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Dual algorithm for L1 isotonic optimization on a partially ordered set

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Let X be a partially ordered finite set with a partial order <<. A real-valued function g on X is called isotone with respect to << if g(x) ≤ g(y) whenever x, y ϵ X and X << y. Let G(X) be the set of all isotone functions on x. The problem being considered here is:(UNFORMATTED TABLE OR EQUATION FOLLOWS)MinimizeΣ[subscript]sp xϵ X | f(x) - g(x) |subject to gϵ G(X).≤qno (D):(TABLE/EQUATION ENDS);This isotone optimization problems originates mainly from order restricted statistical inference. The L[subscript]2 version of this problem has been fully discussed. In considering this L[subscript]1 problem, we try to take advantage of duality. For any x in X, let U(x) and L(x) denote the immediate successors of x and the immediate predecessors of x respectively. Define the set L by L = (x,y) |x ϵ X, y ϵ U(x) . Let w and F be functions on X and L respectively. The dual problem proposed here is:(UNFORMATTED TABLE OR EQUATION FOLLOWS)\line (P)\:Maximize Σ[subscript]sp xϵ X f(x)w(x) subject to\vskip4pt\vbox\halign#\hfil&&#\hfil (P-1):&w(x)\0,+1,-1,&x ϵ X (P-2):&F(x,y) ≥ 0,&(x,y) ϵ L (P-3):&w(x) = Σ[subscript]sp yϵ U(x) F(x,y) - &Σ[subscript]sp zϵ L(x) F(z,x), xϵ X. (TABLE/EQUATION ENDS)A finite algorithm which utilizes network flows is constructed that solves both the primal and the dual simultaneously. This algorithm is also used to prove the duality theorems. The optimal criteria obtained are: (UNFORMATTED TABLE OR EQUATION FOLLOWS)\eqalignCondition A: &For each x in X, w(x) = sgn [f(x) - g(x)] &whenever f(x) ≠ g(x) Condition B: &Σ[subscript]sp xϵ X w(x)g(x) = 0. (TABLE/EQUATION ENDS)