Finding normal solutions in piecewise linear programming

Letf: ℝn → (−∞, ∞] be a convex polyhedral function. We show that if any standard active set method for quadratic programming (QP) findsx(t)= arg minx¦x¦2/2+tf(x) for somet> 0, then its final working set defines a simple equality QP subproblem, whose Lagrange multiplier can be used both for testing ift is large enough forx(t) to coincide with the normal minimizer off, and for increasingt otherwise. The QP subproblem may easily be solved via the matrix factorizations used for findingx(t). This opens up the way for efficient implementations. We also give finite methods for computing the whole trajectory {x(t)}t≥0, minimizingf over an ellipsoid, and choosing penalty parameters inL1QP methods for strictly convex QP.

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