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Deregulated electricity markets in the U.S. currently use an auction mechanism that minimizes total supply bid costs to select bids and their levels. Payments are then settled based on market-clearing prices. Under this setup, the consumer payments could be significantly higher than the minimized bid costs obtained from auctions. This gives rise to ldquopayment cost minimization,rdquo an alternative auction mechanism that minimizes consumer payments. We previously presented an augmented Lagrangian and surrogate optimization framework to solve payment cost minimization problems without considering transmission. This paper extends that approach to incorporate transmission capacity constraints. The consideration of transmission constraints complicates the problem by entailing power flow and introducing locational marginal prices (LMPs). DC power flow is used for simplicity and LMPs are defined by ldquoeconomic dispatchrdquo for the selected supply bids. To characterize LMPs that appear in the payment cost objective function, Karush-Kuhn-Tucker (KKT) conditions of economic dispatch are established and embedded as constraints. The reformulated problem is difficult in view of the complex role of LMPs and the violation of constraint qualifications caused by the complementarity constraints of KKT conditions. Our key idea is to extend the surrogate optimization framework and use a regularization technique. Specific methods to satisfy the ldquosurrogate optimization conditionrdquo in the presence of transmission capacity constraints are highlighted. Numerical testing results of small examples and the IEEE Reliability Test System with randomly generated supply bids demonstrate the quality, effectiveness, and scalability of the method.