Complete Solutions to Mixed Integer Programming

This paper considers a new canonical duality theory for solving mixed integer quadratic programming problem. It shows that this well-known NP-hard problem can be converted into concave maximization dual problems without duality gap. And the dual problems can be solved, under certain conditions, by polynomial algorithms.


Introduction
Mixed integer nonlinear programming refers to optimization problems which involve continuous and discrete variables [8].In this paper, we consider the following constrained mixed integer quadratic programming: 0 ( ) m in ( , ) ( ) where,     c, w, b are given vectors, d is a given scalar, and , 0  0. c  X is a feasible space defined by Problem of the form (1) has a broad spectrum of applications, including process industry (process design [2,13,18], production planning [14], supply chain optimization [1,3], logistics and so on), management science (scheduling problem), financial (portfolio optimization problems [22]), engineering (network design [23]), machine learning (semi-supervised support vector machines), and computational chemistry /biology (solvent design problems).
Various methods have been proposed for solving mixed integer programming, such as branch and bound [4,5,19,21,24], cutting plane, branch and cut [16], branch and reduce, outer approximation [6,7,15], hybrid methods, and penalty method [17].But the difficulty for developing an efficient method for such mixed integer programming lies not only on the nonlinearity of the func-tions involved, but also on existence of both discrete and continuous variables [20].But if we introduce the canonical duality with some strategy, we can find global optima in polynomial time [10,11,12].
The rest of paper is arranged as follows.In section 2, we demonstrate how to rewrite the primal problem as a dual problem by using the canonical dual transformation.In section 3, optimality criterions for global solutions are discussed.Finally, in the last section, we present some conclusions.

Canonical Dual Transformation
Canonical duality theory [9] is a potentially powerful methodology which can be used to solve a large class of non-convex and discrete problems in nonlinear analysis, global optimization, and computational science.
Since { 1,1} n y   , one penalty term is added.Let a be a penalty factor, the original problem can be formulated We choose the geometrically nonlinear operator then, the canonical function associated with this geometrical operator is sponding to  , we have And the Legendre conjugates of the function   Thus the total complementarily function can be defined by Therefore, the canonical dual problem can be proposed as the following: and where e is a vector with its entry 1.Its dual feasible space is defined as The

Global Optimality Condition
Theorem 1The problem is canonical dual to the primal problem in the sense that if x y defined by is a KKT point of , and Proof.By introducing a Lagrange multiplier The criticality conditions   , , , and the KKT conditions   and So, in terms of we have Therefore, be a subset of a S , and we have the following theorem.
Theorem 2 Suppose that the vector   , , x max In this paper, the canonical duality theory n applied to solve mixed integer pro ramming oblem.Theorems show that by al dual nsformation, primal problems can be converted into canonical dual problem.By the fact that the canonical dual function is concave on the dual feasible space, so th can be solved by w tion methods. is concave on the critical point ,

P
By the fact that the canonical dual function is a