Global optimality conditions for some classes of optimization problems (Q970570)

The authors give new necessary and sufficient optimality conditions for global optimization problems. In particular, tractable optimality conditions for the problems of minimizing a weakly convex or concave function subject to standard constraints, such as box constraints, binary constraints, and simplex constraints are established. The main theoretical tool is an explicit representation for the \(L\)-subdifferential \(\partial_L f(\overline x)=\{l \in L: f(x)\geq f(\overline x)+l(x)-l(\overline x), \forall x \in \mathbb{R}^n \}\) of a function \(f(x)=1/2 \langle A_0 x, x\rangle+p(x)\), where \(A_0\) is a symmetric \(n\times n\) matrix, \(p: \mathbb{R}^n \to \mathbb{R}\cup \{+\infty \}\) a convex function, and \(L=\{ l: l(x)=1/2 \langle Q x, x\rangle + \langle \beta, x\rangle, Q=\text{{Diag}}(q), q, \beta \in \mathbb{R}^n \}\).