Question $1.(50$ points$)$ In an electrical network, the power loss that happens when an electrical current of $$ amperes flows through a resistance of $$ ohms is $2$ watts. In the figure below, $710$ amperes of current must be sent from node $1$ to node $4.$ The current flowing through each node must satisfy conservation of flow. For example, the flow entering the network $($node $1),$ which is $710$ amperes, should equal the flow exiting node $1,$ that is$,$ the flow through the $1-$ohm resistor $+$ the flow through the $4-$ohm resistor $($note that there are five total resistors in the network$).$ Amazingly, the physical laws of conservation determine the flow through the network that minimizes the power loss. $1.(30$ points$)$ Formulate a quadratic program $($QP$)$ that, when solved, will yield the current flowing through each resistor. $2.(20$ points$)$ Write out a $.$lp file, and use Gurobi to solve the QP you formulated. What is the optimal flow across the network? What is the optimized power loss? Question $1.(50$ points$)$ In an electrical network, the power loss that happens when an electrical current of I amperes flows through a resistance of $\backslash ($ R $\backslash )$ ohms is $\backslash ($ I$^\{2\}$ R $\backslash )$ watts. In the figure below, $710$ amperes of current must be sent from node $1$ to node $4.$ The current flowing through each node must satisfy conservation of flow. For example, the flow entering the network $($node $1),$ which is $710$ amperes, should equal the flow exiting node $1,$ that is$,$ the flow through the $1-$ohm resistor $+$ the flow through the $4-$ohm resistor $($note that there are five total resistors in the network$).$ Amazingly, the physical laws of conservation determine the flow through the network that minimizes the power loss.

$1.(30$ points$)$ Formulate a quadratic program $\backslash (($Q P$)\backslash )$ that, when solved, will yield the current flowing through each resistor.

$2.(20$ points$)$ Write out a $.$lp file, and use Gurobi to solve the QP you formulated. What is the optimal flow across the network? What is the optimized power loss?