Table 2. Solving (21) is equivalent to solving (22). For notational simplicity, we define new sets: Table 2 An Example of How the Limit Order Logic Constraint Can be Simplified ■ X ( ℓ$_{1}$, ℓ$_{2}$, ..., ℓ$_{m}$ ) (23) =     $^{x}$ ∈ R $^{J}$| x$_{j}$ = 0 if max ξ ∈ E ( ℓ$_{1}$,ℓ$_{2}$,...,ℓ$_{K}$ ) ( A T ξ ) j > B$_{j}$b$_{j}$ and x$_{j}$ ∈ [0 , Q$_{j}$ ] if min ξ ∈ E ( ℓ$_{1}$,ℓ$_{2}$,...,ℓ$_{K}$ ) A T ξ j = max ξ ∈ E ( ℓ$_{1}$,ℓ$_{2}$,...,ℓ$_{K}$ ) ( A T ξ ) j = B$_{j}$b$_{j}$ x$_{j}$ = Q$_{j}$ if min ξ ∈ E ( ℓ$_{1}$,ℓ$_{2}$,...,ℓ$_{K}$ ) ( A T ξ ) j < B$_{j}$b$_{j}$ , ∀ $_{j}$     20 min ξ , x ,µ, ζ , ω ℓ ( µ ) = µ ln ( ∑ 5 i =1 θ$_{i}$e − d i µ ) such that (A) − d$_{i}$ ≥ − z$_{i}$e ζ$_{i}$ for ∀ i (B) ζ$_{i}$ ≥ α ∑ 7 j =1 [ x$_{j}$ A$_{ij}$ − Q$_{j}$ ( A T ξ ) j − B$_{j}$b$_{j}$ ( x$_{j}$ − Q$_{j}$ ) ] , ∀ i (C) ζ$_{i}$ ≥ α 7 j $_{=1}$[ x$_{j}$ A$_{ij}$ − B$_{j}$b$_{j}$x$_{j}$ ] , ∀ i (D) µ ≥ 0 (E) ξ ∈ E (1 , 2 , 2 , 2 , 1) (F) x$_{1}$ = x$_{5}$ = x$_{6}$ = x$_{7}$ = 0 . 001 , 0 ≤ x$_{2}$, x$_{3}$, x$_{4}$ ≤ 0 . 001 (22) order limit security bid market relevant restriction # quantity # price clearing restriction on the b price in E (1 , 2 , 2 , 2 , 1) order fill x$_{j}$ 1 0.001 1 0.18 ξ$_{1}$ ξ$_{1}$ ≤ 0 . 18 x$_{1}$ = 0 . 001 2 0.001 2 0.18 ξ$_{2}$ ξ$_{2}$ = 0 . 18 0 ≤ x$_{2}$ ≤ 0 . 001 3 0.001 3 0.18 ξ$_{3}$ ξ$_{3}$ = 0 . 18 0 ≤ x$_{3}$ ≤ 0 . 001 4 0.001 4 0.18 ξ$_{4}$ ξ$_{4}$ = 0 . 18 0 ≤ x$_{4}$ ≤ 0 . 001 5 0.002 1 0.20 ξ$_{1}$ ξ$_{1}$ ≤ 0 . 18 x$_{5}$ = 0 . 001 6 0.001 1 0.25 ξ$_{1}$ ξ$_{1}$ ≤ 0 . 18 x$_{6}$ = 0 . 001 7 0.001 1 0.20 ξ$_{1}$ ξ$_{1}$ ≤ 0 . 18 x$_{7}$ = 0 . 001