![]() In: Proceedings of the Third International Conference on Operational Research 1963, pp. ![]() Tavistock Publishing (1970)īeale, E.L.M.: Two transportation oroblems. (ed.) Proceedings of the Fifth International Conference on Operational Research 1969, pp. ![]() 45, 69–84 (1988)īeale, E.L.M., Tomlin, J.A.: Special facilities in a general mathematical programming system for nonconvex problem using ordered sets of variables. Tomlin, J.A.: Special ordered sets and an application to gas supply operating planning. 3(3), 221–258 (2012)įrank, S., Steponavice, I., Rebennack, S.: Optimal power flow: a bibliographic survey II. Springer (2010)įrank, S., Steponavice, I., Rebennack, S.: Optimal power flow: a bibliographic survey I. (eds.) Handbook of Power Systems I, chap. In: Rebennack, S., Pardalos, P.M., Pereira, M.V., Iliadis, N.A. Zheng, Q.P., Rebennack, S., Iliadis, N.A., Pardalos, P.M.: Optimization models in the natural gas industry. Kallrath, J., Maindl, T.I.: Real Optimization with SAP-APO. Kallrath, J.: Combined strategic and operational planning-an MILP success story in chemical industry. Our approach computes breakpoint systems with up to one order of magnitude less breakpoints compared to an equidistant approach. We present computational results for 10 univariate functions. Alternatively, we use two heuristics which compute the breakpoints subsequently, solving small non-convex problems. We develop three non-convex optimization models: two yield the minimal number of breakpoints, and another in which, for a fixed number of breakpoints, the breakpoints are placed such that the maximal deviation is minimized. The linear approximators, under-, and over-estimators involve shift variables at the breakpoints allowing for the computation of an optimal piecewise linear, continuous approximator, under-, and over-estimator. For univariate functions, we compute optimal breakpoint systems subject to the condition that the piecewise linear approximator, under-, and over-estimator never deviate more than a given \(\delta \)-tolerance from the original function over a given finite interval.
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