Keep or pay to discard cards for an average payout

A shuffled deck contains $100$ cards valued $1$ to $100$. Cards are turned over one at a time, and for each one you choose to keep it or throw it away, paying $1 for every card you throw away. At the end you receive the average value of the cards you kept. What strategy is optimal, and what is the expected profit?

Keep every card; expected profit $50.50. Any discarding is best done on the lowest cards, and throwing away all cards below $k$ yields an average of $\tfrac{100+k}{2}$ at a cost of $k-1$, i.e. a profit of $51-\tfrac k2$, which falls as $k$ grows. So $k=1$ is optimal, giving $\tfrac{101}{2}$.