See Table 1-4 for more information on what problem formulations are covered and what information must be provided.Currently, if the analytical gradient is provided in fun, the options parameter DerivativeCheck cannot be used with the large-scale method to compare the analytic gradient to the finite-difference gradient. Instead, use the medium-scale method to check the derivative with options parameter MaxIter set to 0 iterations. Then run the problem with the large-scale method.References[1] Han, S.P., "A Globally Convergent Method for Nonlinear Programming," Journal of Optimization Theory and Applications, Vol. 22, p. 297, 1977.[2] Powell, M.J.D., "The Convergence of Variable Metric Methods For Nonlinearly Constrained Optimization Calculations," Nonlinear Programming 3, (O.L. Mangasarian, R.R. Meyer, and S.M. Robinson, eds.) Academic Press, 1978.[3] Powell, M.J.D., "A Fast Algorithm for Nonlineary Constrained Optimization Calculations," Numerical Analysis, ed. G.A. Watson, Lecture Notes in Mathematics, Springer Verlag, Vol. 630, 1978.[4] Gill, P.E., W. Murray, and M.H. Wright, Practical Optimization, Academic Press, London, 1981.[5] Coleman, T.F. and Y. Li, "On the Convergence of Reflective Newton Methods for Large-Scale Nonlinear Minimization Subject to Bounds," Mathematical Programming, Vol. 67, Number 2, pp. 189-224, 1994.[6] Coleman, T.F. and Y. Li, "An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds," SIAM Journal on Optimization, Vol. 6, pp. 418-445, 1996.See Alsofminbnd, fminsearch, fminunc, optimset

thanx i really appreciated it!

Thanx for the tip!i have the initial parameters vector x = [.00000502 .589 .00000132 421.39 .2]so i've created a new M-File that computes nonlconfunction [c ceq] = lcon(x)c=x(2)+x(3)*(x(4)^2)-1;ceq=[];and then I insert this contraint in FMINCON as follows (this is the maximization M-file)function hn_par = hnparametersclear all;close all;format long;%data loaddata = xlsread('c:\ANDREA\aaa.xls');%first guessomega = .00000502;beta = .589;alpha=.00000132;rho = 421.39;lambda = .2;par=[omega beta alpha rho lambda];%contraintsparL=[0.00000001 0.1 0.00000001 150 0.1];parU=[.00005 1 .00005 500 1];cA=[0 1 400^2 0 0];cB=[.99999];% optimization optionsoptopt = optimset('MaxIter',5,'Display','iter','LargeScale','off');[x, fval, exitflag, output, lambda, jacob, hess] = fmincon('mlf', par, cA, cB, [], [], parL, parU, 'lcon', optopt,data);par = xwhere MLF is my objective function! Without the 'lcon' constraint the routine doesn't crash and returns pretty good parameters but when I insert 'lcon' matlab returns??? Error using ==> fminconFMINCON cannot continue because user supplied nonlinear constraint function failed with the following error:Error using ==> lconToo many input arguments.Error in ==> hnparameters at 21[x, fval, exitflag, output, lambda, jacob, hess] = fmincon('mlf', par, cA, cB, [], [], parL, parU, @lcon, optopt,data);this seems strange to me because i think to have created correctly NONLCON as shown in the matlab help (tipe HELP FMINCON) from the command window! I don't get the TOO MANY INPUT ARGUMENTS thing.....Any suggestionbest regards!

I think it is the icon function that fails not minconKyriakos

Hi lisiandreaAre u brazilian too? I'm estimating a garch-like model but I'm having a hard time at matlab.Can u see if u find why the error at my confun.m code?I wanna know how for example to make gammas(i)>0 for all i>2 without putting a for inside the constraints.Thanksfunction [constraints ceq]= confun(alphas, betas, lambdas, gammas, c, data)constraints =[-gammas(i+1) +0.000000000000001; c(i+1)- Inf; -c(i+1) - Inf ; (1/(1+exp( -gammas(i+2)*(mu(t-1)+ (h(t-1))*( e(t-1)^2) -c(i+2)) )))-(1/(1+exp( -gammas(i+1)*(mu(t-1)+ (h(t-1))*( e(t-1)^2) -c(i+1)) ))); -somaalpha(i) 0.0001; -somabeta(i) 0;-somalambda(i) 0; 1/2*(betas(1)+lambdas(1))+ 1/2*(somalambda(H+1) + somabeta(H+1)) -0.9999999999999; (betas(1)) ^2 + betas(1)*betaU + 1/2*(betaU ^2) + fmoment* [ lambdas(1) + lambdas(1)* lambdaU + 1/2* (lambdaU ^2)] + 2 *lambdas(1) *betas(1) + betas(1)*lambdaU + betaU* lambdas(1) + betaU*lambdaU -0.99999999 ]ceq=[]%Constraint optimization part 2 x0 = [0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1,0.1, 0.1, 0.1, 0.1] ; %3 regimes 3 coefic + 2 gamma and 2 for c (3+5H=13)options = optimset('LargeScale', 'off');[alphas, betas, lambdas,gammas, c, fval] = fmincon('objfun', x0,[],[],[],[],[],[], 'confun',options, data)