Problem $3.(25\%)$ Consider $n$ assets, each labeled by $i=1,$dots,$n,$ and the following specification for a portfolio

There is a total budget of $$B$ that must be used up $[$note $B>0$ is a known constant$].$

Each individual asset is divisible, and may be invested in an amount taken from the range $-b,b($a negative$/$positive amount means that the asset is sold$/$bought$)[$note $b>0$ a known constant$].$

Given a portfolio $(x_1,$dots,$x_n)$ where $x_i$ is the amount $($in $ $)$ invested in asset $i,$ the expected return and the risk are given respectively by:

expected return $={\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i(1\frac{+}{b}ar(R{)}_i),$ risk $=\sqrt[\phantom{2}]{{\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i^2{}_{ii}}$

where ${}_{ii}\frac{,}{b}ar(R{)}_i$ are known constants.

$($a$)(10\%)$ Based on the above data, formulate a portfolio optimization problem in order to maximize the expected return while keeping the risk at below $$.$ In the problem formulated, please indicate what are the decision variable$($s$)?$ what is the objective function? what is the constraint?

$($b$)(15\%)$ There is an additional requirement as follows:

For asset $i,$ the bank charges you with a transaction fee of $$c_i$ per dollar of asset i sold$/$bought$.$ Note that if $$1$ of asset $i$ is bought or sold, then the transaction cost would be $$c_i$ for this asset.

Under the new setting, formulate a portfolio optimization problem in order to maximize the expected net return while keeping the risk$/$standard deviation at below $$.$

The expected net return is given by $($expected return$)-($total transaction cost$).$