Directional differentiability of the optimal value function in indefinite quadratic programming (Q696141)

The author considers quadratic programming problems of the form: minimize \(f(x)\) subject to \(x\in \Delta(A, b)\), where \(f(x)\equiv c^Tx+ {1\over 2} (x^TDx)\) and \(\Delta(A, b)\equiv \{x\mid Ax\geq b\}\), \(A\), \(D\) are given matrices and \(b\), \(c\) are given vectors. Optimal value function \(\varphi(.)\) of this problem depends on input parameters \(D\), \(A\), \(c\), \(b\) and is defined as follows: \[ \varphi(D,A,c,b)\equiv \inf\{f(x)\mid x\in\Delta(A, b)\}\quad\text{if }\Delta(A, b)\neq\varnothing, \] \[ \varphi(D,A,c,b)\equiv \infty\quad\text{if }\Delta(A, b)=\varnothing. \] In the paper, an explicit formula for computing the directional derivative of \(\varphi(.)\) is derived. The formula is derived under conditions that do not exclude its application to some indefinite quadratic problems. Therefore, the result obtained in the paper can be used also for investigating differential stability for a class of indefinite quadratic problems, for which the existing stability results from the literature requiring convexity of the objective function cannot be used.