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Let us assume that we have the following (non linear) program :

R = min { ( (c1^T)*x + d1 ) / ( (c2^T)x + d2 ) : Ax<=b , (c2^T)*x+d2>0}

We also assume that the feasible solution area is bounded and that the objective value of the optimal solution belongs to [L,U].

We assume that we know a δ >0 , so that for every feasible solution x , ( c2^T)*x >= δ .

How can I prove that for every ε > 0 there is an ( 1 + ε) approximation of the solution of R? ( We can use a linear programming algorithm as subroutine). In addition , which is the time complexity of that algorithm?

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