Solutions and Optimality Criteria to Box Constrained Nonconvex Minimization Problems

This paper presents a canonical duality theory for solving nonconvex polynomial programming problems subjected to box constraints. It is proved that under certain conditions, the constrained nonconvex problems can be converted to the so-called canonical (perfect) dual problems, which can be solved by deterministic methods. Both global and local extrema of the primal problems can be identified by a triality theory proposed by the author. Applications to nonconvex integer programming and Boolean least squares problems are discussed. Examples are illustrated. A conjecture on NP-hard problems is proposed. 1. Primal problem and its dual form. The box constrained nonconvex minimization problem is proposed as a primal problem (P) given below: (P) : min x∈Xa {P (x) = Q(x) + W (x)} (1) where Xa = {x ∈ Rn | ` ≤ x ≤ `} is a feasible space, Q(x) = 1 2xT Ax − c x is a quadratic function, A = A ∈ Rn×n is a given symmetric matrix, `, `, and c are three given vectors in Rn, W (x) is a nonconvex function. In this paper, we simply assume that W (x) is a so-called double-well fourth order polynomial function defined by W (x) = 1 2 ( 1 2 |Bx|2 − α )2 , (2) where B ∈ Rm×n is a given matrix and α > 0 is a given parameter. The notation |x| used in this paper denotes the Euclidean norm of x. Problems of the form (1) appear frequently in many applications, such as semilinear nonconvex partial differential equations [15], structural limit analysis, discretized optimal control problems with distributed parameters, information theory, and network communication. Particularly, if W (x) = 0, the problem (P) is directly related to certain successive quadratic programming methods ([9, 10, 18]). 2000 Mathematics Subject Classification. 49N15, 49M37, 90C26, 90C20.