A complete characterization of strong duality in nonconvex optimization with a single constraint

Abstract

We first establish sufficient conditions ensuring strong duality for cone constrained nonconvex optimization problems under a generalized Slater-type condition. Such conditions allow us to cover situations where recent results cannot be applied. Afterwards, we provide a new complete characterization of strong duality for a problem with a single constraint: showing, in particular, that strong duality still holds without the standard Slater condition. This yields Lagrange multipliers characterizations of global optimality in case of (not necessarily convex) quadratic homogeneous functions after applying a generalized joint-range convexity result. Furthermore, a result which reduces a constrained minimization problem into one with a single constraint under generalized convexity assumptions, is also presented.

Connections between mean-field game and social welfare optimization

“…Therefore, to apply the result of the theorems, we need to check whether the social welfare optimization problem has strong duality, especially when Assumption 5 is not satisfied. Along this direction, many sufficient conditions have been developed to check the strong duality for nonconvex optimization [43], [44], [45]. Due to space limit, we will not present the technical details of these works.…”

Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap

“…will play an important role in our analysis. These sets or their conic hull arise in a natural way when dealing with duality results or in deriving alternative theorems, see [21,22,25,31,35,40]. Giannessi [32,33] used them in a systematic manner for a constrained extremum problem giving rise to the image space analysis.…”

“…The case P = R + may be found in [21], and if P = {0} the result appears in [13] as a consequence of Theorem 2.4 in [22] (there the assumption A ∩ ri P = ∅, with P being as in [22], was forgotten; see [28] for an alternative proof).…”

“…Theorem 5.4 Set P = {0} ⊂ R and assume μ ∈ R with F {0} being convex. Then, dual strong-duality holds for (21) if, and only if…”

“…Notice, in view of [20], that under (32), problem (21) with P = R m + has a nonempty and compact solution set.…”