# American Institute of Mathematical Sciences

April  2008, 4(2): 213-225. doi: 10.3934/jimo.2008.4.213

## Global extremal conditions for multi-integer quadratic programming

 1 Department of Mathematical Sciences, Tsinghua University, Beijing, China 2 Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 3 Department of Mathematics, Virginia Tech, Blacksburgh, VA

Received  September 2007 Revised  January 2008 Published  April 2008

This paper presents a canonical duality approach to solve an integer quadratic programming problem, in which the objective function is quadratic and each variable may assume the value of one of $p~( \ge 3)$ integers. We first transform the problem into a $\{-1,1\}$ integer quadratic programming problem and then derive its ''canonical dual''. It is shown that, under certain conditions, this nonconvex multi-integer programming problem is equivalent to a concave maximization dual problem over a convex feasible domain. A global optimality condition is derived and some computational examples are provided to illustrate this approach.
Citation: Zhenbo Wang, Shu-Cherng Fang, David Y. Gao, Wenxun Xing. Global extremal conditions for multi-integer quadratic programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 213-225. doi: 10.3934/jimo.2008.4.213
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