An approximate exact penalty in constrained vector optimization on metric spaces (Q467460)

The author studies the following vector optimization problem (P): minimize \(F(x)\) subject to \(\phi(x)=0, x\in X\), where \(F\) is a lower semi-continuous function from a complete metric space \((X, \rho)\) to a normed space \(Y\), partially ordered by a closed, convex and pointed cone \(Y_+\) with a nonempty interior, \(\phi\) is a positive function on \(X\). The main result of this paper states that if \(x_0\) is an \(\epsilon\)-solution of the unconstrained problem (P\(\lambda\)): minimize \(F(x)+\lambda \phi(x) e\) on \(X\), where \(\lambda \geqq \lambda_0\) for some \(\lambda_0\geqq 0\), \(e\in\int(Y_+)\), then there exists an \(\epsilon\)-solution of (P\(\lambda\)) which is feasible for (P) and satisfies \(F(z_0)\leq F(x_0)+\lambda \phi(x_0)e\) and \(\rho(x_0,z_0)\leq \epsilon\). It is regrettable that relevent references on vector optimization were not used, which should help to simplify certain proofs.