solve a convex optimization problem (10) with C replaced with each C$_{m}$ , m ∈ { 1 ,...,M } . Let L$_{m}$ denote the optimal value of the objective function from solving the convex optimization problem (10) with C = C$_{m}$ . Let ( x$_{m}$ , ξ$_{m}$ ) denote the optimal solutions to those problems. Third, we find m ∗ = arg max m L$_{m}$ . ( x$_{m}$ ∗ , ξ$_{m}$ ∗ ) becomes the global optimal solution to the main optimization problem (9). By Lemma 1, ( x$_{m}$ ∗ , ξ$_{m}$ ∗ ) is the global optimal solution to (8). Suppose that a total of K types of contingent claims are traded in the market. Let the vector P$_{k}$ ∈ R N denote the payoff structure of the k th security (1 ≤ k ≤ K ). For example, if the i th outcome is realized, the person holding the claim receives P$_{k,i}$ per share from the market maker. Because there are J outstanding orders in the limit order book, there are J limit prices. Let n$_{k}$ denote the number of distinct limit prices associated with the k th security. If there are multiple orders with the same limit price and the same security, only one is counted toward n$_{k}$ . Sort those bid prices in ascending order. Let B l k denote the l th smallest limit price associated with the k th security. Example 1 For the sake of simplicity, consider a market in which only Arrow-Debreu securities are traded. Suppose that N = 5 . Suppose that there are K = 5 different Arrow-Debreu securities, one for each state of the world. The k th Arrow-Debreu security pays $ 1 per share to its holder if and only if the k th state is realized. Five row vectors in (11) show the payoff structures of Arrow-Debreu securities. For example, the nonzero entry in the first element of P$_{1}$ implies that the first security pays $ 1 per share if the first outcome is realized. Table 1 illustrates seven orders ( J = 7 ) outstanding in the limit order book. For example, the person who submitted the first order wants to buy the first Arrow-Debreu security. The entry 0.18 P$_{1}$ = [ 1 0 0 0 0 ] P$_{2}$ = 0 1 0 0 0 P$_{3}$ = [ 0 0 1 0 0 ] P$_{4}$ = 0 0 0 1 0 P$_{5}$ = [ 0 0 0 0 1 ] (11) 15 4.3.1 Partitioning of the Feasible Set