[Ipopt] A question about the Lagrange multipliers of Ipopt solution

[马平川]

Dear sir or madam,
        Sorry for bothering you with this e-mail. But I actually encounter
a big problem with my Master issue code.
        My current problem lies in the coding based on Ipopt Matlab
interface. I hava a question about the lagrange multipliers of Ipopt
solution.
        We all know that the general nonlinear programming problem form of
Ipopt is the minimal optimization with the nonlinear constraints and
variables bound constraints which both have the lower and upper bounds. And
the value of the Lagrange multipliers at the solution lies in the
info.lambda, info.zl, info.zu fields. But it seems that the value of
info.lambda isn't right.
        My ipopt version is 3.10.0. I run the original matlab example *
examplehs071.m* which lies in the Ipopt source package. This optimization
problem has 4 variables and 2 nonlinear constraints which includes 1
inequality constraint with only lower bounds and 1 equality constraint. So
the number of the multipliers associated with nonlinear constraints is 2.
Then I have a modification on the example by changing the upper bound of
inequality constraint from inf to 100 without any other changes. This
mdification doesn't change the optimal solution. I thought this
modification would make the length of the multipliers corresponding to
nonlinear constraints become 3. But I was wrong. The length of info.lambda
was still 2 and its value didn't change at all.
        I have read the Ipopt original artical "*A, Wachter and L. T.
Biegler. On the implementation of an interior-point filter line-search
algorithm for large-scale nonlinear programming. Mathematical Programming,
106(1):25-57, 2006*". But in the artical the optimization form only has
equality constraints which have the same lower and upper bounds. It also
said in that artical, "problems with general nonlinear inequality
constraints, 'd(x)<=0', can be reformulated in the above form by
introducing slack variables." But I think that even so, 1 inequality
constraint with both lower and upper bounds will generate 2 lagrange
multiplier accordding to KKT condition, just like the situation of variable
bound constraints.
        So please give me the answer if you know it. And I have another
question about the value of the multiplier of that original example *
examplehs071.m*. The multiplier corresponding to the inequality constraint
with only lower bounds is negative, acctually -0.5523. But according to the
usual custom, we will make this multiplier positive by the proper
definition of Lagrangian function. So please tell the definition of
Lagrangian function of ipopt so that I could judge the sign of Lagrange
multipliers.
        Thank you for your reading. I will really appreciate you for your
answer.
        Hoping your reply.
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Regards
Neo Ma 马平川 | Graduate

T:+86-10-6278 2545 | E:chambertinofn at gmail.com
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Dept. of Electrical Engineering, Tsinghua Univ.
BLDG. 28# RM.312 Tsinghua Univ. Beijing 100084 P.R.CHINA
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Remember what should be remembered, and forget what should be forgotten.
Alter what is changeable, and accept what is immutable.
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