By S. S. Kutateladze (auth.), Vladimir F. Demyanov, Panos M. Pardalos, Mikhail Batsyn (eds.)

This quantity includes a choice of papers in keeping with lectures and displays added on the foreign convention on optimistic Nonsmooth research (CNSA) held in St. Petersburg (Russia) from June 18-23, 2012. This convention used to be prepared to mark the fiftieth anniversary of the start of nonsmooth research and nondifferentiable optimization and was once devoted to J.-J. Moreau and the overdue B.N. Pshenichnyi, A.M. Rubinov, and N.Z. Shor, whose contributions to NSA and NDO stay invaluable.

The first 4 chapters of the ebook are dedicated to the speculation of nonsmooth research. Chapters 5-8 comprise new leads to nonsmooth mechanics and calculus of adaptations. Chapters 9-13 are relating to nondifferentiable optimization, and the amount concludes with 4 chapters containing attention-grabbing and demanding historic chapters, together with tributes to 3 giants of nonsmooth research, convexity, and optimization: Alexandr Alexandrov, Leonid Kantorovich, and Alex Rubinov. The final bankruptcy presents an summary and significant snapshots of the 50-year heritage of convex research and optimization.

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**Additional info for Constructive Nonsmooth Analysis and Related Topics**

**Example text**

Set √ g(u, z, , p) = z − (y + v − t p) + ε ( u − x + t p ) +t( u − u + z − z + p − p ) + iL (p). , [2, 9]) in any neighborhood of (u , z , p ) and (u , p ) we can find (u0 , z0 , p0 ) ∈ X × Y × L, (u1 , z1 ) ∈ Graph F, (u∗ , z∗ , p∗ ) ∈ ∂ g(u0 , z0 , p0 ) and (x∗ , y∗ ) ∈ X ∗ × Y ∗ such that x∗ ∈ D∗ F(u1 , z1 )(y∗ ) and u∗ + x∗ < ε , z∗ − y∗ < ε , p∗ < t ε . D. Ioffe We have √ • u∗ = ε u∗1 +tu∗2 , where u∗1 ∈ ∂ · (u0 − x), u∗2 ∈ ∂ · (u0 − u ), so that u∗i ≤ 1; • z∗ = z∗1 + tz∗2 , where z∗1 ∈ ∂ · (z0 − (y + v − t p0 ) so that z∗1 = 1 (as we can choose the neighborhood of (u , z , p ) small enough to guarantee that z0 − (y + v − t p0 ) > 0)√and z∗2 ∈ ∂ · (z0 − z ), so that z∗2 ≤ 1; • p∗ = −tz∗1 + t ε p∗1 + t p∗2 + p∗3 , where p∗1 ∈ ∂ · (p0 ), p∗2 ∈ ∂ · (p0 − p ), so that p∗i ≤ 1, i = 1, 2 and p∗3 ∈ L⊥ .

D. Ioffe Graph F, then F is compactly regular at (x, ¯ y) ¯ if and only if it is ∂H -coderivatively compact at (x, ¯ y). ¯ Proof. If F is ∂H -coderivatively compact, then it is compactly regular by Proposition 2 even without the assumption that F is steady. The proof of the opposite implication is a slight modification of the proof of Proposition 3 and we leave it for the reader. Observe finally that a simple modification of the above proof allows to include the property of being steady into the list of properties inherited by arbitrarily large separable subspaces.

So, we have to use the “density” of B0 and B0 in B and B , respectively. • Let us define < ·, · >K : B × B0 / C by < f , g >K := lim < fn , g >K , /∞ n where f ∈ B, g ∈ B0 , and { fn }∞ n=1 is a Cauchy sequence in B0 such that / f (y) for all y ∈ Y . Using inequality (4), {< fn , g >K }∞ is a Cauchy fn (y) n=1 sequence in C; the limit exits and is well defined. Clearly, the form < ·, · >K is bilinear in B × B0 . Finally, |< f , g >K | = n lim < fn , g >K ≤ lim || fn ||B0 ||g||B = || f ||B ||g||B 0 0 /∞ /∞ n (5) for all f ∈ B and g ∈ B0 , where { fn }∞ n=1 is a Cauchy sequence in B0 such that / f (y) for all y ∈ Y .