MATLAB: Optimization using fmincon where objective function includes the lagrange multiplier

fminconlagrange multiplierMATLAB

my objective function is to maximize sum_i(log(1-lambda(yi-xi*beta))) with respect to beta subject to the constraint sum_i(yi-xi*beta)=0. My problem is lambda in the objective function is the lagrange multiplier of the constraint. How can I set the fmincon optimization then?

whenever I call the objective function there comes an error asking for values of lambda.

Best Answer

It's not clear how lambda can be "the Lagrange multiplier of the constraint" when it is a parameter of your objective function. As this parameter changes, so does the required Lagrange multiplier, and there is no certainty those 2 values would ever coincide.

Possibly, you're trying to solve the following problem in the unknowns beta and c?

min.

    f(beta,c) = -sum_i(log(1-c*(yi-xi*beta)))

s.t.

      g(beta,c) = sum_i(yi-xi*beta)=0
     grad_wrt_beta f(beta,c)+ c * grad_wrt_beta g(beta,c) =0

Anyway, this is sounding like one of those situations where it would be wise to explain why the heck you want to do this.

Related Solutions

MATLAB: What is the ordering of the Lagrange multipliers (Lambda) in the FGOALATTAIN function in the Optimization Toolbox

Ordering of Lagrange multipliers (Lambda) occurs in the same way as it is done for FMINCON. The active constraints listed at the end of the execution of FMINCON are numbers. Here is the ordering used for constraints in FMINCON:

linear equalities

nonlinear equalities

lower bounds

upper bounds

linear inequalities

nonlinear inequalities

A detailed explanation of this can be found at the Related Solution listed at the bottom of the page.

In case of FGOALATTAIN, certain constraints exist for objective functions, in terms of achieving the goal. Note the Optimization problem formulation:

    min                { LAMBDA :  F(X)-WEIGHT.*LAMBDA<=GOAL } 
 X,LAMBDA

This indicates that some nonlinear constraints are being added on the objective function that is different from other nonlinear constraints c(x).

Now, the order of Lagrange multipliers (Lambda) are :

linear equalities

nonlinear equalities

lower bounds

upper bounds

linear inequalities

nonlinear inequalities

Exact goal achieved %% NOTE the additional list

The Lambda structure is obtained as one of the outputs from FGOALATTAIN using the following syntax:

 [x,fval,attainfactor,exitflag,output,lambda] = fgoalattain(...)

returns a structure 'lambda' whose fields contain the Lagrange multipliers at the solution 'x'.

Lambda has the following field(s):

lower: LB

upper: UB

eqlin: Aeq*X = Beq

eqnonlin: Ceq(X)=0

ineqlin: A*X <= B.

ineqnonlin: C(X)<=0

NOTE: Lambda does NOT have the fields for "Exact goal achieved ".

MATLAB: Controlling accuracy of variables in ga

You should definitely not use integer constraints when you don't need them.

The real question is, do you need to use ga? Think carefully about your problem. If it is smooth with smooth constraints (such as the one you mentioned), then use an Optimization Toolbox™ solver such as fmincon for speed and robustness. If your problem is not smooth, try patternsearch for speed and robustness.

It sounds like you don't need high accuracy for your solution. It is much easier to tune patternsearch than ga; for example, for two decimals of accuracy, set MeshTolerance to 3e-3 or something similar.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

Best Answer

Related Solutions

MATLAB: What is the ordering of the Lagrange multipliers (Lambda) in the FGOALATTAIN function in the Optimization Toolbox

MATLAB: Controlling accuracy of variables in ga

Related Question