Coercivity and image of constrained extremum problems (Q2564181)

The author presents a coercivity in the image space approach to solve the problem: \(\min\varphi(x)\), \(x\in R:=\{x\in X:g(x)\geq 0\}\), where \(\varphi:X\to \mathbb{R}\), \(g:X\to \mathbb{R}^m\), and \(X\) is a subset of a Banach space \(Y\). The method of images is based upon the fact that \(\bar x\in R\) is a minimum point iff the system \(\quad\varphi (\bar x)-\varphi (x)>0,\quad g(x)\geq 0,\quad x\in X\) has no solution, or equivalently, \(\mathcal H \cap \mathcal K(\bar x)=\emptyset\), where \(\mathcal K(\bar x):=(\varphi (\bar x)-\varphi(x),g(x)),\;\mathcal H:=\{(u,v)\in \mathbb{R}^{1+m}:u>0,v\geq 0\}\). The set \(\mathcal K(\bar x)\) is called the image of the constrained extremum problem. New results in establishing conditions which guarantee the existence of a separation hyperplane between \(\mathcal H\) and \(\mathcal K(\bar x)\) are obtained by means of a coercivity condition in the image space. The application to variational inequalities is shown.