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Math Help - Tricky constrained maximisation problem

  1. #1
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    Tricky constrained maximisation problem

    the problem is

    Let c∈R, f:R^n →R and h:R^n →R. Suppose that X⊆R^n.

    Consider the following two constrained optimisation problems:

    I) Find x∈R^n to maximise f(x) subject to the constraint h(x)=c.

    II) Find x∈X to maximise f(x) subject to the constraint h(x)=c.


    a) Prove that if x* solves I and x*∈X, then x* solves II.

    b) Suppose X=R^n(+). Provide a counter example to the following (false) claim: "if x* solves II and x(i)*>0 for each i∈{1,2,...,n}, then x* solves I.


    thanks in advance
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  2. #2
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    any ideas?
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  3. #3
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    a) Suppose x^* does solves I but does not solve II. Then there exists \hat{x} \in X with h(\hat{x}) = c and f(\hat{x}) > f(x^*). Since \hat{x} is feasible to I, x^* does not solve I which is a contradiction.

    b) Think of a single variable function which has a peak at some positive value of x but has a larger peak for some negative value of x. Like this guy -x*(0.5x-1)*(x-2)*(x+1 - Wolfram|Alpha)
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