Abstract
In this paper we derive the convex envelope of separable functions obtained as a linear combination of strictly convex coercive one-dimensional functions over compact regions defined by linear combinations of the same one-dimensional functions. As a corollary of the main result, we are able to derive the convex envelope of any quadratic function (not necessarily separable) over any ellipsoid, and the convex envelope of some quadratic functions over a convex region defined by two quadratic constraints.
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References
Ballerstein, M., Michaels, D.: Extended formulations for convex envelopes. J. Glob. Optim. 60, 217–238 (2014)
Ben-Tal, A., Den Hertog, D., Laurent, M.: Hidden convexity in partially separable optimization, Technical Report 2011–70. Tilburg University, Center for Economic Research (2011)
Ben-Tal, A., den Hertog, D.: Hidden conic quadratic representation of some nonconvex quadratic optimization problems. Math. Progr. 143, 1–29 (2014)
Burer, S., Anstreicher, K.: Second-order cone constraints for extended trust-region subproblems. SIAM J. Optim. 23(1), 432–451 (2013)
Flippo, O.E., Jansen, B.: Duality and sensitivity in nonconvex quadratic optimization over an ellipsoid. Eur. J. Oper. Res. 94, 167–178 (1996)
Hager, W.W., Phan, D.T.: An ellipsoidal branch and bound algorithm for global optimization. SIAM J. Optim. 20, 740–758 (2009)
Kwak, N.: Principal component analysis based on L1-norm maximization. IEEE Trans. Pattern Anal. Mach. Intell. 30, 1672–1680 (2008)
Kwak, N.: Principal component analysis by Lp-norm maximization. IEEE Trans. Cybern. 44, 594–609 (2014)
Jeyakumar, V., Li, G.Y.: Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization. Math. Program. 147, 171–206 (2014)
Laraki, R., Lasserre, J.B.: Computing uniform convex approximations for convex envelopes and convex hulls. J. Convex Anal. 15, 635–654 (2008)
Le Thi, H.A.: An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints. Math. Program. 87, 401–426 (2000)
Locatelli, M.: Some results for quadratic problems with one or two quadratic constraints. Oper. Res. Lett. 43, 126–131 (2015)
Locatelli, M.: Convex envelopes of some quadratic functions over the \(n\)-dimensional unit simplex. SIAM J. Optim. 25, 589–621 (2015)
Lovász, L.: Submodular functions and convexity. In: Grötschel, M., Korte, B. (eds.) Mathematical Programming : The State of the Art, pp. 232–257. Springer, Berlin (1982)
Meyer, C.A., Floudas, C.A.: Convex envelopes for edge-concave functions. Math. Program. 103, 207–224 (2005)
Motzkin, T.S., Straus, E.G.: Maxima for graphs and a new proof of a theorem of Turán. Can. J. Math. 17, 533–540 (1965)
Nguyen, T.T., Richard, J.-P.P., Tawarmalani, M.: Deriving convex hulls through lifting and projection. Math. Program. 169, 377–415 (2018)
Scozzari, A., Tardella, F.: A clique algorithm for standard quadratic programming. Discrete Appl. Math. 156, 2439–2448 (2008)
Sturm, J.F., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28, 246–267 (2003)
Tawarmalani, M., Richard, J.-P.P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program. 138, 531–577 (2013)
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Locatelli, M. Convex envelopes of separable functions over regions defined by separable functions of the same type. Optim Lett 12, 1725–1739 (2018). https://doi.org/10.1007/s11590-018-1291-5
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DOI: https://doi.org/10.1007/s11590-018-1291-5