AnO(n ³ logn) strong polynomial algorithm for an isotonic regression knapsack problem

Abstract

We introduce the isotonic regression knapsack problem

$$\begin{gathered} \min (1/2)\sum\limits_{i = 1}^n {\{ d_i x_i^2 - 2\alpha _i x_i \} } , \hfill \\ s.t. \sum\limits_{i = 1}^n {a_i x_i } = c, x_1 \leqslant x_2 \leqslant \cdots \leqslant x_{n - 1} \leqslant x_n , \hfill \\ \end{gathered} $$

where eachd _i is positive and each α _i ,a _i ,i=1, ...,n, andc are arbitrary scalars. This problem is the natural extension of the isotonic regression problem which permits a strong polynomial solution algorithm. In this paper, an approach is developed from the Karush-Kuhn-Tucker conditions. By considering the Lagrange function without the inequalities, we establish a way to find the proper Lagrange multiplier associated with the equation using a parametric program, which yields optimality. We show that such a procedure can be performed in strong polynomial time, and an example is demonstrated.