A Maximization Problem Related to Parametric Linear Complementarity
Ikuyo Kaneko · SIAM Journal on Control and Optimization · 1978
The programming problem considered here is that of finding the maximal value of $\alpha $ such that the solution z of $q + \alpha p + Mz \geqq 0$, $z \geqq 0$ and $z^T (q + \alpha p + Mz) = 0$ satisfies $z \leqq a$. In this problem, q, p, a are n-vectors such that $q \geqq 0$, $a > 0$, M is an $n \times n$P-matrix and $\alpha $ is a scalar. This problem has an important application in structural mechanics. In this paper it is first explained that a certain local optimum of the above problem can be obtained more easily than the global optimum, and then necessary and sufficient conditions are determined under which the local and global optima coincide. Relationships are examined between these conditions and those on the isotonicity of the solutions of the parametric linear complementarity problem.