The problem is like m a x x u ( x 1 , x 2 , . . . , x L ) = − ∑ i = 1 L ∣ x i − a i ∣, s . t . ∑ i L x i ≤ Cfor each i, a i &gt; 0 is a scalar; C is a constant that is strictly greater than 0; x = ( x 1 , x 2 , . . . , x L ) ∈ R + L . Characterize the optimal x as a function of C or a i .Hint: to solve the problem we should discuss the cases when C ≤ ∑ i a i and C ≥ ∑ i a i .Thank you!

Question

The problem is like  m  a      x                  x                u  (      x    1    ,      x    2    ,  .  .  .  ,      x    L    )  =  −      ∑          i      =      1              L        ∣      x    i    −      a    i    ∣,  s  .  t  .      ∑    i    L        x    i    ≤  Cfor each   i,       a    i    &amp;gt;  0 is a scalar;  C is a constant that is strictly greater than 0;      x    =  (      x    1    ,      x    2    ,  .  .  .  ,      x    L    )  ∈                    R                    +              L      . Characterize the optimal       x   as a function of   C or       a    i  .Hint: to solve the problem we should discuss the cases when   C  ≤      ∑    i        a    i   and   C  ≥      ∑    i        a    i  .Thank you!

pulpasqsltl · Accepted Answer

The solution to your problem is easy in the case   C  ≥      ∑    i        a    i  . Then you can choose       x    i    =      a    i   and you are done with the maximiation since   0  ≥  u  (      x    )  =  0.For the case   C  &amp;lt;      ∑    i        a    i   the choice       x    i    &amp;gt;      a    i   is not feasible. In this case it does not make sence to choose xi&amp;gt;ai and we can reformulate   u  (      x    )  =  −  ∑  (      a    i    −      x    i    ). From the latter formulation it follows that the goal is to choose   ∑      x    i   as large as possible to maximize   u  (      x    ). As   ∑      x    i    ≤  C we have   u  (      x    )  =  C  −  ∑      a    i  .

The problem is like m a x <mrow class="MJX-TeXAtom-ORD"> <mrow class="M

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