# 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 and Continuous Dynamical Systems - B, 2012, 17 (6) : 1693-1706. doi: 10.3934/dcdsb.2012.17.1693
##### References:
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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), Article 81, 10 pp. (electronic). [2] P. T. An, Stability of generalized monotone maps with respect to their characterizations, Optimization, 55 (2006), 289-299. doi: 10.1080/02331930600705242. [3] D. Aussel, Subdifferential properties of quasiconvex and pseudoconvex functions: Unified approach, J. Optim. Theory Appl., 97 (1998), 29-45. doi: 10.1023/A:1022618915698. [4] D. Aussel, J.-N. Corvellec and M. Lassonde, Subdifferential characterization of quasiconvexity and convexity, J. Convex Anal., 1 (1994), 195-201. [5] D. Aussel and A. Daniilidis, Normal characterization of the main classes of quasiconvex functions, Set-Valued Anal., 8 (2000), 219-236. [6] M. Avriel, W. E. Diewert, S. Schaible and I. Zang, "Generalized Concavity," Mathematical Concepts and Methods in Science and Engineering, 36, Plenum Press, New York, 1988. [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, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. [8] E. N. Barron, R. Goebel and R. Jensen, Functions which are quasiconvex under small linear perturbations, submitted. [9] E. N. Barron, R. Goebel and R. Jensen, Quasiconvex functions and viscosity solutions of partial differential equations, Trans. Amer. Math. Soc., accepted. [10] Hitoshi Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Engrg. Chuo Univ., 28 (1985), 33-77. [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), 343-356. doi: 10.1023/A:1022659230603. [12] H. X. Phu and P. T. An, Stable generalization of convex functions, Optimization, 38 (1996), 309-318. doi: 10.1080/02331939608844259. [13] R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317, Springer-Verlag, Berlin, 1998. [14] M. Soleimani-damaneh, Characterization of nonsmooth quasiconvex and pseudoconvex functions, J. Math. Anal. Appl., 330 (2007), 1387-1392. doi: 10.1016/j.jmaa.2006.08.033. [15] L. Thibault and D. Zagrodny, Integration of subdifferentials of lower semicontinuous functions on Banach spaces, J. Math. Anal. Appl., 189 (1995), 33-58. doi: 10.1006/jmaa.1995.1003.
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