By J.-B. Hiriart-Urruty (auth.), Prof. Dr. Jacob Ponstein (eds.)

The research and optimization of convex services have re ceived loads of awareness over the last twenty years. If we needed to select key-words from those advancements, we'd continue the concept that of ~ubdi66~e~ and the duality theo~y. because it ordinary within the improvement of mathematical theories, humans had in view that attempted to increase the identified defi nitions and houses to new periods of features, together with the convex ones. For what issues the generalization of the thought of subdifferential, great achievements were performed long ago decade and any rna·· thematician who's confronted with a nondifferentiable nonconvex functionality has now a panoply of generalized subdifferentials or derivatives at his disposal. much is still performed during this sector, specifically pertaining to vecto~-valued features ; in spite of the fact that we predict the golden age for those researches is in the back of us. Duality thought has additionally involved many mathematicians because the underlying mathematical framework has been laid down within the context of Convex research. some of the duality schemes that have emerged within the re cent years, regardless of in their mathematical attractiveness, haven't continually proved as robust as expected.

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**Sample text**

3. that is - 0'. 19) Another variational formulation for AM is : 0'. x<. 3 .. In successive papers ([49J. [50J. 15). on. The dual ity schemes TOLAND proposed in the context of Calculus of Variations have been further developed by AUCHMUTY ([6]). Relationships with LEGENDRE's transformation are laid out in EKELAND's papers ([15J, [16J). 13) in its most general and abstract setting has recently been derived by the author ([24J) . 15) concerns with the ~eg~zation of f. Given the convex function g or h.

12 ) for an non-null d in T(S;x o )' Suppose there is a sequence (x n) c S converging to xo ' xn ~ xo ' such that f(x n) ~ f(x o )' Subsequencing if necessary, we may suppose that xn - Xo I xn - Xo II does converge to a 1 imit d, d ~ 0 • d belongs to T(S;x o ) since xn = Xo + II xn - Xo I d converges to Xo inS. 0 ~ 0, which is in contra- I V• 2. VuaLUtj Given f = g-h E DCQRn), the conjugate f* of f can be expressed in terms of the conjugate of the (convex) functions g and h. c. functions actually take root in this basic formula we state now.

This state of affairs is somewhat baffling since. as recalled in the next section. c. functions on the real line bear an easy characterization. c. c .. Exampte. 4. Let A be a Borel set of R satisfying the next property: for all nonempty open interval I of R. A(A n I) > 0 and X(ln I) > O. The function f defined on R by f(x) = f: lA(t) dt is strictly increasing and locally Lipschitz onR. c. onlR? Although there are various reasons for answering "no". e .. II. C. c. ). c. onR if and only if f' is of bounded variation on compact intervals oflR.