Canonical duality for solving general nonconvex constrained problems

On Constrained Optimization Problems Solved Using the Canonical Duality Theory

“…Because the quadratic problems (with quadratic constraints) are discussed in [18], we discuss only those constrained optimization problems with non quadratic objective function or with at least one non quadratic constraint. In the survey paper [4] (the same as [3]) there are mentioned the following papers: [6], [5], [7], [10] (with its preprint version [8]); besides these papers we add the retracted version [9] of [10], and [13].…”

“…The problem considered by Latorre and Gao in [7] is of type (P J ) in which Λ k are quadratic and V k are "differentiable canonical functions". In our framework (which, apparently, is more restrictive) and with our notations, the following set is used in [7]:…”

“…The dual feasible spaces should be defined as S 0 ...". 3 Besides the set S 0 mentioned above, the following sets are also considered in [7]:…”

“…The problem is a particular case of the one considered in [7, Ex. 1], "which is very simple, but important in both theoretical study and real-world applications since the constraint is a socalled double-well function, the most commonly used nonconvex potential in physics and engineering sciences [7]"; 1 more precisely, q := 1, c := 6, d := 4, e := 2.…”

“…In fact the positivity of the Lagrange multipliers λ j is needed for recovering the Lagrangian L from Ξ [see (7)], while the non vanishing condition on λ j is needed to get L(x, λ) = Ξ(x, λ, σ) and ∇ x L(x, λ) = ∇ x Ξ(x, λ, σ) when ∇ σ Ξ(x, λ, σ) = 0, as seen in Lemma 5. Of course, such conditions are not needed in quadratic minimization problems, as observed after Corollary 15.…”