Sufficient Conditions and Perfect Duality in Nonconvex Minimization with Inequality Constraints

This paper presents a duality theory for solving concave minimization problem and nonconvex quadratic programming problem subjected to nonlinear inequality constraints. By use of the canonical dual transformation developed recently, two canonical dual problems are formulated, respectively. These two dual problems are perfectly dual to the primal problems with zero duality gap. It is proved that the sufficient conditions for global minimizers and local extrema (both minima and maxima) are controlled by the triality theory discovered recently [5]. This triality theory can be used to develop certain useful primal-dual methods for solving difficult nonconvex minimization problems. Results shown that the difficult quadratic minimization problem with quadratic constraint can be converted into a one-dimensional dual problem, which can be solved completely to obtain all KKT points and global minimizer. 1. Concave Minimization Problem and Parametrization. The concave minimization problem to be discussed in this paper is denoted as the primal problem ((P) in short) (P) : minP (x) ∀x ∈ Xf , (1.1) where P (x) is a real-valued concave function defined on a suitable convex set Xa ⊂ R, and Xf ⊂ R is the feasible space, defined by Xf = {x ∈ Xa ⊂ Rn| Bx ≤ b}, (1.2) in which, B ∈ Rm×n is given matrix such that rankB = m, and b ∈ R is a given vector. The goal in this problem is to find both global and local minimum values that P can achieve in the feasible space and, if this value is not −∞, to find, if it exists, at least one vector x̄ ∈ Xf that achieves this value. The concave minimization problem (P) appears in many applications. Methods and solutions to this very difficult problem are fundamentally important in both mathematics and engineering science. Mathematically speaking, if the function P is continuous on its domain Xa and the feasible set Xf is compact, then by the well-known Weierstrass Theorem, the global minimum value is finite, and at least one point in Xf exists which attains this value. From point view of convex analysis, if the convex subset Xf ⊂ R 2000 Mathematics Subject Classification. 49N15, 49M37, 90C26, 90C20.