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L. 5) the 1 compression region for the ith sensor. 5) deﬁne an injective and surjective relationship between a pair of local regions { i , ci } and a pair of simple polynomials {Ii , 1 − Ii }. Furthermore, a local message (I1 (y1 ) = d1 , I2 (y2 ) = d2 , . . 6) of the above simple polynomials, where for any j ≤ L, Pj (yj ) = 1 − Ij (yj ) if dj = 0, Ij (yj ) if dj = 1. 6) the local message polynomial. Obviously, {(y1 , y2 , . . , yL ) : (I1 (y1 ) = d1 , I2 (y2 ) = d2 , . . , IL (yL ) = dL )} = {(y1 , y2 , .

50), which is a very useful tool for integer, mixed integer, large-scale, and nonlinear programming. 67) Introduction ■ 23 for λ ∈ Rm , ν ∈ Rp . 50): For any λ 0 and any ν, we have g(λ, ν) ≤ p∗ . t. λj ≥ 0, j = 1, . . , m. 50) is sometimes called the primal problem. 68). 68) is a convex optimization problem, since the objective to be maximized is concave and the constraint is convex. 50) is convex. , d∗ ≤ p∗ , which holds even if the original problem is not convex. This property is called weak duality.

YL )d y1 d y3 · · · dyL , ··· IL (yL ) = I PL1 L(y1 , y2 , . . 14) Parallel Statistical Binary Decision Fusion ■ 35 where I [·] is deﬁned by I [x] = 1, 0, if x ≥ 0, if x < 0. 13), we know that the optimal local compression rules (I1 , I2 , . . 10, which can be written in the following diﬀerent equivalent forms: [1 − PH1 (I1 (y1 ), . . , IL (yL ))]L(y1 , . . , yL )d y1 · · · dyL = [(1 − I1 (y1 ))P11 + P12 ]L(y1 , . . , yL )d y1 · · · dyL = P11 L(y1 , . . , yL )d y2 d y3 · · · dyL d y1 + CI1 I1 (y1 )=0 = [(1 − I2 (y2 ))P21 + P22 ]L(y1 , .