Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap
This paper deals with cone-constrained optimization problems of the form \[ \mu :=\inf_{x\in K}f\left( x\right), \tag{1} \] where \(f:X\longrightarrow \mathbb{R\cup }\left\{ +\infty \right\}\), \( K=\left\{ x\in C:g\left( x\right) \in -P\right\}\), with \(C\subset X\), \( P\subset Y\), \(g:X\longrightarrow Y\), \(X,Y\) normed spaces (even though the results could be straightforwardly extended to locally convex Hausdorff topological vector spaces by using nets instead of sequences), \(P\) is a convex cone, and \(K\neq \emptyset\). The Lagrange dual of (1) is \[ \upsilon :=\sup_{\lambda \in P^{\ast }}\inf_{x\in C}\left\{ f\left( x\right) +\left\langle \lambda ,g\left( x\right) \right\rangle \right\}. \tag{2} \] The paper provides known and new results guaranteeing zero-duality gap together with the solvability of either (1) or (2), situations described as primal or dual strong duality (primal strong duality is also called reverse strong duality in the literature). The authors appeal to a perturbational scheme based on the following parametric problem (to be interpreted as a perturbation of (1)): \[ \psi \left( a\right) :=\inf_{x\in K(a)}f\left( x\right) , \tag{3} \] where \(K(a)=\left\{ x\in C:g\left( x\right) \in a-P\right\}\). So, (1) is embedded into (3) as \(K(0)=K\) and \(\psi \left( 0\right) =\mu\). The main results are Theorem 4.4, characterizing the primal strong duality in terms of the closedness of the set \(\left\{ \left( f\left( x\right) ,g\left( x\right) \right) :x\in C\right\} +\mathbb{R}_{+}\times P\) (expression depending on the data) relative to vertical half-lines contained in the line \(\mathbb{R}\times \left\{ 0\right\} \subset \mathbb{R}\times Y\) (or, equivalently, \(\overline{\psi }\left( 0\right) =\psi \left( 0\right) \) together with the solvability of (1), by Proposition 2.2), and Theorem 4.2, characterizing the dual strong duality by the condition that \(\partial \psi \left( 0\right) \neq \emptyset\). The last two sections provide, thanks to Theorem 5.5 (5.6), checkable conditions for particular types of inequality (equality, respectively) constrained problems involving linear fractional and quadratic functions.
scientific article; zbMATH DE number 6818683
scientific article