Subdifferential monotonicity as characterization of convex functions
It is known that the monotonicity of the Clarke subdifferential of a locally Lipschitz real valued function is equivalent to the convexity of this function. In order to prove the same result for a lower semicontinuous function f : E {yields} R {union} {l_brace}+{infinity}{r_brace} we have considered in a previous work the Moreau-Yosida proximal approximation f{lambda} (x) : = inf {sub y{element_of}E}[f (y) + {sub 2{lambda}}{sup 1}{parallel}x - y{parallel}{sup 2}] since under some general conditions f{sub {lambda}} is locally Lipschitz f(x) = {sub {lambda}>0}f{sub {lambda}}(x). Thus our procedure consisted in deriving the monotonicity of {partial_derivative}f{sub {lambda}} from that of {partial_derivative}f. This method is required the reflexivity of the space E because it depended heavily on the fact that the above infimum is attained whenever the Frechet subdifferential {partial_derivative}{sup F} f(x) of f at x is nonempty. This has been obtained by supposing (thanks to the reflexivity of E) that the norm of E is Kadec and by showing that (when {partial_derivative}F f(x) {ne} {phi}) there exists some minimizing sequence of weakly converging to some point z and whose norms converge to {parallel}z{parallel}, which implies the strong convergence of the sequence. The same result was proved before by R.A. Poliquin for E = R{sup n} with the help of his notion of quadratic conjugate function. In this paper, by a completely different approach, we avoid the use of the Moreau-Yosida approximation f{sub {lambda}} in order to get the result for any Banach space E. In fact we prove that the monotonicity of any classical subdifferential {partial_derivative}f of a lower semicontinuous function f : E {yields} R {union} {l_brace}+{infinity}{r_brace} defined on a Banach space E is equivalent to the convexity of this function f.
- OSTI ID:
- 35920
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
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