# American Institute of Mathematical Sciences

September  2012, 17(6): 1693-1706. doi: 10.3934/dcdsb.2012.17.1693

## The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions

 1 Department of Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois, 60660, United States, United States, United States

Received  July 2011 Revised  December 2011 Published  May 2012

Necessary and sufficient conditions for quasiconvexity, also called level-set convexity, of a function are given in terms of first-order partial differential equations. Solutions to the equations are understood in the viscosity sense and the conditions apply to nonsmooth and semicontinuous functions. A comparison principle, implying uniqueness of solutions, is shown for a related partial differential equation. This equation is then used in an iterative construction of the quasiconvex envelope of a function. The results are then extended to robustly quasiconvex functions, that is, functions which are quasiconvex under small linear perturbations.
Citation: Emmanuel N. Barron, Rafal Goebel, Robert R. Jensen. The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1693-1706. doi: 10.3934/dcdsb.2012.17.1693
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##### References:
 [1] P. T. An, A new type of stable generalized convex functions,, JIPAM. J. Inequal. Pure Appl. Math., 7 (2006). Google Scholar [2] P. T. An, Stability of generalized monotone maps with respect to their characterizations,, Optimization, 55 (2006), 289. doi: 10.1080/02331930600705242. Google Scholar [3] D. Aussel, Subdifferential properties of quasiconvex and pseudoconvex functions: Unified approach,, J. Optim. Theory Appl., 97 (1998), 29. doi: 10.1023/A:1022618915698. Google Scholar [4] D. Aussel, J.-N. Corvellec and M. Lassonde, Subdifferential characterization of quasiconvexity and convexity,, J. Convex Anal., 1 (1994), 195. Google Scholar [5] D. Aussel and A. Daniilidis, Normal characterization of the main classes of quasiconvex functions,, Set-Valued Anal., 8 (2000), 219. Google Scholar [6] M. Avriel, W. E. Diewert, S. Schaible and I. Zang, "Generalized Concavity," Mathematical Concepts and Methods in Science and Engineering, 36,, Plenum Press, (1988). Google Scholar [7] M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,", With appendices by Maurizio Falcone and Pierpaolo Soravia, (1997). Google Scholar [8] E. N. Barron, R. Goebel and R. Jensen, Functions which are quasiconvex under small linear perturbations,, submitted., (). Google Scholar [9] E. N. Barron, R. Goebel and R. Jensen, Quasiconvex functions and viscosity solutions of partial differential equations,, Trans. Amer. Math. Soc., (). Google Scholar [10] Hitoshi Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets,, Bull. Fac. Sci. Engrg. Chuo Univ., 28 (1985), 33. Google Scholar [11] J.-P. Penot and P. H. Quang, Generalized convexity of functions and generalized monotonicity of set-valued maps,, J. Optim. Theory Appl., 92 (1997), 343. doi: 10.1023/A:1022659230603. Google Scholar [12] H. X. Phu and P. T. An, Stable generalization of convex functions,, Optimization, 38 (1996), 309. doi: 10.1080/02331939608844259. Google Scholar [13] R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317 (1998). Google Scholar [14] M. Soleimani-damaneh, Characterization of nonsmooth quasiconvex and pseudoconvex functions,, J. Math. Anal. Appl., 330 (2007), 1387. doi: 10.1016/j.jmaa.2006.08.033. Google Scholar [15] L. Thibault and D. Zagrodny, Integration of subdifferentials of lower semicontinuous functions on Banach spaces,, J. Math. Anal. Appl., 189 (1995), 33. doi: 10.1006/jmaa.1995.1003. Google Scholar
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