  The problem of obtaining market-clearing prices for markets with
non-convexities has been widely studied in the literature. This is particularly
the case in electricity markets, where worldwide deregulation leads to markets
in which non-convexities arise from the decisions of market operators regarding
which generators are committed to provide electricity power. Here, we extend
seminal results in this area to address the problem of obtaining
market-clearing prices for markets in which beyond non-convexities, it is
relevant to account for convex quadratic market costs. In a general market,
such costs arise from quadratic commodity costs or transactions costs. In an
electricity market, such quadratic costs arise when ramping costs need to be
considered due to the presence of renewable energy sources, which continue to
increase their participation in electricity markets. To illustrate our results,
we compute and analyze the clearing prices of a classical market problem with
the addition of ramping costs.
