Integer Isotone Optimization

Consider the following integer isotone optimization problem. Given an $n$-vector $x$ find an $n$-vector $y$ with integer components so as to minimize $\max\{w_j|x_j - y_j| : 1 \leq j \leq n\}$ subject to $y_1 \leq y_2 \leq \cdots \leq y_n$, where each weight $w_j > 0$. In this article, the dual of this problem is defined, a strong duality theorem is established, and the set of all optimal solutions is shown to be all monotonic integer vectors lying in a vector interval. In addition, algorithms are obtained for computation of optimal solutions having the worst-case time complexity $O(n^2)$, when $w_j$ are arbitrary, and $O(n)$, when $w_j = 1$ for all $j$. The problem considered is of isotonic regression type and has practical applications, for example, to estimation and curve fitting. It is also of independent mathematical interest. The problem and the results can be easily extended to a partially ordered set.

[1]  Vasant A. Ubhaya,et al.  Duality and Lipschitzian selections in best approximation from nonconvex cones , 1991 .

[2]  L. Rogge,et al.  Natural choice of L1-approximants , 1981 .

[3]  J. B. Kruskal,et al.  Least-Squares Fitting by Monotonic Functions Having Integer Values , 1976 .

[4]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[5]  Michael J. Best,et al.  Active set algorithms for isotonic regression; A unifying framework , 1990, Math. Program..

[6]  F. T. Wright,et al.  Order restricted statistical inference , 1988 .

[7]  V. Ubhaya,et al.  Duality in approximation and conjugate cones in normed linear spaces , 1975 .

[8]  Nilotpal Chakravarti,et al.  Isotonic Median Regression: A Linear Programming Approach , 1989, Math. Oper. Res..

[9]  Ulf Strömberg,et al.  An algorithm for isotonic regression with arbitrary convex distance function , 1991 .

[10]  Jacob Ponstein,et al.  Approaches to the Theory of Optimization , 1980 .

[11]  F. Deutsch,et al.  Applications of the Hahn-Banach Theorem in Approximation Theory , 1967 .

[12]  D. Landers,et al.  Best approximants in LΦ-spaces , 1980 .

[13]  A. Money,et al.  Nonlinear Lp-Norm Estimation , 2020 .

[14]  Tim Robertson,et al.  Algorithms in Order Restricted Statistical Inference and the Cauchy Mean Value Property , 1980 .

[15]  Vasant A. Ubhaya,et al.  Lipschitzian selections in best approximation by continuous functions , 1990 .

[16]  Constance van Eeden,et al.  Testing and estimating ordered parameters of probability distribution , 1958 .

[17]  David A. Ratkowsky,et al.  Handbook of nonlinear regression models , 1990 .

[18]  Joyce Snell,et al.  6. Alternative Methods of Regression , 1996 .

[19]  F. Deutsch,et al.  Some applications of functional analysis to approximation theory , 1966 .