Local minima for indefinite quadratic knapsack problems

We consider the complexity of finding a local minimum for the nonconvex Quadratic Knapsack Problem. Global minimization for this example of quadratic programming is NP-hard. Moré and Vavasis have investigated the complexity of local minimization for the strictly concave case of QKP; here we extend their algorithm to the general indefinite case. Our main result is an algorithm that computes a local minimum in O(n(logn)2) steps. Our approach involves eliminating all but one of the convex variables through parametrization, yielding a nondifferentiable problem. We use a technique from computational geometry to address the nondifferentiable problem.

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